diff --git a/parse/train/0HW7A5YZjq7/0HW7A5YZjq7.md b/parse/train/0HW7A5YZjq7/0HW7A5YZjq7.md new file mode 100644 index 0000000000000000000000000000000000000000..8582942fcccb8b29c73fe528fc8ff58e179695fe --- /dev/null +++ b/parse/train/0HW7A5YZjq7/0HW7A5YZjq7.md @@ -0,0 +1,352 @@ +# 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. + +
Top-1(%)Top-5(%)
Supervised [54]75.992.8
SimCLR[19]69.389.0
MoCo v2 [20]71.190.1
BYOL [25]74.391.6
SwAV[18]75.3-
Barlow Twins [22]73.291.0
SSL-HSIC (w/o target)72.290.7
SSL-HSIC (w/ target)74.892.2
+ +Table 2: Fine-tuning on $1 \%$ , $10 \%$ and $100 \%$ of the ImageNet training set and evaluating on the validation set. + +
Top-1(%)Top-5(%)
1%10%100%1%10%100%
Supervised [54]25.456.475.948.480.492.8
SimCLR [19]48.365.676.075.587.893.1
BYOL [25]53.268.877.778.489.093.9
SwAV[18]53.970.2178.589.9-
Barlow Twins [22]55.069.7-79.289.3-
SSL-HSIC (w/o target)45.365.576.472.787.593.2
SSL-HSIC (w/ target)52.167.977.277.788.693.6
+ +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 $\mathrm { I o U } { = } 0 . 5$ . + +
Dataset MetricBirdsnap Caltechl01 Cifarl0 Cifarl001 Top-1MPCATop-1DTD Top-1Aircraft Food FlowersMPCA Top-1MPCA MPCA Top-1PetsCarsTop-1SUN397 VOC2007 AP50
Linear:Top-1
Supervised-IN [19]53.794.593.678.361.072.394.791.567.861.982.8
SimCLR[19]37.490.390.671.674.9 74.550.368.490.383.650.358.880.5
BYOL [25]57.294.291.378.475.560.675.396.190.466.762.282.5
SSL-HSIC (w/o target)50.692.391.575.975.357.973.695.088.259.361.081.4
SSL-HSIC (w/ target)57.893.592.377.076.258.575.695.491.262.661.883.3
Fine-tuned:
Supervised-IN [19]85.0
Random init [19]75.893.397.586.474.686.0 85.988.397.692.192.194.3 53.667.3
SimCLR[19]76.1 75.972.6 92.195.9 97.780.2 85.964.8 73.288.186.9 88.292.0 97.081.5 89.291.4 91.363.584.1
BYOL [25]76.393.897.886.176.288.188.597.091.791.663.785.4
SSL-HSIC (w/o target)73.191.597.485.375.387.187.596.490.691.662.284.1
SSL-HSIC (w/ target)74.993.897.884.775.488.987.797.391.791.861.784.1
+ +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. + +
MethodVOC2012 mIoUNYU v2
pct.<1.25pct.<1.252pct.<1.253rmsrel
Supervised-IN74.481.195.398.80.5730.127
SimCLR75.283.396.599.10.5570.134
BYOL76.384.696.799.10.5410.129
SSL-HSIC(w/o target)74.984.196.799.20.5390.130
SSL-HSIC(w/ target)76.083.896.899.10.5480.130
+ +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. + +
MethodAPbbApmk
Supervised39.635.6
SimCLR39.735.8
MoCo v240.136.3
BYOL41.637.2
SwAV41.637.8
SSL-HSIC(w/o target)40.536.3
SSL-HSIC(w/ target)41.336.8
+ +# 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. + +
SSL-HSICBYOL[25]Sup.[25]
ResNetTop1Top5Top1Top5Top1Top5
50 (1x)74.892.274.391.676.492.9
50 (2x)77.994.077.493.679.995.0
50(4x)79.194.578.694.280.795.3
200 (2x)79.694.879.694.980.195.2
+ +Table 7: Top-1 and top-5 accuracies for different ResNet architectures using semisupervised fine-tuning. + +
Top1Top5
ResNet1%10%100%1%10%100%
50 (1x)52.167.977.277.788.693.6
50 (2x)61.272.679.383.891.294.7
50 (4x)67.075.479.787.492.594.8
200(2x)69.076.380.588.392.995.2
+ +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 + +
Top-1Top-5
SSL-HSIC66.787.6
InfoNCE66.086.9
+ +Table 8: Linear evaluation results when varying different hyperparameters. +(d) Projector/predictor size + +
Output DimTop-1(%)
6465.4
12866.0
25666.4
51266.6
102466.6
+ +(b) # Fourier features + +
#RFFsTop-1(%)
6466.0
12866.2
25666.2
51266.4
102466.5
204866.5
No Approx.66.7
+ +(c) Batch size + +
Top-1(%)
Batch SizeSSL-HSICSimCLR
25663.757.5
51265.660.7
102466.762.8
204867.164.0
409666.764.6
+ +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 + +[1] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. 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Belongie. “Feature Pyramid Networks for Object Detection.” CVPR. 2017. arXiv: 1612.03144. \ No newline at end of file diff --git a/parse/train/0HW7A5YZjq7/0HW7A5YZjq7_content_list.json b/parse/train/0HW7A5YZjq7/0HW7A5YZjq7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..99110a45ccd15a0c50cae5ede9ea99fbc74eb2ad --- /dev/null +++ b/parse/train/0HW7A5YZjq7/0HW7A5YZjq7_content_list.json @@ -0,0 +1,1424 @@ +[ + { + "type": "text", + "text": "Self-Supervised Learning with Kernel Dependence Maximization ", + "text_level": 1, + "bbox": [ + 261, + 122, + 738, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yazhe Li∗ DeepMind and Gatsby Unit, UCL yazhe@google.com ", + "bbox": [ + 238, + 226, + 462, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Roman Pogodin∗ Gatsby Unit, UCL roman.pogodin.17@ucl.ac.uk ", + "bbox": [ + 534, + 226, + 758, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Danica J. Sutherland UBC and Amii† dsuth@cs.ubc.ca ", + "bbox": [ + 279, + 289, + 429, + 332 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Arthur Gretton Gatsby Unit, UCL arthur.gretton@gmail.com ", + "bbox": [ + 540, + 289, + 746, + 332 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 367, + 535, + 383 + ], + "page_idx": 0 + }, + { + "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. ", + "bbox": [ + 233, + 400, + 766, + 648 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 676, + 310, + 694 + ], + "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. ", + "bbox": [ + 174, + 708, + 825, + 861 + ], + "page_idx": 0 + }, + { + "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]. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 475, + 275 + ], + "page_idx": 1 + }, + { + "type": "image", + "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). " + ], + "image_footnote": [], + "bbox": [ + 509, + 89, + 815, + 242 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 380, + 825, + 450 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 455, + 825, + 540 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 545, + 825, + 726 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 731, + 825, + 842 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our main contributions are as follows: ", + "bbox": [ + 176, + 848, + 426, + 863 + ], + "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": [ + 174, + 869, + 825, + 911 + ], + "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. ", + "bbox": [ + 176, + 92, + 825, + 133 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 174, + 154, + 308, + 171 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Self-supervised learning ", + "text_level": 1, + "bbox": [ + 174, + 185, + 380, + 200 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 212, + 825, + 281 + ], + "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": [ + 174, + 287, + 825, + 481 + ], + "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. ", + "bbox": [ + 173, + 487, + 826, + 570 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 570, + 601, + 736 + ], + "page_idx": 2 + }, + { + "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. " + ], + "image_footnote": [], + "bbox": [ + 614, + 580, + 808, + 656 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 186, + 742, + 604, + 757 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 757, + 825, + 811 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Hilbert-Schmidt Independence Criterion (HSIC) ", + "text_level": 1, + "bbox": [ + 174, + 829, + 552, + 844 + ], + "page_idx": 2 + }, + { + "type": "text", + "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]). ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "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: ", + "bbox": [ + 174, + 152, + 823, + 195 + ], + "page_idx": 3 + }, + { + "type": "equation", + "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", + "bbox": [ + 290, + 200, + 705, + 224 + ], + "page_idx": 3 + }, + { + "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": [ + 173, + 229, + 825, + 272 + ], + "page_idx": 3 + }, + { + "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 ", + "bbox": [ + 171, + 279, + 825, + 308 + ], + "page_idx": 3 + }, + { + "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", + "bbox": [ + 183, + 314, + 795, + 332 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 338, + 823, + 367 + ], + "page_idx": 3 + }, + { + "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", + "bbox": [ + 362, + 372, + 633, + 405 + ], + "page_idx": 3 + }, + { + "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]. ", + "bbox": [ + 174, + 411, + 825, + 455 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Self-supervised learning with Kernel Dependence Maximization ", + "text_level": 1, + "bbox": [ + 174, + 473, + 736, + 492 + ], + "page_idx": 3 + }, + { + "type": "text", + "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: ", + "bbox": [ + 173, + 503, + 826, + 631 + ], + "page_idx": 3 + }, + { + "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": [ + 318, + 637, + 679, + 657 + ], + "page_idx": 3 + }, + { + "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": [ + 173, + 669, + 825, + 712 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 173, + 717, + 826, + 792 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Due to the one-hot encoded labels, we can re-write HSIC $( Z , Y )$ as (see Appendix A) ", + "bbox": [ + 173, + 796, + 736, + 813 + ], + "page_idx": 3 + }, + { + "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", + "bbox": [ + 299, + 818, + 696, + 835 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 90, + 825, + 161 + ], + "page_idx": 4 + }, + { + "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). ", + "bbox": [ + 173, + 166, + 825, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 Connection to InfoNCE ", + "text_level": 1, + "bbox": [ + 174, + 267, + 377, + 282 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 171, + 294, + 823, + 323 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 259, + 330, + 736, + 348 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 356, + 825, + 398 + ], + "page_idx": 4 + }, + { + "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: ", + "bbox": [ + 176, + 404, + 825, + 446 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/148d20d2934fb617453a770c46e3d6152b49681716cb2763777add25e07a7604.jpg", + "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", + "bbox": [ + 194, + 454, + 782, + 500 + ], + "page_idx": 4 + }, + { + "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": [ + 171, + 515, + 823, + 544 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1f185702c2be46b3c39e79c4f9490be60c34e4feadce71afbc2161c341e2afff.jpg", + "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": [ + 281, + 551, + 714, + 569 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For $\\mathrm { H S I C } ( Z , Z ) \\le 1$ , we also have that ", + "bbox": [ + 174, + 577, + 436, + 592 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7e6b291b6bf5748716ae3884e71aff0e541c35dd259cdadb0e91261d691af218.jpg", + "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", + "bbox": [ + 326, + 599, + 669, + 617 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 625, + 825, + 654 + ], + "page_idx": 4 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 659, + 826, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Connection to metric learning ", + "text_level": 1, + "bbox": [ + 176, + 815, + 421, + 830 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 825, + 119 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/cf68eb4fe4081b6ab5f7838fa93ef24888e35baa7e17177109b15d47e260c0a9.jpg", + "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", + "bbox": [ + 191, + 125, + 781, + 204 + ], + "page_idx": 5 + }, + { + "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 $\\mathrm { H S I C } ( Z , \\bar { Y } )$ . ", + "bbox": [ + 176, + 205, + 823, + 263 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 173, + 270, + 826, + 436 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 Estimator of SSL-HSIC ", + "text_level": 1, + "bbox": [ + 174, + 450, + 379, + 465 + ], + "page_idx": 5 + }, + { + "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": [ + 173, + 477, + 825, + 518 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 523, + 823, + 594 + ], + "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": [ + 174, + 647, + 825, + 718 + ], + "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, + 723, + 781, + 738 + ], + "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": [ + 320, + 741, + 676, + 767 + ], + "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": [ + 173, + 776, + 825, + 864 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 89, + 312, + 107 + ], + "page_idx": 6 + }, + { + "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": [ + 173, + 128, + 825, + 213 + ], + "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": [], + "bbox": [ + 174, + 218, + 823, + 363 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Implementation ", + "text_level": 1, + "bbox": [ + 174, + 513, + 323, + 529 + ], + "page_idx": 6 + }, + { + "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": [ + 173, + 545, + 825, + 712 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 718, + 825, + 787 + ], + "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, + 825, + 849 + ], + "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": [ + 174, + 856, + 825, + 912 + ], + "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": "
Top-1(%)Top-5(%)
Supervised [54]75.992.8
SimCLR[19]69.389.0
MoCo v2 [20]71.190.1
BYOL [25]74.391.6
SwAV[18]75.3-
Barlow Twins [22]73.291.0
SSL-HSIC (w/o target)72.290.7
SSL-HSIC (w/ target)74.892.2
", + "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": "
Top-1(%)Top-5(%)
1%10%100%1%10%100%
Supervised [54]25.456.475.948.480.492.8
SimCLR [19]48.365.676.075.587.893.1
BYOL [25]53.268.877.778.489.093.9
SwAV[18]53.970.2178.589.9-
Barlow Twins [22]55.069.7-79.289.3-
SSL-HSIC (w/o target)45.365.576.472.787.593.2
SSL-HSIC (w/ target)52.167.977.277.788.693.6
", + "bbox": [ + 450, + 292, + 820, + 412 + ], + "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": "
Dataset MetricBirdsnap Caltechl01 Cifarl0 Cifarl001 Top-1MPCATop-1DTD Top-1Aircraft Food FlowersMPCA Top-1MPCA MPCA Top-1PetsCarsTop-1SUN397 VOC2007 AP50
Linear:Top-1
Supervised-IN [19]53.794.593.678.361.072.394.791.567.861.982.8
SimCLR[19]37.490.390.671.674.9 74.550.368.490.383.650.358.880.5
BYOL [25]57.294.291.378.475.560.675.396.190.466.762.282.5
SSL-HSIC (w/o target)50.692.391.575.975.357.973.695.088.259.361.081.4
SSL-HSIC (w/ target)57.893.592.377.076.258.575.695.491.262.661.883.3
Fine-tuned:
Supervised-IN [19]85.0
Random init [19]75.893.397.586.474.686.0 85.988.397.692.192.194.3 53.667.3
SimCLR[19]76.1 75.972.6 92.195.9 97.780.2 85.964.8 73.288.186.9 88.292.0 97.081.5 89.291.4 91.363.584.1
BYOL [25]76.393.897.886.176.288.188.597.091.791.663.785.4
SSL-HSIC (w/o target)73.191.597.485.375.387.187.596.490.691.662.284.1
SSL-HSIC (w/ target)74.993.897.884.775.488.987.797.391.791.861.784.1
", + "bbox": [ + 179, + 542, + 815, + 728 + ], + "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, + 825, + 821 + ], + "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": "
MethodVOC2012 mIoUNYU v2
pct.<1.25pct.<1.252pct.<1.253rmsrel
Supervised-IN74.481.195.398.80.5730.127
SimCLR75.283.396.599.10.5570.134
BYOL76.384.696.799.10.5410.129
SSL-HSIC(w/o target)74.984.196.799.20.5390.130
SSL-HSIC(w/ target)76.083.896.899.10.5480.130
", + "bbox": [ + 174, + 167, + 596, + 246 + ], + "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": "
MethodAPbbApmk
Supervised39.635.6
SimCLR39.735.8
MoCo v240.136.3
BYOL41.637.2
SwAV41.637.8
SSL-HSIC(w/o target)40.536.3
SSL-HSIC(w/ target)41.336.8
", + "bbox": [ + 607, + 180, + 825, + 282 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Ablation Studies ", + "text_level": 1, + "bbox": [ + 174, + 314, + 343, + 332 + ], + "page_idx": 8 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 351, + 821, + 378 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 385, + 825, + 468 + ], + "page_idx": 8 + }, + { + "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": "
SSL-HSICBYOL[25]Sup.[25]
ResNetTop1Top5Top1Top5Top1Top5
50 (1x)74.892.274.391.676.492.9
50 (2x)77.994.077.493.679.995.0
50(4x)79.194.578.694.280.795.3
200 (2x)79.694.879.694.980.195.2
", + "bbox": [ + 178, + 525, + 488, + 607 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/f3fccfe523b95e9b192709865cc8dea79b37b22cdf8e7aaec72c340fcee31090.jpg", + "table_caption": [ + "Table 7: Top-1 and top-5 accuracies for different ResNet architectures using semisupervised fine-tuning. " + ], + "table_footnote": [], + "table_body": "
Top1Top5
ResNet1%10%100%1%10%100%
50 (1x)52.167.977.277.788.693.6
50 (2x)61.272.679.383.891.294.7
50 (4x)67.075.479.787.492.594.8
200(2x)69.076.380.588.392.995.2
", + "bbox": [ + 531, + 525, + 823, + 608 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 614, + 825, + 712 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 717, + 825, + 801 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 808, + 825, + 863 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/9035c053c539fa3c75259cd473bf307a2eeffeb6c29d7088e0a836195d641218.jpg", + "table_caption": [ + "(a) Regularization " + ], + "table_footnote": [], + "table_body": "
Top-1Top-5
SSL-HSIC66.787.6
InfoNCE66.086.9
", + "bbox": [ + 176, + 140, + 323, + 186 + ], + "page_idx": 9 + }, + { + "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": "
Output DimTop-1(%)
6465.4
12866.0
25666.4
51266.6
102466.6
", + "bbox": [ + 681, + 138, + 805, + 217 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/1d5b2f05662825431d99f28568973bde10e24f890522e49407153fb772150317.jpg", + "table_caption": [ + "(b) # Fourier features " + ], + "table_footnote": [], + "table_body": "
#RFFsTop-1(%)
6466.0
12866.2
25666.2
51266.4
102466.5
204866.5
No Approx.66.7
", + "bbox": [ + 331, + 140, + 464, + 237 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/f748bd0018c5619e5b3b530a607f0d56d93a2079015f5736d08550c3905c2787.jpg", + "table_caption": [ + "(c) Batch size " + ], + "table_footnote": [], + "table_body": "
Top-1(%)
Batch SizeSSL-HSICSimCLR
25663.757.5
51265.660.7
102466.762.8
204867.164.0
409666.764.6
", + "bbox": [ + 478, + 140, + 661, + 233 + ], + "page_idx": 9 + }, + { + "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": [ + 171, + 268, + 823, + 297 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusions ", + "text_level": 1, + "bbox": [ + 174, + 324, + 305, + 340 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 358, + 825, + 455 + ], + "page_idx": 9 + }, + { + "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]. ", + "bbox": [ + 174, + 462, + 825, + 546 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Broader impact ", + "text_level": 1, + "bbox": [ + 174, + 571, + 307, + 589 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 607, + 825, + 690 + ], + "page_idx": 9 + }, + { + "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]. ", + "bbox": [ + 174, + 696, + 825, + 808 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 88, + 553, + 107 + ], + "page_idx": 10 + }, + { + "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": [ + 174, + 119, + 825, + 176 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This work was supported by DeepMind, the Gatsby Charitable Foundation, the Wellcome Trust, NSERC, and the Canada CIFAR AI Chairs program. 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Measuring dependence with HSIC, this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 307, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 307, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "pushes different images’ representation distribu-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 307, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 307, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "tions apart (black arrows) and pulls representa-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 307, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 307, + 264, + 506, + 279 + ], + "score": 1.0, + "content": "tions of the same image together (colored shapes).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "source image, and push apart those of “negative” pairs (from different images). These methods are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "either understood from an information theoretic perspective as estimating the mutual information", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "between the “positives” [15], or explained as aligning features subject to a uniformity constraint [23].", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "Another line of research [24, 25] attempts to learn representation without the “negative” pairs, but", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 344, + 420, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 420, + 360 + ], + "score": 1.0, + "content": "requires either a target network or stop-gradient operation to avoid collapsing.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 507, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 507, + 374 + ], + "score": 1.0, + "content": "We examine the contrastive framework from a statistical dependence point of view: feature represen-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 373, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 384 + ], + "score": 1.0, + "content": "tations for a given transformed image should be highly dependent on the image identity (Figure 2).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "To measure dependence, we turn to the Hilbert-Schmidt Independence Criterion (HSIC) [26], and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "propose a new loss for self-supervised learning which we call SSL-HSIC. 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This perspective is closely related to the objective of BYOL [25], and can", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 563, + 467, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 467, + 577 + ], + "score": 1.0, + "content": "explain properties such as alignment and uniformity [23] observed in contrastive learning.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "Our perspective brings additional advantages, in computation and in simplicity of the algorithm,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "compared with existing approaches. 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Measuring dependence with HSIC, this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 307, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 307, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "pushes different images’ representation distribu-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 307, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 307, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "tions apart (black arrows) and pulls representa-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 307, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 307, + 264, + 506, + 279 + ], + "score": 1.0, + "content": "tions of the same image together (colored shapes).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 357 + ], + "lines": [], + "index": 34, + "bbox_fs": [ + 105, + 302, + 506, + 360 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 507, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 507, + 374 + ], + "score": 1.0, + "content": "We examine the contrastive framework from a statistical dependence point of view: feature represen-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 373, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 384 + ], + "score": 1.0, + "content": "tations for a given transformed image should be highly dependent on the image identity (Figure 2).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "To measure dependence, we turn to the Hilbert-Schmidt Independence Criterion (HSIC) [26], and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "propose a new loss for self-supervised learning which we call SSL-HSIC. Our loss is inspired by", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "HSIC Bottleneck [27, 28], an alternative to Information Bottleneck [29], where we use the image", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 415, + 331, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 331, + 430 + ], + "score": 1.0, + "content": "identity as the label, but change the regularization term.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 361, + 507, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "Through the dependence maximization perspective, we present a unified view of various self-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "supervised losses. Previous work [30] has shown that the success of InfoNCE cannot be solely", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "attributed to properties of mutual information, in particular because mutual information (unlike kernel", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "measures of dependence) has no notion of geometry in feature space: for instance, all invertible", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "encoders achieve maximal mutual information, but they can output dramatically different representa-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "tions with very different downstream performance [30]. Variational bounds on mutual information", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "do impart notions of locality that allow them to succeed in practice, departing from the mutual", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "information quantity that they try to estimate. We prove that InfoNCE, a popular such bound, in fact", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "approximates SSL-HSIC with a variance-based regularization. Thus, InfoNCE can be thought of as", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "working because it implicitly estimates a kernel-based notion of dependence. We additionally show", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "SSL-HSIC is related to metric learning, where the features learn to align to the structure induced by", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "the self-supervised labels. This perspective is closely related to the objective of BYOL [25], and can", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 563, + 467, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 467, + 577 + ], + "score": 1.0, + "content": "explain properties such as alignment and uniformity [23] observed in contrastive learning.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 432, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "Our perspective brings additional advantages, in computation and in simplicity of the algorithm,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "compared with existing approaches. 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MoCo [17] increases the number of negative examples by using a memory bank.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "BYOL [25] learns solely on positive image pairs, training so that representations of one view match", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "that of the other under a moving average of the featurizer. 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[30] observe that InfoNCE performance cannot be explained solely by the proper-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "ties of the mutual information, however, but is influenced more by other factors, such as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 507, + 443 + ], + "score": 1.0, + "content": "the formulation of the estimator and the architecture of the feature extractor. 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The square root makes the two terms", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 591, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 183, + 606 + ], + "score": 1.0, + "content": "on the same scale:", + "type": "text" + }, + { + "bbox": [ + 183, + 592, + 235, + 605 + ], + "score": 0.91, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 591, + 365, + 606 + ], + "score": 1.0, + "content": "is effectively a dot product, and", + "type": "text" + }, + { + "bbox": [ + 366, + 591, + 426, + 605 + ], + "score": 0.92, + "content": "\\sqrt { \\mathrm { H S I C } ( Z , Z ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 591, + 507, + 606 + ], + "score": 1.0, + "content": "a norm, so that e.g.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "scaling the kernel by a constant does not change the relative amount of regularization;2 this also gives", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 616, + 231, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 231, + 628 + ], + "score": 1.0, + "content": "better performance in practice.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 569, + 507, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 451, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 451, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 338, + 646 + ], + "score": 1.0, + "content": "Due to the one-hot encoded labels, we can re-write HSIC", + "type": "text" + }, + { + "bbox": [ + 338, + 632, + 365, + 644 + ], + "score": 0.9, + "content": "( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 630, + 451, + 646 + ], + "score": 1.0, + "content": "as (see Appendix A)", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 630, + 451, + 646 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 648, + 426, + 662 + ], + "lines": [ + { + "bbox": [ + 183, + 648, + 426, + 662 + ], + "spans": [ + { + "bbox": [ + 183, + 648, + 426, + 662 + ], + "score": 0.87, + "content": "\\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}", + "type": "interline_equation", + "image_path": "0208221dffe7712d774ffaf6c530bb7b763087d9478695aa452ece0cf6179e2e.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 183, + 648, + 426, + 662 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "where the first expectation is over the distribution of “positive” pairs (those from the same source", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "image), and the second one is a sum over all image pairs, including their transformations. The first", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "term in (5) pushes representations belonging to the same image identity together, while the second", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 453, + 119 + ], + "score": 1.0, + "content": "term keeps mean representations for each identity apart (as in Figure 2). The scaling of", + "type": "text" + }, + { + "bbox": [ + 453, + 105, + 505, + 118 + ], + "score": 0.66, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 429, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 270, + 128 + ], + "score": 1.0, + "content": "depends on the choice of the kernel over", + "type": "text" + }, + { + "bbox": [ + 270, + 117, + 279, + 127 + ], + "score": 0.7, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 117, + 429, + 128 + ], + "score": 1.0, + "content": ", and is irrelevant to the optimization.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "This form also reveals three key theoretical results. Section 3.1 shows that InfoNCE is better", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "understood as an HSIC-based loss than a mutual information between views. Section 3.2 reveals", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 258, + 167 + ], + "score": 1.0, + "content": "that the dependence maximization in", + "type": "text" + }, + { + "bbox": [ + 258, + 154, + 309, + 167 + ], + "score": 0.89, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "can also be viewed as a form of distance metric", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 389, + 177 + ], + "score": 1.0, + "content": "learning, where the cluster structure is defined by the labels. Finally,", + "type": "text" + }, + { + "bbox": [ + 390, + 165, + 441, + 177 + ], + "score": 0.88, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "is proportional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 504, + 189 + ], + "score": 1.0, + "content": "to the average kernel-based distance between the distribution of views for each source image (the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 334, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 334, + 199 + ], + "score": 1.0, + "content": "maximum mean discrepancy, MMD; see Appendix B.2).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 212, + 231, + 224 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 232, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 232, + 226 + ], + "score": 1.0, + "content": "3.1 Connection to InfoNCE", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 233, + 504, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "In this section we show the connection between InfoNCE and our loss; see Appendix B for full", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 470, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 470, + 256 + ], + "score": 1.0, + "content": "details. We first write the latter in its infinite sample size limit (see [23] for a derivation) as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 262, + 451, + 276 + ], + "lines": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "spans": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "score": 0.9, + "content": "\\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] ,", + "type": "interline_equation", + "image_path": "3f0f16bba114be5d8c5c03e090df921e66943d3f2aee43cefc166db7c1941606.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "where the last two expectations are taken over all points, and the first is over the distribution of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 208, + 306 + ], + "score": 1.0, + "content": "positive pairs. The kernel", + "type": "text" + }, + { + "bbox": [ + 209, + 293, + 245, + 305 + ], + "score": 0.94, + "content": "k ( z _ { 1 } , z _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "was originally formulated as a scoring function in a form of a dot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 454, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 454, + 316 + ], + "score": 1.0, + "content": "product [15], and then a scaled cosine similarity [19]. Both functions are valid kernels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 320, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 176, + 333 + ], + "score": 1.0, + "content": "Now assume that", + "type": "text" + }, + { + "bbox": [ + 176, + 320, + 213, + 332 + ], + "score": 0.92, + "content": "k ( z _ { 1 } , z _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 318, + 321, + 333 + ], + "score": 1.0, + "content": "doesn’t deviate much from", + "type": "text" + }, + { + "bbox": [ + 321, + 320, + 380, + 332 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 318, + 505, + 333 + ], + "score": 1.0, + "content": ", Taylor-expand the exponent in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 331, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 151, + 345 + ], + "score": 1.0, + "content": "(6) around", + "type": "text" + }, + { + "bbox": [ + 151, + 332, + 210, + 344 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 331, + 266, + 345 + ], + "score": 1.0, + "content": ", then expand", + "type": "text" + }, + { + "bbox": [ + 266, + 332, + 389, + 344 + ], + "score": 0.9, + "content": "\\log ( 1 + \\mathbb { E } _ { z _ { 2 } } ( . . . ) ) \\approx \\mathbb { E } _ { z _ { 2 } } ( . . . )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 331, + 450, + 345 + ], + "score": 1.0, + "content": ". We obtain an", + "type": "text" + }, + { + "bbox": [ + 450, + 331, + 502, + 344 + ], + "score": 0.87, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 331, + 507, + 345 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 342, + 173, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 173, + 355 + ], + "score": 1.0, + "content": "based objective:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 360, + 479, + 396 + ], + "lines": [ + { + "bbox": [ + 119, + 360, + 479, + 396 + ], + "spans": [ + { + "bbox": [ + 119, + 360, + 479, + 396 + ], + "score": 0.94, + "content": "\\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 } } .", + "type": "interline_equation", + "image_path": "148d20d2934fb617453a770c46e3d6152b49681716cb2763777add25e07a7604.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 119, + 360, + 479, + 372.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 119, + 372.0, + 479, + 384.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 119, + 384.0, + 479, + 396.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 408, + 504, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 504, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 189, + 422 + ], + "score": 1.0, + "content": "Since the scaling of", + "type": "text" + }, + { + "bbox": [ + 189, + 408, + 241, + 420 + ], + "score": 0.91, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 406, + 495, + 422 + ], + "score": 1.0, + "content": "is irrelevant to the optimization, we assume scaling to replace", + "type": "text" + }, + { + "bbox": [ + 495, + 410, + 504, + 419 + ], + "score": 0.62, + "content": "\\propto", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 390, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 125, + 433 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 421, + 136, + 429 + ], + "score": 0.77, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 417, + 379, + 433 + ], + "score": 1.0, + "content": ". In the small variance regime, we can show that for the right", + "type": "text" + }, + { + "bbox": [ + 379, + 421, + 386, + 431 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 417, + 390, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 437, + 437, + 451 + ], + "lines": [ + { + "bbox": [ + 172, + 437, + 437, + 451 + ], + "spans": [ + { + "bbox": [ + 172, + 437, + 437, + 451 + ], + "score": 0.89, + "content": "\\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}", + "type": "interline_equation", + "image_path": "1f185702c2be46b3c39e79c4f9490be60c34e4feadce71afbc2161c341e2afff.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 172, + 437, + 437, + 451 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 267, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 268, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 123, + 470 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 457, + 192, + 469 + ], + "score": 0.88, + "content": "\\mathrm { H S I C } ( Z , Z ) \\le 1", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 456, + 268, + 470 + ], + "score": 1.0, + "content": ", we also have that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 475, + 410, + 489 + ], + "lines": [ + { + "bbox": [ + 200, + 475, + 410, + 489 + ], + "spans": [ + { + "bbox": [ + 200, + 475, + 410, + 489 + ], + "score": 0.88, + "content": "- \\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 )", + "type": "interline_equation", + "image_path": "7e6b291b6bf5748716ae3884e71aff0e541c35dd259cdadb0e91261d691af218.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 200, + 475, + 410, + 489 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "due to the square root. InfoNCE and SSL-HSIC in general don’t quite bound each other due to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 397, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 397, + 518 + ], + "score": 1.0, + "content": "discrepancy in the variance terms, but in practice the difference is small.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "Why should we prefer the HSIC interpretation of InfoNCE? Initially, InfoNCE was suggested as a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 534, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 506, + 545 + ], + "score": 1.0, + "content": "variational approximation to the mutual information between two views [15]. It has been observed,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "however, that using tighter estimators of mutual information leads to worse performance [30]. It is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "also simple to construct examples where InfoNCE finds different representations while the underlying", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "MI remains constant [30]. Alternative theories suggest that InfoNCE balances alignment of “positive”", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "examples and uniformity of the overall feature representation [23], or that (under strong assumptions)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "it can identify the latent structure in a hypothesized data-generating process, akin to nonlinear ICA", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 598, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 612 + ], + "score": 1.0, + "content": "[50]. Our view is consistent with these theories, but doesn’t put restrictive assumptions on the input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "data or learned representations. In Section 5 (summarized in Table 8a), we show that our interpretation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 622, + 274, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 274, + 633 + ], + "score": 1.0, + "content": "gives rise to a better objective in practice.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 258, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 260, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 260, + 661 + ], + "score": 1.0, + "content": "3.2 Connection to metric learning", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Our SSL-HSIC objective is closely related to kernel alignment [48], especially centered kernel", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "alignment [49]. As a kernel method for distance metric learning, kernel alignment measures the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 494, + 701 + ], + "score": 1.0, + "content": "agreement between a kernel function and a target function. Intuitively, the self-supervised labels", + "type": "text" + }, + { + "bbox": [ + 495, + 689, + 504, + 699 + ], + "score": 0.74, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 227, + 712 + ], + "score": 1.0, + "content": "imply a cluster structure, and", + "type": "text" + }, + { + "bbox": [ + 228, + 700, + 279, + 712 + ], + "score": 0.84, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "estimates the degree of agreement between the learned", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "features and this cluster structure in the kernel space. This relationship with clustering is also", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "where the first expectation is over the distribution of “positive” pairs (those from the same source", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "image), and the second one is a sum over all image pairs, including their transformations. The first", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "term in (5) pushes representations belonging to the same image identity together, while the second", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 453, + 119 + ], + "score": 1.0, + "content": "term keeps mean representations for each identity apart (as in Figure 2). The scaling of", + "type": "text" + }, + { + "bbox": [ + 453, + 105, + 505, + 118 + ], + "score": 0.66, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 429, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 270, + 128 + ], + "score": 1.0, + "content": "depends on the choice of the kernel over", + "type": "text" + }, + { + "bbox": [ + 270, + 117, + 279, + 127 + ], + "score": 0.7, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 117, + 429, + 128 + ], + "score": 1.0, + "content": ", and is irrelevant to the optimization.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 72, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "This form also reveals three key theoretical results. Section 3.1 shows that InfoNCE is better", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "understood as an HSIC-based loss than a mutual information between views. Section 3.2 reveals", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 258, + 167 + ], + "score": 1.0, + "content": "that the dependence maximization in", + "type": "text" + }, + { + "bbox": [ + 258, + 154, + 309, + 167 + ], + "score": 0.89, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "can also be viewed as a form of distance metric", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 389, + 177 + ], + "score": 1.0, + "content": "learning, where the cluster structure is defined by the labels. Finally,", + "type": "text" + }, + { + "bbox": [ + 390, + 165, + 441, + 177 + ], + "score": 0.88, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "is proportional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 504, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 504, + 189 + ], + "score": 1.0, + "content": "to the average kernel-based distance between the distribution of views for each source image (the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 334, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 334, + 199 + ], + "score": 1.0, + "content": "maximum mean discrepancy, MMD; see Appendix B.2).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 132, + 506, + 199 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 212, + 231, + 224 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 232, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 232, + 226 + ], + "score": 1.0, + "content": "3.1 Connection to InfoNCE", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 233, + 504, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "In this section we show the connection between InfoNCE and our loss; see Appendix B for full", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 470, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 470, + 256 + ], + "score": 1.0, + "content": "details. We first write the latter in its infinite sample size limit (see [23] for a derivation) as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 233, + 505, + 256 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 262, + 451, + 276 + ], + "lines": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "spans": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "score": 0.9, + "content": "\\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] ,", + "type": "interline_equation", + "image_path": "3f0f16bba114be5d8c5c03e090df921e66943d3f2aee43cefc166db7c1941606.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 159, + 262, + 451, + 276 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "where the last two expectations are taken over all points, and the first is over the distribution of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 208, + 306 + ], + "score": 1.0, + "content": "positive pairs. The kernel", + "type": "text" + }, + { + "bbox": [ + 209, + 293, + 245, + 305 + ], + "score": 0.94, + "content": "k ( z _ { 1 } , z _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "was originally formulated as a scoring function in a form of a dot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 454, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 454, + 316 + ], + "score": 1.0, + "content": "product [15], and then a scaled cosine similarity [19]. Both functions are valid kernels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 282, + 506, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 320, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 176, + 333 + ], + "score": 1.0, + "content": "Now assume that", + "type": "text" + }, + { + "bbox": [ + 176, + 320, + 213, + 332 + ], + "score": 0.92, + "content": "k ( z _ { 1 } , z _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 318, + 321, + 333 + ], + "score": 1.0, + "content": "doesn’t deviate much from", + "type": "text" + }, + { + "bbox": [ + 321, + 320, + 380, + 332 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 318, + 505, + 333 + ], + "score": 1.0, + "content": ", Taylor-expand the exponent in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 331, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 151, + 345 + ], + "score": 1.0, + "content": "(6) around", + "type": "text" + }, + { + "bbox": [ + 151, + 332, + 210, + 344 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { z _ { 2 } } \\left[ k ( z _ { 1 } , z _ { 2 } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 331, + 266, + 345 + ], + "score": 1.0, + "content": ", then expand", + "type": "text" + }, + { + "bbox": [ + 266, + 332, + 389, + 344 + ], + "score": 0.9, + "content": "\\log ( 1 + \\mathbb { E } _ { z _ { 2 } } ( . . . ) ) \\approx \\mathbb { E } _ { z _ { 2 } } ( . . . )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 331, + 450, + 345 + ], + "score": 1.0, + "content": ". We obtain an", + "type": "text" + }, + { + "bbox": [ + 450, + 331, + 502, + 344 + ], + "score": 0.87, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 331, + 507, + 345 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 342, + 173, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 173, + 355 + ], + "score": 1.0, + "content": "based objective:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 318, + 507, + 355 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 360, + 479, + 396 + ], + "lines": [ + { + "bbox": [ + 119, + 360, + 479, + 396 + ], + "spans": [ + { + "bbox": [ + 119, + 360, + 479, + 396 + ], + "score": 0.94, + "content": "\\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 } } .", + "type": "interline_equation", + "image_path": "148d20d2934fb617453a770c46e3d6152b49681716cb2763777add25e07a7604.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 119, + 360, + 479, + 372.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 119, + 372.0, + 479, + 384.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 119, + 384.0, + 479, + 396.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 408, + 504, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 504, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 189, + 422 + ], + "score": 1.0, + "content": "Since the scaling of", + "type": "text" + }, + { + "bbox": [ + 189, + 408, + 241, + 420 + ], + "score": 0.91, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 406, + 495, + 422 + ], + "score": 1.0, + "content": "is irrelevant to the optimization, we assume scaling to replace", + "type": "text" + }, + { + "bbox": [ + 495, + 410, + 504, + 419 + ], + "score": 0.62, + "content": "\\propto", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 390, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 125, + 433 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 421, + 136, + 429 + ], + "score": 0.77, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 417, + 379, + 433 + ], + "score": 1.0, + "content": ". 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InfoNCE and SSL-HSIC in general don’t quite bound each other due to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 397, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 397, + 518 + ], + "score": 1.0, + "content": "discrepancy in the variance terms, but in practice the difference is small.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 495, + 505, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "Why should we prefer the HSIC interpretation of InfoNCE? 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It is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "also simple to construct examples where InfoNCE finds different representations while the underlying", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "MI remains constant [30]. 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Our view is consistent with these theories, but doesn’t put restrictive assumptions on the input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "data or learned representations. 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The assumption of centered and normalized features for (10) is important in the case", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "of BYOL: without it, BYOL can find trivial solutions where all the features are collapsed to the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "feature vector far away from the origin. The target network is used to prevent the collapse. 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Both points are non-trivial: the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "score": 1.0, + "content": "self-supervised framework implies non-i.i.d. batches (due to positive examples), while the estimator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 465, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 465, + 412 + ], + "score": 1.0, + "content": "in (3) assumes i.i.d. data; moreover, the time to compute (3) is quadratic in the batch size.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 144, + 428 + ], + "score": 1.0, + "content": "First, for", + "type": "text" + }, + { + "bbox": [ + 144, + 416, + 195, + 428 + ], + "score": 0.82, + "content": "\\mathrm { H S I C } ( Z , Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "we use the biased estimator in (3). 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For", + "type": "text" + }, + { + "bbox": [ + 315, + 438, + 366, + 450 + ], + "score": 0.85, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "the situation is more delicate: the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 462, + 461 + ], + "score": 1.0, + "content": "i.i.d. estimator needs re-scaling, and its bias depends on the number of positive examples", + "type": "text" + }, + { + "bbox": [ + 463, + 451, + 474, + 459 + ], + "score": 0.83, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 449, + 505, + 461 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 460, + 393, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 393, + 473 + ], + "score": 1.0, + "content": "is typically very small (usually 2). 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See Appendix A for the derivations.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 478, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 478, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 223, + 587 + ], + "score": 1.0, + "content": "For convenience, we assume", + "type": "text" + }, + { + "bbox": [ + 223, + 574, + 258, + 583 + ], + "score": 0.89, + "content": "\\Delta l = N", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 572, + 321, + 587 + ], + "score": 1.0, + "content": "(any scaling of", + "type": "text" + }, + { + "bbox": [ + 321, + 574, + 325, + 583 + ], + "score": 0.72, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 572, + 409, + 587 + ], + "score": 1.0, + "content": "can be subsumed by", + "type": "text" + }, + { + "bbox": [ + 410, + 576, + 417, + 585 + ], + "score": 0.77, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 572, + 478, + 587 + ], + "score": 1.0, + "content": "), and optimize", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 587, + 414, + 608 + ], + "lines": [ + { + "bbox": [ + 196, + 587, + 414, + 608 + ], + "spans": [ + { + "bbox": [ + 196, + 587, + 414, + 608 + ], + "score": 0.92, + "content": "\\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 ) } .", + "type": "interline_equation", + "image_path": "b66475b9da98befa9943a5613b26dbb795453ced10e1f5b98427f81b5627b47d.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 196, + 587, + 414, + 608 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 342, + 628 + ], + "score": 1.0, + "content": "The computational complexity of the proposed estimators is", + "type": "text" + }, + { + "bbox": [ + 343, + 615, + 387, + 628 + ], + "score": 0.92, + "content": "O ( B ^ { 2 } M ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 615, + 494, + 628 + ], + "score": 1.0, + "content": "for each mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 495, + 617, + 504, + 626 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 126, + 639 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 628, + 138, + 637 + ], + "score": 0.79, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 627, + 334, + 639 + ], + "score": 1.0, + "content": "augmentations. 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Moreover, the self-supervised label", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "imposes that the features from transformations of the same image are gathered in the same cluster.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "Equation (10) also allows us to connect SSL-HSIC to non-contrastive objectives such as BYOL,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "although the connection is subtle because of its use of predictor and target networks. If each image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 342, + 282 + ], + "score": 1.0, + "content": "is augmented with two views, we can compute (10) using", + "type": "text" + }, + { + "bbox": [ + 342, + 268, + 414, + 281 + ], + "score": 0.91, + "content": "\\bar { z _ { i } } \\approx ( z _ { i } ^ { 1 } + \\bar { z _ { i } ^ { 2 } } ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 268, + 506, + 282 + ], + "score": 1.0, + "content": ", so the clustering loss", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 143, + 293 + ], + "score": 1.0, + "content": "becomes", + "type": "text" + }, + { + "bbox": [ + 144, + 279, + 216, + 292 + ], + "score": 0.9, + "content": "\\propto \\textstyle \\sum _ { i } | | z _ { i } ^ { 1 } - z _ { i } ^ { 2 } | | _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 279, + 399, + 293 + ], + "score": 1.0, + "content": ". This is exactly the BYOL objective, only that", + "type": "text" + }, + { + "bbox": [ + 399, + 280, + 410, + 291 + ], + "score": 0.87, + "content": "z _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "in BYOL comes from a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 303 + ], + "score": 1.0, + "content": "target network. The assumption of centered and normalized features for (10) is important in the case", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "of BYOL: without it, BYOL can find trivial solutions where all the features are collapsed to the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "feature vector far away from the origin. The target network is used to prevent the collapse. SSL-HSIC,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "on the other hand, rules out such a solution by building the centering into the loss function, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 461, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 461, + 347 + ], + "score": 1.0, + "content": "therefore can be trained successfully without a target network or stop gradient operation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 214, + 506, + 347 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 357, + 232, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 232, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 232, + 371 + ], + "score": 1.0, + "content": "3.3 Estimator of SSL-HSIC", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "To use SSL-HSIC, we need to correctly and efficiently estimate (4). Both points are non-trivial: the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 401 + ], + "score": 1.0, + "content": "self-supervised framework implies non-i.i.d. batches (due to positive examples), while the estimator", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 465, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 465, + 412 + ], + "score": 1.0, + "content": "in (3) assumes i.i.d. data; moreover, the time to compute (3) is quadratic in the batch size.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 378, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 144, + 428 + ], + "score": 1.0, + "content": "First, for", + "type": "text" + }, + { + "bbox": [ + 144, + 416, + 195, + 428 + ], + "score": 0.82, + "content": "\\mathrm { H S I C } ( Z , Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "we use the biased estimator in (3). Although the i.i.d. estimator (3) results in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 118, + 440 + ], + "score": 1.0, + "content": "an", + "type": "text" + }, + { + "bbox": [ + 119, + 427, + 153, + 439 + ], + "score": 0.93, + "content": "O ( 1 / B )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 426, + 187, + 440 + ], + "score": 1.0, + "content": "bias for", + "type": "text" + }, + { + "bbox": [ + 187, + 428, + 197, + 437 + ], + "score": 0.84, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 426, + 463, + 440 + ], + "score": 1.0, + "content": "original images in the batch size (see Appendix A), the batch size", + "type": "text" + }, + { + "bbox": [ + 463, + 429, + 472, + 437 + ], + "score": 0.85, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "is large", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 315, + 451 + ], + "score": 1.0, + "content": "in our case and therefore the bias is negligible. For", + "type": "text" + }, + { + "bbox": [ + 315, + 438, + 366, + 450 + ], + "score": 0.85, + "content": "\\mathrm { H S I C } ( Z , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "the situation is more delicate: the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 462, + 461 + ], + "score": 1.0, + "content": "i.i.d. estimator needs re-scaling, and its bias depends on the number of positive examples", + "type": "text" + }, + { + "bbox": [ + 463, + 451, + 474, + 459 + ], + "score": 0.83, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 449, + 505, + 461 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 460, + 393, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 393, + 473 + ], + "score": 1.0, + "content": "is typically very small (usually 2). We propose the following estimator:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 416, + 506, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 473, + 475, + 512 + ], + "lines": [ + { + "bbox": [ + 118, + 473, + 475, + 512 + ], + "spans": [ + { + "bbox": [ + 118, + 473, + 475, + 512 + ], + "score": 0.93, + "content": "\\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) ,", + "type": "interline_equation", + "image_path": "e8584c88a53b312adc015e59cabde66c5f77f99650cb96ea0d2496add414a696.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 118, + 473, + 475, + 486.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 118, + 486.0, + 475, + 499.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 118, + 499.0, + 475, + 512.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 133, + 525 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 514, + 138, + 523 + ], + "score": 0.79, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 513, + 156, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 514, + 162, + 525 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 513, + 272, + 525 + ], + "score": 1.0, + "content": "index original images, and", + "type": "text" + }, + { + "bbox": [ + 272, + 515, + 279, + 525 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 513, + 297, + 525 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 514, + 301, + 523 + ], + "score": 0.73, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 513, + 425, + 525 + ], + "score": 1.0, + "content": "their random transformations;", + "type": "text" + }, + { + "bbox": [ + 425, + 514, + 431, + 523 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "is the kernel used", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 524, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 146, + 536 + ], + "score": 1.0, + "content": "for latent", + "type": "text" + }, + { + "bbox": [ + 146, + 525, + 163, + 535 + ], + "score": 0.27, + "content": "Z , l", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 524, + 313, + 536 + ], + "score": 1.0, + "content": "is the kernel used for the labels, and", + "type": "text" + }, + { + "bbox": [ + 313, + 524, + 397, + 536 + ], + "score": 0.91, + "content": "\\Delta l = l ( i , i ) - l ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 524, + 402, + 536 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 402, + 525, + 406, + 534 + ], + "score": 0.62, + "content": "\\it l", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 524, + 500, + 536 + ], + "score": 1.0, + "content": "for same labels minus", + "type": "text" + }, + { + "bbox": [ + 500, + 526, + 504, + 534 + ], + "score": 0.74, + "content": "l", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 441, + 547 + ], + "score": 1.0, + "content": "for different labels). 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The output of the encoder is a 2048-dimension embedding vector, which", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 464, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 104, + 464, + 434, + 479 + ], + "score": 1.0, + "content": "is the representation used for downstream tasks. As in BYOL [25], our projector", + "type": "text" + }, + { + "bbox": [ + 434, + 467, + 441, + 477 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 464, + 497, + 479 + ], + "score": 1.0, + "content": "and predictor", + "type": "text" + }, + { + "bbox": [ + 498, + 467, + 504, + 477 + ], + "score": 0.63, + "content": "q", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "networks are 2-layer MLPs with 4096 hidden dimensions and 256 output dimensions. The outputs", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "of the networks are batch-normalized and rescaled to unit norm before computing the loss. We use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "an inverse multiquadric kernel (IMQ) for the latent representation (approximated with 512 random", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "Fourier features that are resampled at each step; see Appendix C for details) and a linear kernel for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 136, + 532 + ], + "score": 1.0, + "content": "labels.", + "type": "text" + }, + { + "bbox": [ + 137, + 521, + 145, + 531 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "in (4) is set to 3. When training without a target network, unlike SimSiam [24], we do", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "not stop gradients for either branch. If the target network is used, its weights are an exponential", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 541, + 507, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 507, + 554 + ], + "score": 1.0, + "content": "moving average of the online network weights. We employ the same schedule as BYOL [25],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 551, + 458, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 246, + 565 + ], + "score": 0.9, + "content": "\\tau = 1 \\stackrel { - } { - } 0 . 0 1 \\stackrel { - } { \\cdot } ( \\cos ( \\pi t / T ) + 1 ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 551, + 267, + 566 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 268, + 553, + 273, + 562 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 551, + 354, + 566 + ], + "score": 1.0, + "content": "the current step and", + "type": "text" + }, + { + "bbox": [ + 355, + 553, + 363, + 562 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 551, + 458, + 566 + ], + "score": 1.0, + "content": "the total training steps.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 433, + 507, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "Image augmentation Our method uses the same data augmentation scheme as BYOL (see Ap-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 503, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 460, + 592 + ], + "score": 1.0, + "content": "pendix D.1). Briefly, we first draw a random patch from the original image and resize it to", + "type": "text" + }, + { + "bbox": [ + 460, + 580, + 503, + 590 + ], + "score": 0.91, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "Then, we apply a random horizontal flip, followed by color jittering, consisting of a random sequence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "score": 1.0, + "content": "of brightness, contrast, saturation, hue adjustments, and an optional grayscale conversion. Finally", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 612, + 490, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 490, + 626 + ], + "score": 1.0, + "content": "Gaussian blur and solarization are applied, and the view is normalized with ImageNet statistics.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 569, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 507, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 507, + 641 + ], + "score": 1.0, + "content": "Optimization We train the model with a batch size of 4096 on 128 Cloud TPU v4 cores. Again,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "following [19, 25], we use the LARS optimizer [52] with a cosine decay learning rate schedule over", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "1000 epochs. The base learning rate to all of our experiments is 0.4 and it is scaled linearly [53] with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 103, + 659, + 456, + 675 + ], + "spans": [ + { + "bbox": [ + 103, + 659, + 163, + 675 + ], + "score": 1.0, + "content": "the batch size", + "type": "text" + }, + { + "bbox": [ + 163, + 662, + 277, + 674 + ], + "score": 0.81, + "content": "l r = 0 . 4 \\times b a t c h { \\_ } s i z e / 2 5 6 .", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 659, + 429, + 675 + ], + "score": 1.0, + "content": ". All experiments use weight decay of", + "type": "text" + }, + { + "bbox": [ + 429, + 661, + 451, + 672 + ], + "score": 0.9, + "content": "1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 659, + 456, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 103, + 628, + 507, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "Learning kernel parameters We use a linear kernel for labels, since the type of kernel only scales", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 306, + 702 + ], + "score": 1.0, + "content": "(12). Our inverse multiquadric kernel for the latent", + "type": "text" + }, + { + "bbox": [ + 306, + 689, + 315, + 699 + ], + "score": 0.82, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "has an additional kernel scale parameter. We op-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "timize this along with all other parameters, but regularize it to maximize the entropy of the distribution", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 477, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 130, + 723 + ], + "score": 0.91, + "content": "k _ { \\sigma } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 711, + 161, + 723 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 162, + 710, + 231, + 723 + ], + "score": 0.94, + "content": "s _ { i j } = \\lVert { z } _ { i } - { z } _ { j } \\rVert ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 711, + 348, + 723 + ], + "score": 1.0, + "content": "; this amounts to maximizing", + "type": "text" + }, + { + "bbox": [ + 348, + 710, + 400, + 723 + ], + "score": 0.93, + "content": "\\log \\lVert k _ { \\sigma } ^ { \\prime } ( s ) \\rVert ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 711, + 477, + 723 + ], + "score": 1.0, + "content": "(Appendix D.1.2).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 111 + ], + "score": 1.0, + "content": "Linear evaluation on ImageNet Learned features are evaluated with the standard linear evaluation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "protocol commonly used in evaluating self-supervised learning methods [15–22, 25]. Table 1 reports", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "the top-1 and top-5 accuracies obtained with SSL-HSIC on ImageNet validation set, and compares", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 477, + 145 + ], + "score": 1.0, + "content": "to previous self-supervised learning methods. Without a target network, our method reaches", + "type": "text" + }, + { + "bbox": [ + 477, + 133, + 505, + 143 + ], + "score": 0.86, + "content": "7 2 . 2 \\%", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 148, + 156 + ], + "score": 1.0, + "content": "top-1 and", + "type": "text" + }, + { + "bbox": [ + 149, + 144, + 176, + 154 + ], + "score": 0.87, + "content": "9 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "top-5 accuracies. Unlike BYOL, the SSL-HSIC objective prevents the network", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "from finding trivial solutions as explained in Section 3.2. Adding the target network, our method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 503, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 361, + 178 + ], + "score": 1.0, + "content": "outperforms most previous methods, achieving top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 362, + 165, + 389, + 176 + ], + "score": 0.88, + "content": "7 4 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 165, + 476, + 178 + ], + "score": 1.0, + "content": "and top-5 accuracy of", + "type": "text" + }, + { + "bbox": [ + 476, + 165, + 503, + 176 + ], + "score": 0.89, + "content": "9 2 . 2 \\%", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "The fact that we see performance gains from adopting a target network suggests that its effect is not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 467, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 467, + 200 + ], + "score": 1.0, + "content": "yet well understood, although note discussion in [25] which points to its stabilizing effect.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 109, + 232, + 257, + 321 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 210, + 259, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 209, + 261, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 261, + 222 + ], + "score": 1.0, + "content": "Table 1: Linear evaluation on the Ima-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 191, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 191, + 232 + ], + "score": 1.0, + "content": "geNet validation set.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 232, + 257, + 321 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 232, + 257, + 321 + ], + "spans": [ + { + "bbox": [ + 109, + 232, + 257, + 321 + ], + "score": 0.967, + "html": "
Top-1(%)Top-5(%)
Supervised [54]75.992.8
SimCLR[19]69.389.0
MoCo v2 [20]71.190.1
BYOL [25]74.391.6
SwAV[18]75.3-
Barlow Twins [22]73.291.0
SSL-HSIC (w/o target)72.290.7
SSL-HSIC (w/ target)74.892.2
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Top-1(%)Top-5(%)
1%10%100%1%10%100%
Supervised [54]25.456.475.948.480.492.8
SimCLR [19]48.365.676.075.587.893.1
BYOL [25]53.268.877.778.489.093.9
SwAV[18]53.970.2178.589.9-
Barlow Twins [22]55.069.7-79.289.3-
SSL-HSIC (w/o target)45.365.576.472.787.593.2
SSL-HSIC (w/ target)52.167.977.277.788.693.6
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Dataset MetricBirdsnap Caltechl01 Cifarl0 Cifarl001 Top-1MPCATop-1DTD Top-1Aircraft Food FlowersMPCA Top-1MPCA MPCA Top-1PetsCarsTop-1SUN397 VOC2007 AP50
Linear:Top-1
Supervised-IN [19]53.794.593.678.361.072.394.791.567.861.982.8
SimCLR[19]37.490.390.671.674.9 74.550.368.490.383.650.358.880.5
BYOL [25]57.294.291.378.475.560.675.396.190.466.762.282.5
SSL-HSIC (w/o target)50.692.391.575.975.357.973.695.088.259.361.081.4
SSL-HSIC (w/ target)57.893.592.377.076.258.575.695.491.262.661.883.3
Fine-tuned:
Supervised-IN [19]85.0
Random init [19]75.893.397.586.474.686.0 85.988.397.692.192.194.3 53.667.3
SimCLR[19]76.1 75.972.6 92.195.9 97.780.2 85.964.8 73.288.186.9 88.292.0 97.081.5 89.291.4 91.363.584.1
BYOL [25]76.393.897.886.176.288.188.597.091.791.663.785.4
SSL-HSIC (w/o target)73.191.597.485.375.387.187.596.490.691.662.284.1
SSL-HSIC (w/ target)74.993.897.884.775.488.987.797.391.791.861.784.1
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Table 3 shows the top-1 accuracy of the linear", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "evaluation and fine-tuning performance on the test set. SSL-HSIC gets state-of-the-art performance on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "3 of the classification tasks and reaches strong performance on others for this benchmark, indicating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 639, + 343, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 343, + 653 + ], + "score": 1.0, + "content": "the learned representations are robust for transfer learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "Transfer to other vision tasks To test the ability of transferring to tasks other than classification,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "we fine-tune the network on semantic segmentation, depth estimation and object detection tasks.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "We use Pascal VOC2012 dataset [58] for semantic segmentation, NYU v2 dataset [66] for depth", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "estimation and COCO [67] for object detection. Object detection outputs either bounding box or", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "object segmentation (instance segmentation). Details of the evaluations setup is in Appendix D.2.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "Table 4 and Table 5 shows that SSL-HSIC achieves competitive performance on all three vision tasks.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 72, + 210, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 211, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 211, + 85 + ], + "score": 1.0, + "content": "4.2 Evaluation Results", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 111 + ], + "score": 1.0, + "content": "Linear evaluation on ImageNet Learned features are evaluated with the standard linear evaluation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "protocol commonly used in evaluating self-supervised learning methods [15–22, 25]. Table 1 reports", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "the top-1 and top-5 accuracies obtained with SSL-HSIC on ImageNet validation set, and compares", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 477, + 145 + ], + "score": 1.0, + "content": "to previous self-supervised learning methods. 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Adding the target network, our method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 165, + 503, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 361, + 178 + ], + "score": 1.0, + "content": "outperforms most previous methods, achieving top-1 accuracy of", + "type": "text" + }, + { + "bbox": [ + 362, + 165, + 389, + 176 + ], + "score": 0.88, + "content": "7 4 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 165, + 476, + 178 + ], + "score": 1.0, + "content": "and top-5 accuracy of", + "type": "text" + }, + { + "bbox": [ + 476, + 165, + 503, + 176 + ], + "score": 0.89, + "content": "9 2 . 2 \\%", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "The fact that we see performance gains from adopting a target network suggests that its effect is not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 467, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 467, + 200 + ], + "score": 1.0, + "content": "yet well understood, although note discussion in [25] which points to its stabilizing effect.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 101, + 506, + 200 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 232, + 257, + 321 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 210, + 259, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 209, + 261, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 261, + 222 + ], + "score": 1.0, + "content": "Table 1: Linear evaluation on the Ima-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 191, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 191, + 232 + ], + "score": 1.0, + "content": "geNet validation set.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 232, + 257, + 321 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 232, + 257, + 321 + ], + "spans": [ + { + "bbox": [ + 109, + 232, + 257, + 321 + ], + "score": 0.967, + "html": "
Top-1(%)Top-5(%)
Supervised [54]75.992.8
SimCLR[19]69.389.0
MoCo v2 [20]71.190.1
BYOL [25]74.391.6
SwAV[18]75.3-
Barlow Twins [22]73.291.0
SSL-HSIC (w/o target)72.290.7
SSL-HSIC (w/ target)74.892.2
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Top-1(%)Top-5(%)
1%10%100%1%10%100%
Supervised [54]25.456.475.948.480.492.8
SimCLR [19]48.365.676.075.587.893.1
BYOL [25]53.268.877.778.489.093.9
SwAV[18]53.970.2178.589.9-
Barlow Twins [22]55.069.7-79.289.3-
SSL-HSIC (w/o target)45.365.576.472.787.593.2
SSL-HSIC (w/ target)52.167.977.277.788.693.6
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Dataset MetricBirdsnap Caltechl01 Cifarl0 Cifarl001 Top-1MPCATop-1DTD Top-1Aircraft Food FlowersMPCA Top-1MPCA MPCA Top-1PetsCarsTop-1SUN397 VOC2007 AP50
Linear:Top-1
Supervised-IN [19]53.794.593.678.361.072.394.791.567.861.982.8
SimCLR[19]37.490.390.671.674.9 74.550.368.490.383.650.358.880.5
BYOL [25]57.294.291.378.475.560.675.396.190.466.762.282.5
SSL-HSIC (w/o target)50.692.391.575.975.357.973.695.088.259.361.081.4
SSL-HSIC (w/ target)57.893.592.377.076.258.575.695.491.262.661.883.3
Fine-tuned:
Supervised-IN [19]85.0
Random init [19]75.893.397.586.474.686.0 85.988.397.692.192.194.3 53.667.3
SimCLR[19]76.1 75.972.6 92.195.9 97.780.2 85.964.8 73.288.186.9 88.292.0 97.081.5 89.291.4 91.363.584.1
BYOL [25]76.393.897.886.176.288.188.597.091.791.663.785.4
SSL-HSIC (w/o target)73.191.597.485.375.387.187.596.490.691.662.284.1
SSL-HSIC (w/ target)74.993.897.884.775.488.987.797.391.791.861.784.1
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Table 3 shows the top-1 accuracy of the linear", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "evaluation and fine-tuning performance on the test set. SSL-HSIC gets state-of-the-art performance on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "3 of the classification tasks and reaches strong performance on others for this benchmark, indicating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 639, + 343, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 343, + 653 + ], + "score": 1.0, + "content": "the learned representations are robust for transfer learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 585, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "Transfer to other vision tasks To test the ability of transferring to tasks other than classification,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "we fine-tune the network on semantic segmentation, depth estimation and object detection tasks.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "We use Pascal VOC2012 dataset [58] for semantic segmentation, NYU v2 dataset [66] for depth", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "estimation and COCO [67] for object detection. Object detection outputs either bounding box or", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "object segmentation (instance segmentation). Details of the evaluations setup is in Appendix D.2.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "Table 4 and Table 5 shows that SSL-HSIC achieves competitive performance on all three vision tasks.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 655, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 133, + 365, + 195 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 366, + 132 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 367, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 367, + 90 + ], + "score": 1.0, + "content": "Table 4: Fine-tuning performance on semantic segmentation and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 368, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 368, + 101 + ], + "score": 1.0, + "content": "depth estimation. Mean Intersection over Union (mIoU) is re-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 367, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 367, + 112 + ], + "score": 1.0, + "content": "ported for semantic segmentation. Relative error (rel), root mean", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 367, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 367, + 122 + ], + "score": 1.0, + "content": "squared error (rms), and the percent of pixels (pct) where the error", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 346, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 142, + 133 + ], + "score": 1.0, + "content": "is below", + "type": "text" + }, + { + "bbox": [ + 142, + 121, + 167, + 131 + ], + "score": 0.86, + "content": "1 . 2 5 ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 120, + 346, + 133 + ], + "score": 1.0, + "content": "thresholds are reported for depth estimation.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 107, + 133, + 365, + 195 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 133, + 365, + 195 + ], + "spans": [ + { + "bbox": [ + 107, + 133, + 365, + 195 + ], + "score": 0.979, + "html": "
MethodVOC2012 mIoUNYU v2
pct.<1.25pct.<1.252pct.<1.253rmsrel
Supervised-IN74.481.195.398.80.5730.127
SimCLR75.283.396.599.10.5570.134
BYOL76.384.696.799.10.5410.129
SSL-HSIC(w/o target)74.984.196.799.20.5390.130
SSL-HSIC(w/ target)76.083.896.899.10.5480.130
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MethodAPbbApmk
Supervised39.635.6
SimCLR39.735.8
MoCo v240.136.3
BYOL41.637.2
SwAV41.637.8
SSL-HSIC(w/o target)40.536.3
SSL-HSIC(w/ target)41.336.8
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SSL-HSICBYOL[25]Sup.[25]
ResNetTop1Top5Top1Top5Top1Top5
50 (1x)74.892.274.391.676.492.9
50 (2x)77.994.077.493.679.995.0
50(4x)79.194.578.694.280.795.3
200 (2x)79.694.879.694.980.195.2
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Top1Top5
ResNet1%10%100%1%10%100%
50 (1x)52.167.977.277.788.693.6
50 (2x)61.272.679.383.891.294.7
50 (4x)67.075.479.787.492.594.8
200(2x)69.076.380.588.392.995.2
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This suggests that the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 552, + 336, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 336, + 564 + ], + "score": 1.0, + "content": "regularization employed by SSL-HSIC is more effective.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 413, + 582 + ], + "score": 1.0, + "content": "Kernel type We investigate the effect of using different a kernel on latents", + "type": "text" + }, + { + "bbox": [ + 413, + 569, + 422, + 579 + ], + "score": 0.69, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 569, + 506, + 582 + ], + "score": 1.0, + "content": ". 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MethodVOC2012 mIoUNYU v2
pct.<1.25pct.<1.252pct.<1.253rmsrel
Supervised-IN74.481.195.398.80.5730.127
SimCLR75.283.396.599.10.5570.134
BYOL76.384.696.799.10.5410.129
SSL-HSIC(w/o target)74.984.196.799.20.5390.130
SSL-HSIC(w/ target)76.083.896.899.10.5480.130
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MethodAPbbApmk
Supervised39.635.6
SimCLR39.735.8
MoCo v240.136.3
BYOL41.637.2
SwAV41.637.8
SSL-HSIC(w/o target)40.536.3
SSL-HSIC(w/ target)41.336.8
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SSL-HSICBYOL[25]Sup.[25]
ResNetTop1Top5Top1Top5Top1Top5
50 (1x)74.892.274.391.676.492.9
50 (2x)77.994.077.493.679.995.0
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Top1Top5
ResNet1%10%100%1%10%100%
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Non-linear", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "kernels indeed improve the performance; Gaussian and IMQ kernels reach very similar performance", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "for 100 epochs. 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The RFF approximation has a minor impact on the overall performance, as long as", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "we resample them; fixed sets of features performed poorly. 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Top-1Top-5
SSL-HSIC66.787.6
InfoNCE66.086.9
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Output DimTop-1(%)
6465.4
12866.0
25666.4
51266.6
102466.6
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#RFFsTop-1(%)
6466.0
12866.2
25666.2
51266.4
102466.5
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In this paper, we only explored", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "learning image representations, but in future work SSL-HSIC can be extended to learning structure", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 422, + 304, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 120, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 422, + 130, + 431 + ], + "score": 0.77, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 422, + 304, + 433 + ], + "score": 1.0, + "content": "as well, building on existing work [41, 45].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 188, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 190, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 190, + 469 + ], + "score": 1.0, + "content": "Broader impact", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "Our work concentrates on providing a more theoretically grounded and interpretable loss function for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "self-supervised learning. 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An important concern for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "fairness of ML algorithms is dataset bias. ImageNet is known for a number of problems such as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "offensive annotations, non-visual concepts and lack of diversity, in particular for underrepresented", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "groups. Existing works and remedies typically focus on label bias. Since SSL doesn’t use labels,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "however, the type and degree of bias could be very different from that of supervised learning. To", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "mitigate the risk of dataset bias, one could employ dataset re-balancing to correct sampling bias [68]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 629, + 502, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 502, + 642 + ], + "score": 1.0, + "content": "or completely exclude human images from the dataset while achieving the same performance [69].", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "A new topic to investigate for self-supervised learning is how the bias/unbiased representation could", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "be transferred to downstream tasks. 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Top-1Top-5
SSL-HSIC66.787.6
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Output DimTop-1(%)
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#RFFsTop-1(%)
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102466.5
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No Approx.66.7
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Top-1(%)
Batch SizeSSL-HSICSimCLR
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The performance saturates at 512 dimensions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 213, + 505, + 237 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 257, + 187, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 254, + 190, + 273 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 190, + 273 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "We introduced SSL-HSIC, a loss function for self-supervised representation learning based on kernel", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "dependence maximization. We provided a unified view on various self-supervised learning losses:", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "we proved that InfoNCE, a lower bound of mutual information, actually approximates SSL-HSIC", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "with a variance-based regularization, and we can also interpret SSL-HSIC as metric learning where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "the cluster structure is imposed by the self-supervised label, of which the BYOL objective is a special", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "case. We showed that training with SSL-HSIC achieves performance on par with the state-of-the-art", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 286, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 286, + 363 + ], + "score": 1.0, + "content": "on the standard self-supervised benchmarks.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 284, + 506, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "Although using the image identity as self-supervised label provides a good inductive bias, it might", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "not be wholly satisfactory; we expect that some images pairs are in fact more similar than others,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "based e.g. on their ImageNet class label. It will be interesting to explore methods that combine label", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 400, + 504, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 504, + 411 + ], + "score": 1.0, + "content": "structure discovery with representation learning (as in SwAV [18]). In this paper, we only explored", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "learning image representations, but in future work SSL-HSIC can be extended to learning structure", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 422, + 304, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 120, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 422, + 130, + 431 + ], + "score": 0.77, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 422, + 304, + 433 + ], + "score": 1.0, + "content": "as well, building on existing work [41, 45].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 366, + 506, + 433 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 188, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 190, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 190, + 469 + ], + "score": 1.0, + "content": "Broader impact", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "Our work concentrates on providing a more theoretically grounded and interpretable loss function for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "self-supervised learning. A better understanding of self-supervised learning, especially through more", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "interpretable learning dynamics, is likely to lead to better and more explicit control over societal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "biases of these algorithms. SSL-HSIC yields an alternative, clearer understanding of existing self-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "supervised methods. As such, it is unlikely that our method introduces further biases than those", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 284, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 284, + 550 + ], + "score": 1.0, + "content": "already present for self-supervised learning.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 481, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "The broader impacts of the self-supervised learning framework is an area that has not been studied", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "by the AI ethics community, but we think it calls for closer inspection. An important concern for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "fairness of ML algorithms is dataset bias. ImageNet is known for a number of problems such as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "offensive annotations, non-visual concepts and lack of diversity, in particular for underrepresented", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "groups. Existing works and remedies typically focus on label bias. Since SSL doesn’t use labels,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "however, the type and degree of bias could be very different from that of supervised learning. To", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "mitigate the risk of dataset bias, one could employ dataset re-balancing to correct sampling bias [68]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 629, + 502, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 502, + 642 + ], + "score": 1.0, + "content": "or completely exclude human images from the dataset while achieving the same performance [69].", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 552, + 507, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "A new topic to investigate for self-supervised learning is how the bias/unbiased representation could", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "be transferred to downstream tasks. We are not aware of any work in this direction. Another area", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "of concern is security and robustness. Compared to supervised learning, self-supervised learning", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "typically involves more intensive data augmentation such as color jittering, brightness adjustment, etc.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "There is some initial evidence suggesting self-supervised learning improves model robustness [70].", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "However, since data augmentation can either be beneficial [71] or detrimental [72] depending on the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 496, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 496, + 725 + ], + "score": 1.0, + "content": "type of adversarial attacks, more studies are needed to assess its role for self-supervised learning.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 645, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 70, + 339, + 85 + ], + "lines": [ + { + "bbox": [ + 105, + 69, + 341, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 341, + 88 + ], + "score": 1.0, + "content": "Acknowledgments and Disclosure of Funding", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 95, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "score": 1.0, + "content": "The authors would like to thank Olivier J. Hénaff for valuable feedback on the manuscript and the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "help for evaluating object detection task. We thank Aaron Van den Oord and Oriol Vinyals for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "score": 1.0, + "content": "providing valuable feedback on the manuscript. We are grateful to Yonglong Tian, Ting Chen and the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 462, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 462, + 142 + ], + "score": 1.0, + "content": "BYOL authors for the help with reproducing baselines and evaluating downstream tasks.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 504, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "score": 1.0, + "content": "This work was supported by DeepMind, the Gatsby Charitable Foundation, the Wellcome Trust,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 319, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 319, + 169 + ], + "score": 1.0, + "content": "NSERC, and the Canada CIFAR AI Chairs program.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 183, + 163, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 165, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 165, + 197 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 506, + 718 + ], + "lines": [ + { + "bbox": [ + 111, + 202, + 507, + 215 + ], + "spans": [ + { + "bbox": [ + 111, + 202, + 128, + 215 + ], + "score": 1.0, + "content": "[1]", + "type": "text" + }, + { + "bbox": [ + 131, + 202, + 507, + 214 + ], + "score": 1.0, + "content": "O. 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Dataset MetricBirdsnap Caltechl01 Cifarl0 Cifarl001 Top-1MPCATop-1DTD Top-1Aircraft Food FlowersMPCA Top-1MPCA MPCA Top-1PetsCarsTop-1SUN397 VOC2007 AP50
Linear:Top-1
Supervised-IN [19]53.794.593.678.361.072.394.791.567.861.982.8
SimCLR[19]37.490.390.671.674.9 74.550.368.490.383.650.358.880.5
BYOL [25]57.294.291.378.475.560.675.396.190.466.762.282.5
SSL-HSIC (w/o target)50.692.391.575.975.357.973.695.088.259.361.081.4
SSL-HSIC (w/ target)57.893.592.377.076.258.575.695.491.262.661.883.3
Fine-tuned:
Supervised-IN [19]85.0
Random init [19]75.893.397.586.474.686.0 85.988.397.692.192.194.3 53.667.3
SimCLR[19]76.1 75.972.6 92.195.9 97.780.2 85.964.8 73.288.186.9 88.292.0 97.081.5 89.291.4 91.363.584.1
BYOL [25]76.393.897.886.176.288.188.597.091.791.663.785.4
SSL-HSIC (w/o target)73.191.597.485.375.387.187.596.490.691.662.284.1
SSL-HSIC (w/ target)74.993.897.884.775.488.987.797.391.791.861.784.1
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Top-1(%)Top-5(%)
1%10%100%1%10%100%
Supervised [54]25.456.475.948.480.492.8
SimCLR [19]48.365.676.075.587.893.1
BYOL [25]53.268.877.778.489.093.9
SwAV[18]53.970.2178.589.9-
Barlow Twins [22]55.069.7-79.289.3-
SSL-HSIC (w/o target)45.365.576.472.787.593.2
SSL-HSIC (w/ target)52.167.977.277.788.693.6
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Top-1(%)Top-5(%)
Supervised [54]75.992.8
SimCLR[19]69.389.0
MoCo v2 [20]71.190.1
BYOL [25]74.391.6
SwAV[18]75.3-
Barlow Twins [22]73.291.0
SSL-HSIC (w/o target)72.290.7
SSL-HSIC (w/ target)74.892.2
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MethodVOC2012 mIoUNYU v2
pct.<1.25pct.<1.252pct.<1.253rmsrel
Supervised-IN74.481.195.398.80.5730.127
SimCLR75.283.396.599.10.5570.134
BYOL76.384.696.799.10.5410.129
SSL-HSIC(w/o target)74.984.196.799.20.5390.130
SSL-HSIC(w/ target)76.083.896.899.10.5480.130
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SSL-HSICBYOL[25]Sup.[25]
ResNetTop1Top5Top1Top5Top1Top5
50 (1x)74.892.274.391.676.492.9
50 (2x)77.994.077.493.679.995.0
50(4x)79.194.578.694.280.795.3
200 (2x)79.694.879.694.980.195.2
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Top1Top5
ResNet1%10%100%1%10%100%
50 (1x)52.167.977.277.788.693.6
50 (2x)61.272.679.383.891.294.7
50 (4x)67.075.479.787.492.594.8
200(2x)69.076.380.588.392.995.2
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MethodAPbbApmk
Supervised39.635.6
SimCLR39.735.8
MoCo v240.136.3
BYOL41.637.2
SwAV41.637.8
SSL-HSIC(w/o target)40.536.3
SSL-HSIC(w/ target)41.336.8
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Output DimTop-1(%)
6465.4
12866.0
25666.4
51266.6
102466.6
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Top-1(%)
Batch SizeSSL-HSICSimCLR
25663.757.5
51265.660.7
102466.762.8
204867.164.0
409666.764.6
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#RFFsTop-1(%)
6466.0
12866.2
25666.2
51266.4
102466.5
204866.5
No Approx.66.7
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Top-1Top-5
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InfoNCE66.086.9
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0000000000000000000000000000000000000000..cc612e0fa74e702dedd82be1245731566d4bbb16 --- /dev/null +++ b/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. + +
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Zhang et al. (SumTransfer) [42]40.941.338.5-=1
Zhang et al. (LSTM) [43]38.642.941.854.759.658.7
Mahasseni et al. (SUM-GANsup) [22]41.743.6-56.361.2=
Rochan et al. (SUM-FCN) [31]47.551.144.156.859.258.2
Rochan et al. (SUM-DeepLab) [31]48.850.245.058.459.157.4
Zhou et al. [47]42.143.942.658.159.858.9
Zhang et al. [44]=44.91=63.91
Fajtl et al. [3]49.751.1-61.462.41
Rochan et al. [30]148.041.656.155.7
Chen et al. (V2TS) [1]-1162.111
He et al. [9]47.2--59.4--
Park et al. (SumGraph) [26]51.452.948.763.965.860.5
GoogleNet+bi-LSTM38.542.440.753.959.658.6
ResNet+bi-LSTM39.444.042.655.061.059.9
CLIP-Image+bi-LSTM41.145.944.956.863.761.6
CLIP-Image+Video Caption+bi-LSTM41.246.145.557.164.362.4
GoogleNet+Transformer51.653.549.464.266.361.3
ResNet+Transformer52.854.950.365.067.562.8
CLIP-Image+Transformer53.555.351.065.568.163.4
CLIP-It: CLIP-Image+Video Caption+Transformer54.256.451.966.369.065.5
+ +# 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. + +
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Mahasseni et al. [22]39.143.451.759.5
Yuan et al. [41]41.91=57.61
Rochan et al. (SUM-FCNun sup) [31]41.5139.552.71-
Rochan et al. [30]47.5=41.655.6=55.7
He et al. [9]46.047.044.558.558.957.8
Park et al. (SumGraph) [26]49.852.147.059.361.257.6
GoogleNet+bi-LSTM33.138.036.547.754.952.3
ResNet+bi-LSTM34.540.139.651.056.253.8
CLIP-Image+bi-LSTM35.741.041.452.858.756.0
CLIP-Image+Video Caption+bi-LSTM36.942.442.553.559.457.6
GoogleNet+Transformer50.052.747.659.962.158.4
ResNet+Transformer50.853.949.361.163.059.9
CLIP-Image+Transformer51.253.649.261.964.060.6
CLIP-It: CLIP-Image+Video Caption+Transformer52.554.750.063.065.762.8
+ +Table 3: Comparing F1 Scores for generic video summarization on the QFVS datase + +
SupervisedVid 1Vid 2Vid 3Vid 4Avg
SubMod [6]49.5151.0364.5235.8250.22
QFVS [33]62.6646.1158.8533.5050.29
CLIP-Image + bi-LSTM ResNet +Transformer65.4356.5568.6340.0657.67
66.9758.3270.1043.3159.67
CLIP-Image + Transformer70.861.6772.4347.4863.11
CLIP-It (Gen. Caption)74.1363.4475.8650.2365.92
CLIP-It (GT Caption)84.9871.2682.5561.4675.06
UnsupervisedVid 1Vid 2Vid 3Vid 4Avg
Quasi [46]53.0653.8049.9122.3144.77
CLIP-Image + Transformer65.4457.2165.1041.6357.35
CLIP-It (Gen. Caption)67.0259.4866.4344.1959.28
CLIP-It (GT Caption)73.9066.8375.4452.3167.12
+ +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] + +
MethodF-MeasureRecall
Supervised
Submod-V+Both et al. [27]34.1531.59
CLIP Image + Transformer41.5839.96
CLIP-It: CLIP Image + Gen. Video Caption + Transformer44.7043.28
CLIP-It: CLIP Image + GT Video Caption + Transformer52.1050.76
Unsupervised
CLIP Image + Transformer39.2237.46
CLIP-It: CLIP Image + Gen.Video Caption + Transformer42.1040.65
CLIP-It: CLIP Image + GT Video Caption + Transformer49.9847.91
Table 5: Results on TV Episodes dataset [16]
MethodF-MeasureRecall
Supervised
Submod-V+Sem. Rep. et al. [27]40.9037.02
CLIP Image+ Transformer47.8246.02
CLIP-It: CLIP Image + GT Video Caption + Transformer55.3453.90
Unsupervised
CLIP Image + Transformer45.7744.01
CLIP-It: CLIP Image + GT Video Caption+ Transformer53.4252.50
+ +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. + +
Vid 1Vid 2Vid 3Vid 4Avg
SeqDPP[5136.5943.6725.2618.1530.92
SH-DPP [32]35.6742.7436.5118.6233.38
QFVS [33]48.6841.6656.4729.9644.19
CLIP-Image + Query + bi-LSTM54.4748.5962.8138.6451.13
ResNet + Query + Transformer55.1951.0364.2639.4752.49
CLIP-It: CLIP-Image + Query + Transformer57.1353.6066.0841.4154.55
+ +![](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. 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CRC Press (1999) 8 \ No newline at end of file diff --git a/parse/train/3ccoZ40Us0N/3ccoZ40Us0N_content_list.json b/parse/train/3ccoZ40Us0N/3ccoZ40Us0N_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..efe3a308016201ccd86902d5b16129a2c3e79185 --- /dev/null +++ b/parse/train/3ccoZ40Us0N/3ccoZ40Us0N_content_list.json @@ -0,0 +1,1005 @@ +[ + { + "type": "text", + "text": "CLIP-It! Language-Guided Video Summarization ", + "text_level": 1, + "bbox": [ + 251, + 122, + 745, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Medhini Narasimhan Anna Rohrbach Trevor Darrell ", + "bbox": [ + 297, + 226, + 704, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "University of California, Berkeley {medhini, anna.rohrbach, trevordarrell}@berkeley.edu https://medhini.github.io/clip_it ", + "bbox": [ + 297, + 241, + 700, + 281 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d7d642572e87e426a8435e58d17586d42b74fd301a04f4a0c71e972de49e802f.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 174, + 290, + 825, + 469 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 560, + 535, + 577 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 233, + 590, + 766, + 825 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 844, + 312, + 862 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 176, + 867, + 826, + 909 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 147 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 154, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 313, + 825, + 518 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 525, + 825, + 733 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 738, + 825, + 835 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 847, + 321, + 863 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 285 + ], + "page_idx": 2 + }, + { + "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]. ", + "bbox": [ + 174, + 291, + 825, + 361 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 173, + 367, + 825, + 449 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 457, + 825, + 622 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 628, + 825, + 712 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 CLIP-It: Language-Guided Video Summarization ", + "text_level": 1, + "bbox": [ + 174, + 722, + 622, + 739 + ], + "page_idx": 2 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 746, + 826, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/5cf0ec307f855e688d0cd7ddc67c06c5416e4e5a59a1eb7b6e6c867ced503bf9.jpg", + "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": [], + "bbox": [ + 178, + 90, + 807, + 315 + ], + "page_idx": 3 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 404, + 826, + 585 + ], + "page_idx": 3 + }, + { + "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, ", + "bbox": [ + 176, + 590, + 823, + 660 + ], + "page_idx": 3 + }, + { + "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": [ + 254, + 694, + 745, + 780 + ], + "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": [ + 173, + 787, + 826, + 820 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 825, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 823, + 119 + ], + "page_idx": 4 + }, + { + "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": [ + 171, + 126, + 823, + 155 + ], + "page_idx": 4 + }, + { + "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). ", + "bbox": [ + 173, + 160, + 825, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 Learning ", + "text_level": 1, + "bbox": [ + 174, + 279, + 279, + 294 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 297, + 826, + 339 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Classification loss. We use a weighted binary cross entropy loss for classifying each frame: ", + "bbox": [ + 171, + 344, + 774, + 361 + ], + "page_idx": 4 + }, + { + "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": [ + 285, + 367, + 710, + 411 + ], + "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": [ + 173, + 422, + 826, + 472 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 477, + 825, + 561 + ], + "page_idx": 4 + }, + { + "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": [ + 171, + 566, + 823, + 595 + ], + "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": [ + 411, + 602, + 584, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\hat { \\mathbf { y } }$ denotes the reconstructed features. ", + "bbox": [ + 174, + 647, + 457, + 661 + ], + "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": [ + 174, + 667, + 823, + 708 + ], + "page_idx": 4 + }, + { + "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": [ + 346, + 707, + 650, + 747 + ], + "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": [ + 176, + 750, + 714, + 765 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The final loss function for supervised learning is then, ", + "bbox": [ + 173, + 770, + 529, + 785 + ], + "page_idx": 4 + }, + { + "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": [ + 387, + 792, + 609, + 809 + ], + "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": [ + 173, + 815, + 825, + 871 + ], + "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": [ + 410, + 871, + 588, + 887 + ], + "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": [ + 171, + 890, + 823, + 920 + ], + "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": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Zhang et al. (SumTransfer) [42]40.941.338.5-=1
Zhang et al. (LSTM) [43]38.642.941.854.759.658.7
Mahasseni et al. (SUM-GANsup) [22]41.743.6-56.361.2=
Rochan et al. (SUM-FCN) [31]47.551.144.156.859.258.2
Rochan et al. (SUM-DeepLab) [31]48.850.245.058.459.157.4
Zhou et al. [47]42.143.942.658.159.858.9
Zhang et al. [44]=44.91=63.91
Fajtl et al. [3]49.751.1-61.462.41
Rochan et al. [30]148.041.656.155.7
Chen et al. (V2TS) [1]-1162.111
He et al. [9]47.2--59.4--
Park et al. (SumGraph) [26]51.452.948.763.965.860.5
GoogleNet+bi-LSTM38.542.440.753.959.658.6
ResNet+bi-LSTM39.444.042.655.061.059.9
CLIP-Image+bi-LSTM41.145.944.956.863.761.6
CLIP-Image+Video Caption+bi-LSTM41.246.145.557.164.362.4
GoogleNet+Transformer51.653.549.464.266.361.3
ResNet+Transformer52.854.950.365.067.562.8
CLIP-Image+Transformer53.555.351.065.568.163.4
CLIP-It: CLIP-Image+Video Caption+Transformer54.256.451.966.369.065.5
", + "bbox": [ + 171, + 117, + 828, + 424 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 440, + 313, + 458 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 464, + 823, + 491 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Generic Video Summarization ", + "text_level": 1, + "bbox": [ + 176, + 500, + 423, + 515 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 518, + 825, + 573 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 580, + 825, + 800 + ], + "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. ", + "bbox": [ + 176, + 808, + 825, + 849 + ], + "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": [ + 176, + 856, + 825, + 911 + ], + "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": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Mahasseni et al. [22]39.143.451.759.5
Yuan et al. [41]41.91=57.61
Rochan et al. (SUM-FCNun sup) [31]41.5139.552.71-
Rochan et al. [30]47.5=41.655.6=55.7
He et al. [9]46.047.044.558.558.957.8
Park et al. (SumGraph) [26]49.852.147.059.361.257.6
GoogleNet+bi-LSTM33.138.036.547.754.952.3
ResNet+bi-LSTM34.540.139.651.056.253.8
CLIP-Image+bi-LSTM35.741.041.452.858.756.0
CLIP-Image+Video Caption+bi-LSTM36.942.442.553.559.457.6
GoogleNet+Transformer50.052.747.659.962.158.4
ResNet+Transformer50.853.949.361.163.059.9
CLIP-Image+Transformer51.253.649.261.964.060.6
CLIP-It: CLIP-Image+Video Caption+Transformer52.554.750.063.065.762.8
", + "bbox": [ + 173, + 117, + 828, + 348 + ], + "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": "
SupervisedVid 1Vid 2Vid 3Vid 4Avg
SubMod [6]49.5151.0364.5235.8250.22
QFVS [33]62.6646.1158.8533.5050.29
CLIP-Image + bi-LSTM ResNet +Transformer65.4356.5568.6340.0657.67
66.9758.3270.1043.3159.67
CLIP-Image + Transformer70.861.6772.4347.4863.11
CLIP-It (Gen. Caption)74.1363.4475.8650.2365.92
CLIP-It (GT Caption)84.9871.2682.5561.4675.06
UnsupervisedVid 1Vid 2Vid 3Vid 4Avg
Quasi [46]53.0653.8049.9122.3144.77
CLIP-Image + Transformer65.4457.2165.1041.6357.35
CLIP-It (Gen. Caption)67.0259.4866.4344.1959.28
CLIP-It (GT Caption)73.9066.8375.4452.3167.12
", + "bbox": [ + 253, + 371, + 740, + 590 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 607, + 825, + 676 + ], + "page_idx": 6 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 683, + 825, + 766 + ], + "page_idx": 6 + }, + { + "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", + "bbox": [ + 174, + 772, + 825, + 897 + ], + "page_idx": 6 + }, + { + "type": "image", + "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. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 823, + 376 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/3c1b4747417c0ee5ce5f793081633a4249bd4d0f0c0be562dedda1199fe5b01d.jpg", + "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. " + ], + "image_footnote": [], + "bbox": [ + 183, + 449, + 823, + 672 + ], + "page_idx": 7 + }, + { + "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. ", + "bbox": [ + 174, + 733, + 820, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 767, + 825, + 892 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/14764633006cbbd0468cb459a47e92d35cb3c23edd4e58b2a63fd00bb96c188e.jpg", + "table_caption": [ + "Table 4: Results on UT Egocentric dataset [16] " + ], + "table_footnote": [], + "table_body": "
MethodF-MeasureRecall
Supervised
Submod-V+Both et al. [27]34.1531.59
CLIP Image + Transformer41.5839.96
CLIP-It: CLIP Image + Gen. Video Caption + Transformer44.7043.28
CLIP-It: CLIP Image + GT Video Caption + Transformer52.1050.76
Unsupervised
CLIP Image + Transformer39.2237.46
CLIP-It: CLIP Image + Gen.Video Caption + Transformer42.1040.65
CLIP-It: CLIP Image + GT Video Caption + Transformer49.9847.91
Table 5: Results on TV Episodes dataset [16]
MethodF-MeasureRecall
Supervised
Submod-V+Sem. Rep. et al. [27]40.9037.02
CLIP Image+ Transformer47.8246.02
CLIP-It: CLIP Image + GT Video Caption + Transformer55.3453.90
Unsupervised
CLIP Image + Transformer45.7744.01
CLIP-It: CLIP Image + GT Video Caption+ Transformer53.4252.50
", + "bbox": [ + 214, + 104, + 779, + 448 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 173, + 463, + 825, + 589 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 173, + 595, + 825, + 693 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 698, + 825, + 755 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/bf1ab8fad45ed422cb09bb2d772b5e4e5900b4e6dbf40f63aa8eb4fe634018e7.jpg", + "table_caption": [ + "Table 6: Comparing F1 Scores of different methods on the QFVS dataset. " + ], + "table_footnote": [], + "table_body": "
Vid 1Vid 2Vid 3Vid 4Avg
SeqDPP[5136.5943.6725.2618.1530.92
SH-DPP [32]35.6742.7436.5118.6233.38
QFVS [33]48.6841.6656.4729.9644.19
CLIP-Image + Query + bi-LSTM54.4748.5962.8138.6451.13
ResNet + Query + Transformer55.1951.0364.2639.4752.49
CLIP-It: CLIP-Image + Query + Transformer57.1353.6066.0841.4154.55
", + "bbox": [ + 192, + 786, + 802, + 907 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/8598abec91315572d62925e9e2ddce1b0bafa26350dcb4469546767f520beafd.jpg", + "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\". " + ], + "image_footnote": [], + "bbox": [ + 186, + 88, + 826, + 234 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 301, + 825, + 426 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 433, + 825, + 558 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.2 Query-Focused Video Summarization ", + "text_level": 1, + "bbox": [ + 176, + 602, + 473, + 617 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 176, + 636, + 823, + 664 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 670, + 825, + 712 + ], + "page_idx": 9 + }, + { + "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). ", + "bbox": [ + 173, + 718, + 825, + 829 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 835, + 825, + 919 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Discussion and Broader Impacts ", + "text_level": 1, + "bbox": [ + 174, + 89, + 475, + 107 + ], + "page_idx": 10 + }, + { + "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. ", + "bbox": [ + 174, + 111, + 825, + 237 + ], + "page_idx": 10 + }, + { + "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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Sequence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "labeling methods [5, 42, 43, 44, 18] model variable-range dependencies between frames but fail to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "capture relationships across all frames simultaneously. 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However, in their work the query can only be chosen from a fixed set", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "score": 1.0, + "content": "of predefined concepts which limits the flexibility for the user.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "It is important to note that efforts in generic and query-focused video summarization have so far", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "been disjoint, with no single method for both. 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In case of generic video summarization, the natural language text is a video description obtained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "using an off-the-shelf dense video captioning method. Alternatively, in the case of query-focused", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "video summarization, the language input is a user-defined query. Unlike existing generic methods", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "which only use visual cues, we show that adding language as an input leads to the “discovery” of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "relevant concepts and actions resulting in better summaries. Our method uses a Transformer with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 346, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 357 + ], + "score": 1.0, + "content": "positional encoding, which can attend to all frames at once (unlike LSTM based methods) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "also keep track of the ordering of frames (unlike graph based methods). In contrast to existing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "query-focused methods [32] which only allow keyword based queries, we aim to enable open-ended", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "natural language queries for users to customize video summaries. For example, as seen in Fig. 1,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "using our method users can specify long and descriptive queries such as, “All water bodies such as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 403, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 403, + 412 + ], + "score": 1.0, + "content": "lakes, rivers, and waterfalls” which is not possible with previous methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Given an input video, CLIP-It generates a video summary guided by either a user-defined natural", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "language query or a system generated description. It uses a Language-Guided Attention head", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 436, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 506, + 453 + ], + "score": 1.0, + "content": "to compute a joint representation of the image and language embeddings, and a Frame-Scoring", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "Transformer to assign scores to individual frames in the video using the fused representations.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 459, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 507, + 474 + ], + "score": 1.0, + "content": "Following [43, 42], the summary video is constructed from high scoring frames by converting frame-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "level scores to shot-level scores and using knapsack algorithm to fit the maximum number of high", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "scoring shots in a timed window. Fig. 1 shows an output from our method. Given an hour long video", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "of a national park tour, we generate a 2 minute generic summary comprising of all the important", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "scenes in the video. Given the two language queries, our method picks the matching keyframes in both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "cases. We can train CLIP-It without ground-truth supervision by leveraging the reconstruction and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "diversity losses [9, 31]. For generic video summarization, we evaluate our approach on the standard", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 479, + 549 + ], + "score": 1.0, + "content": "benchmarks, TVSum and SumMe. We achieve performance improvement on F1 score of nearly", + "type": "text" + }, + { + "bbox": [ + 480, + 537, + 494, + 547 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 547, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 209, + 560 + ], + "score": 1.0, + "content": "the supervised setting and", + "type": "text" + }, + { + "bbox": [ + 210, + 547, + 225, + 558 + ], + "score": 0.86, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 547, + 484, + 560 + ], + "score": 1.0, + "content": "in the unsupervised setting on both datasets. We show large gains", + "type": "text" + }, + { + "bbox": [ + 484, + 547, + 504, + 559 + ], + "score": 0.8, + "content": "( 5 \\% )", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 556, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 507, + 572 + ], + "score": 1.0, + "content": "in the Transfer setting, where CLIP-It is trained and evaluated on disjoint sets of data. For the query-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 570, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 581 + ], + "score": 1.0, + "content": "focused scenario we evaluate on the QFVS dataset [33], where we also achieve state-of-the-art results.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "To summarize our contributions, we introduce CLIP-It, a language-guided model that unifies generic", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "and query-focused video summarization. Our approach uses language conditioning in the form", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "of off-the-shelf video descriptions (for generic summarization) or user-defined natural language", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "queries (for query-focused summarization). We show that the Transformer design enables effective", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "contextualization across frames, benefiting our tasks. We also demonstrate the impact of language", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "guidance on generic summarization. Finally, we establish the new state-of-the-art on both generic", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 651, + 380, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 380, + 665 + ], + "score": 1.0, + "content": "and query-focused datasets in supervised and unsupervised settings.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48 + }, + { + "type": "title", + "bbox": [ + 107, + 671, + 197, + 684 + ], + "lines": [ + { + "bbox": [ + 104, + 669, + 198, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 198, + 686 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "Generic Video Summarization. A video summary is a short synopsis created by stitching together", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "important clips from the original video. Early works [21, 35, 34] referred to it as Video Skimming or", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "Dynamic Video Summarization and used hand-designed features to generate summaries. Likewise,", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 505, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "Generic video summarization datasets such as TVSum [36] and SumMe [7] provide ground-truth", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "annotations in the form of frame or shot-level importance scores specified by multiple annotators.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "Several learning-based methods reduce the task to a frame-wise score prediction problem. Sequence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "labeling methods [5, 42, 43, 44, 18] model variable-range dependencies between frames but fail to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "capture relationships across all frames simultaneously. While attention [3] and graph based [26]", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "methods address this partly, they disregard ordering of the input frames which is useful when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "predicting scores. Moreover, these methods use only visual cues to produce the summaries and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "cannot be customized with a natural language input. Another line of work, query-focused video", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "summarization [32], allows the users to customize the summary by specifying a query containing two", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "concepts (eg., food and drinks). However, in their work the query can only be chosen from a fixed set", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 357, + 243 + ], + "score": 1.0, + "content": "of predefined concepts which limits the flexibility for the user.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 122, + 506, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "It is important to note that efforts in generic and query-focused video summarization have so far", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "been disjoint, with no single method for both. Our key innovation is to unify these tasks in one", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "language-guided framework. We introduce CLIP-It, a multimodal summarization model which takes", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "two inputs, a video and a natural language text, and synthesizes a summary video conditioned on the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "text. In case of generic video summarization, the natural language text is a video description obtained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "using an off-the-shelf dense video captioning method. Alternatively, in the case of query-focused", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "video summarization, the language input is a user-defined query. Unlike existing generic methods", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "which only use visual cues, we show that adding language as an input leads to the “discovery” of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "relevant concepts and actions resulting in better summaries. Our method uses a Transformer with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 346, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 357 + ], + "score": 1.0, + "content": "positional encoding, which can attend to all frames at once (unlike LSTM based methods) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "also keep track of the ordering of frames (unlike graph based methods). In contrast to existing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "query-focused methods [32] which only allow keyword based queries, we aim to enable open-ended", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "natural language queries for users to customize video summaries. For example, as seen in Fig. 1,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "using our method users can specify long and descriptive queries such as, “All water bodies such as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 403, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 403, + 412 + ], + "score": 1.0, + "content": "lakes, rivers, and waterfalls” which is not possible with previous methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 247, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "Given an input video, CLIP-It generates a video summary guided by either a user-defined natural", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "language query or a system generated description. It uses a Language-Guided Attention head", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 436, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 506, + 453 + ], + "score": 1.0, + "content": "to compute a joint representation of the image and language embeddings, and a Frame-Scoring", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "Transformer to assign scores to individual frames in the video using the fused representations.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 459, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 507, + 474 + ], + "score": 1.0, + "content": "Following [43, 42], the summary video is constructed from high scoring frames by converting frame-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "level scores to shot-level scores and using knapsack algorithm to fit the maximum number of high", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "scoring shots in a timed window. Fig. 1 shows an output from our method. Given an hour long video", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "of a national park tour, we generate a 2 minute generic summary comprising of all the important", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "scenes in the video. Given the two language queries, our method picks the matching keyframes in both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "cases. We can train CLIP-It without ground-truth supervision by leveraging the reconstruction and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "diversity losses [9, 31]. For generic video summarization, we evaluate our approach on the standard", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 479, + 549 + ], + "score": 1.0, + "content": "benchmarks, TVSum and SumMe. We achieve performance improvement on F1 score of nearly", + "type": "text" + }, + { + "bbox": [ + 480, + 537, + 494, + 547 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 547, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 209, + 560 + ], + "score": 1.0, + "content": "the supervised setting and", + "type": "text" + }, + { + "bbox": [ + 210, + 547, + 225, + 558 + ], + "score": 0.86, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 547, + 484, + 560 + ], + "score": 1.0, + "content": "in the unsupervised setting on both datasets. We show large gains", + "type": "text" + }, + { + "bbox": [ + 484, + 547, + 504, + 559 + ], + "score": 0.8, + "content": "( 5 \\% )", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 556, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 507, + 572 + ], + "score": 1.0, + "content": "in the Transfer setting, where CLIP-It is trained and evaluated on disjoint sets of data. For the query-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 570, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 581 + ], + "score": 1.0, + "content": "focused scenario we evaluate on the QFVS dataset [33], where we also achieve state-of-the-art results.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 416, + 507, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "To summarize our contributions, we introduce CLIP-It, a language-guided model that unifies generic", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "and query-focused video summarization. Our approach uses language conditioning in the form", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "of off-the-shelf video descriptions (for generic summarization) or user-defined natural language", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "queries (for query-focused summarization). We show that the Transformer design enables effective", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "contextualization across frames, benefiting our tasks. We also demonstrate the impact of language", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "guidance on generic summarization. Finally, we establish the new state-of-the-art on both generic", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 651, + 380, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 380, + 665 + ], + "score": 1.0, + "content": "and query-focused datasets in supervised and unsupervised settings.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 586, + 506, + 665 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 671, + 197, + 684 + ], + "lines": [ + { + "bbox": [ + 104, + 669, + 198, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 198, + 686 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "Generic Video Summarization. A video summary is a short synopsis created by stitching together", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "important clips from the original video. Early works [21, 35, 34] referred to it as Video Skimming or", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "Dynamic Video Summarization and used hand-designed features to generate summaries. Likewise,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "non-parametric unsupervised methods [13, 16, 23, 19, 20, 28] used various heuristics to represent", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "the importance of frames. Introduction of benchmark datasets such as TVSum [36] and SumMe [7]", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "provided relevance scores for frames in videos annotated by users, resulting in multiple human", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "generated summaries. This enabled automatic evaluation of video summarization techniques and", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "has lead to the development of many supervised learning based methods [6, 8, 22, 26, 31, 30, 41, 42,", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "43, 44, 45]. These approaches capture high-level semantic information and outperform the heuristic", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "unsupervised methods. Fully convolutional sequence networks [31] treat video summarization", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "as a binary label prediction problem. Determinantal point processes [15] and LSTM [10] based", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "approaches [5, 18, 42, 43, 44] model variable-range dependencies between frames. However, these", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "are sequential and fail to capture relationships across all frames simultaneously. Attention [3] and", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "graph based methods [26] address this issue by modeling relationships across all frames, but they", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 192, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 504, + 204 + ], + "score": 1.0, + "content": "disregard the ordering of frames in a video, which is also useful for summarization. Our method uses", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "a Transformer [38] with positional encoding, which allows for joint attention across all frames while", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 215, + 335, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 335, + 227 + ], + "score": 1.0, + "content": "maintaining an ordering of the input sequence of frames.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 54, + "bbox_fs": [ + 105, + 687, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "non-parametric unsupervised methods [13, 16, 23, 19, 20, 28] used various heuristics to represent", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "the importance of frames. Introduction of benchmark datasets such as TVSum [36] and SumMe [7]", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "provided relevance scores for frames in videos annotated by users, resulting in multiple human", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "generated summaries. This enabled automatic evaluation of video summarization techniques and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "has lead to the development of many supervised learning based methods [6, 8, 22, 26, 31, 30, 41, 42,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "43, 44, 45]. These approaches capture high-level semantic information and outperform the heuristic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "unsupervised methods. Fully convolutional sequence networks [31] treat video summarization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "as a binary label prediction problem. Determinantal point processes [15] and LSTM [10] based", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "approaches [5, 18, 42, 43, 44] model variable-range dependencies between frames. However, these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "are sequential and fail to capture relationships across all frames simultaneously. Attention [3] and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "graph based methods [26] address this issue by modeling relationships across all frames, but they", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 192, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 504, + 204 + ], + "score": 1.0, + "content": "disregard the ordering of frames in a video, which is also useful for summarization. Our method uses", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "a Transformer [38] with positional encoding, which allows for joint attention across all frames while", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 215, + 335, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 335, + 227 + ], + "score": 1.0, + "content": "maintaining an ordering of the input sequence of frames.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "Some of the above works have supervised and unsupervised variants with modifications to the objec-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "tive function. Specifically, for the unsupervised variant, they use reconstruction and diversity losses", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "which do not require ground truth. We follow prior work in terms of using the same loss functions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "Other notable unsupervised approaches include an adversarial LSTM based method [22], a generative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "adversarial network to learn from unpaired data [30], and a cycle consistent learning objective [41].", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 304 + ], + "score": 1.0, + "content": "Video-Text Summarization. Plummer et al. [27] use image-language embeddings for generating", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "video summaries but evaluate their approach in the text domain, and not on the generic video", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "score": 1.0, + "content": "summarization benchmarks. Furthermore, they require text to be provided as input and don’t have a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 325, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 335 + ], + "score": 1.0, + "content": "mechanism to generate captions if absent. On the other hand, our method works well with off-the-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "shelf captions and we evaluate on both generic and query-focused benchmark datasets. Chen et al. [1]", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 482, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 482, + 357 + ], + "score": 1.0, + "content": "jointly train a text and video summarization network but rely on ground-truth text summaries.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "Query-Focused Video Summarization. Oftentimes users browsing videos on YouTube are looking", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "for something specific so a generic summary might not suffice. In this case, there should be an option", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "to customize the generated summary using a natural language query. Sharghi et al. 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Other works [12, 39] similarly condition the summary", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 483, + 381, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 381, + 494 + ], + "score": 1.0, + "content": "generation on keyword based queries but haven’t released their data.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "Transformers. Transformers [38] were introduced for neural machine translation and have since", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "been applied to video-language tasks such as video-retrieval [4] and video captioning [17, 48]. In this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "work we adapt transformers for video summarization. We modify self-attention [38] to a Language-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "Guided Attention block that accepts inputs from two modalities. Additionally, our method also relies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "on CLIP [29] for extracting image and text features and a Bi-Modal Transformer [11] for dense video", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 552, + 375, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 375, + 565 + ], + "score": 1.0, + "content": "caption generation, both of which also have transformer backbone.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 107, + 572, + 381, + 586 + ], + "lines": [ + { + "bbox": [ + 104, + 571, + 383, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 383, + 589 + ], + "score": 1.0, + "content": "3 CLIP-It: Language-Guided Video Summarization", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "CLIP-It is a unified language-guided framework for both generic and query-focused video summa-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "rization. It takes an input video and a user-defined query or a system generated dense video caption", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "and constructs a summary by selecting key frames from the video. First, we explain the intuition", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "behind our approach. In the case of query-focused summarization, clearly, it is necessary to model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "score": 1.0, + "content": "the user query as input in order to produce an appropriate summary. 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The dog runs after it.), we could leverage its semantic embedding to match", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 114, + 699 + ], + "score": 0.43, + "content": "i t", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "to the most relevant portions of the video. Such a description could be obtained automatically", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "by generating dense video captions [11]. We first present an overview of our approach, CLIP-It,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 365, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 365, + 723 + ], + "score": 1.0, + "content": "followed by a detailed description of the individual components.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 226 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 104, + 72, + 506, + 227 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "Some of the above works have supervised and unsupervised variants with modifications to the objec-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "tive function. Specifically, for the unsupervised variant, they use reconstruction and diversity losses", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "which do not require ground truth. We follow prior work in terms of using the same loss functions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "Other notable unsupervised approaches include an adversarial LSTM based method [22], a generative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 288 + ], + "score": 1.0, + "content": "adversarial network to learn from unpaired data [30], and a cycle consistent learning objective [41].", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 231, + 506, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 304 + ], + "score": 1.0, + "content": "Video-Text Summarization. Plummer et al. [27] use image-language embeddings for generating", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "video summaries but evaluate their approach in the text domain, and not on the generic video", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "score": 1.0, + "content": "summarization benchmarks. Furthermore, they require text to be provided as input and don’t have a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 325, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 335 + ], + "score": 1.0, + "content": "mechanism to generate captions if absent. 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Additionally, our method also relies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "on CLIP [29] for extracting image and text features and a Bi-Modal Transformer [11] for dense video", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 552, + 375, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 375, + 565 + ], + "score": 1.0, + "content": "caption generation, both of which also have transformer backbone.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 497, + 506, + 565 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 572, + 381, + 586 + ], + "lines": [ + { + "bbox": [ + 104, + 571, + 383, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 383, + 589 + ], + "score": 1.0, + "content": "3 CLIP-It: Language-Guided Video Summarization", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "CLIP-It is a unified language-guided framework for both generic and query-focused video summa-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "rization. It takes an input video and a user-defined query or a system generated dense video caption", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "and constructs a summary by selecting key frames from the video. First, we explain the intuition", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "behind our approach. In the case of query-focused summarization, clearly, it is necessary to model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "score": 1.0, + "content": "the user query as input in order to produce an appropriate summary. In the case of generic video sum-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "marization no user query is available; nonetheless, we show here that we can leverage the semantic", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "information contained in associated natural language descriptions to guide the summarization process.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Assuming we had a generic description that accompanies a video (e.g., A person is walking a dog.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "The person throws a ball. The dog runs after it.), we could leverage its semantic embedding to match", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 114, + 699 + ], + "score": 0.43, + "content": "i t", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "to the most relevant portions of the video. Such a description could be obtained automatically", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "by generating dense video captions [11]. We first present an overview of our approach, CLIP-It,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 365, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 365, + 723 + ], + "score": 1.0, + "content": "followed by a detailed description of the individual components.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 590, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 72, + 494, + 250 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 72, + 494, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 72, + 494, + 250 + ], + "spans": [ + { + "bbox": [ + 109, + 72, + 494, + 250 + ], + "score": 0.975, + "type": "image", + "image_path": "5cf0ec307f855e688d0cd7ddc67c06c5416e4e5a59a1eb7b6e6c867ced503bf9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 72, + 494, + 131.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 131.33333333333334, + 494, + 190.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 190.66666666666669, + 494, + 250.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 256, + 505, + 307 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "score": 1.0, + "content": "Figure 2: Overview of CLIP-It. Given an input video, CLIP-It generates a summary conditioned on either a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "user-defined natural language query or an automatically generated dense video caption. The Language-Guided", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "Attention head fuses the image and language embeddings and the Frame-Scoring Transformer jointly attends to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "all frames to predict their relevance scores. During inference, the video summary is constructed by converting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 296, + 419, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 419, + 308 + ], + "score": 1.0, + "content": "frame scores to shot scores and using Knapsack algorithm to select high scoring shots.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 506, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 300, + 333 + ], + "score": 1.0, + "content": "Overview. Our approach, CLIP-It, is outlined in", + "type": "text" + }, + { + "bbox": [ + 301, + 321, + 323, + 332 + ], + "score": 0.33, + "content": "\\mathrm { F i g } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 320, + 430, + 333 + ], + "score": 1.0, + "content": ". Given a video, we extract", + "type": "text" + }, + { + "bbox": [ + 430, + 321, + 441, + 331 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "frames denoted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 118, + 344 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 332, + 129, + 343 + ], + "score": 0.79, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 331, + 133, + 344 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 187, + 343 + ], + "score": 0.91, + "content": "i \\in [ 1 , \\dots \\bar { N } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 331, + 505, + 344 + ], + "score": 1.0, + "content": ". We formulate the video summarization task as a per-frame binary classification", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 342, + 356 + ], + "score": 1.0, + "content": "problem. We embed the frames using a pretrained network", + "type": "text" + }, + { + "bbox": [ + 342, + 343, + 363, + 355 + ], + "score": 0.91, + "content": "f _ { i m g }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 343, + 506, + 356 + ], + "score": 1.0, + "content": ". If a query is provided (in the form", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 426, + 366 + ], + "score": 1.0, + "content": "of a natural language string), we embed the query using a pretrained network", + "type": "text" + }, + { + "bbox": [ + 426, + 354, + 443, + 365 + ], + "score": 0.9, + "content": "f _ { t x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 353, + 506, + 366 + ], + "score": 1.0, + "content": ". Alternatively,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "as seen in the figure, we use an off-the-shelf video captioning model to generate a dense video", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 162, + 388 + ], + "score": 1.0, + "content": "caption with", + "type": "text" + }, + { + "bbox": [ + 163, + 375, + 174, + 385 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 375, + 270, + 388 + ], + "score": 1.0, + "content": "sentences denoted by", + "type": "text" + }, + { + "bbox": [ + 270, + 375, + 282, + 388 + ], + "score": 0.77, + "content": "C _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 375, + 288, + 388 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 288, + 375, + 351, + 387 + ], + "score": 0.9, + "content": "j \\in [ 1 , \\bar { \\dots { M } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "and embed each sentence using the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 185, + 399 + ], + "score": 1.0, + "content": "pretrained network", + "type": "text" + }, + { + "bbox": [ + 185, + 387, + 202, + 398 + ], + "score": 0.91, + "content": "f _ { t x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 386, + 436, + 399 + ], + "score": 1.0, + "content": ". Next, we compute language attended image embeddings", + "type": "text" + }, + { + "bbox": [ + 436, + 387, + 447, + 397 + ], + "score": 0.85, + "content": "I ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "using learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 396, + 507, + 413 + ], + "spans": [ + { + "bbox": [ + 104, + 396, + 272, + 413 + ], + "score": 1.0, + "content": "Language-Guided Multi-head Attention", + "type": "text" + }, + { + "bbox": [ + 272, + 398, + 306, + 410 + ], + "score": 0.92, + "content": "f _ { i m g \\_ t x t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 396, + 507, + 413 + ], + "score": 1.0, + "content": ". Finally, we train a Frame-Scoring Transformer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "score": 1.0, + "content": "which assigns scores to each frame in the video (green indicating a high score and red indicating", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "a low score). To construct the video summary during inference, we convert frame-level scores to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "shot-level scores and finally, use 0/1 knapsack algorithm to pick the key shots [43]. In the following,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "we describe the Language-Guided Attention and the Frame-Selection Transformer modules, followed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 452, + 300, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 300, + 465 + ], + "score": 1.0, + "content": "by discussing the visual and language encoders.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 468, + 504, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Language-Guided Attention. We use Language-Guided Multi-head Attention to efficiently fuse", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "information across the video and language modalities and to infer long-term dependencies across", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "both. Using a single attention head does not suffice as our goal is to allow all captions to attend to all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "frames in the video. We modify the Multi-Head Attention described in Vaswani et al. [38] to take in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 511, + 423, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 279, + 525 + ], + "score": 1.0, + "content": "inputs from both modalities. We set Query", + "type": "text" + }, + { + "bbox": [ + 279, + 512, + 288, + 523 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 511, + 310, + 525 + ], + "score": 1.0, + "content": ", Key", + "type": "text" + }, + { + "bbox": [ + 311, + 512, + 321, + 522 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 511, + 366, + 525 + ], + "score": 1.0, + "content": ", and Value", + "type": "text" + }, + { + "bbox": [ + 367, + 512, + 376, + 522 + ], + "score": 0.77, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 511, + 423, + 525 + ], + "score": 1.0, + "content": "as follows,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 550, + 456, + 618 + ], + "lines": [ + { + "bbox": [ + 156, + 550, + 456, + 618 + ], + "spans": [ + { + "bbox": [ + 156, + 550, + 456, + 618 + ], + "score": 0.56, + "content": "\\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}", + "type": "interline_equation", + "image_path": "cb199205e7555bb1154bf2fb4a6e5141233c44b2977fd77e0a7e8f33b9c60c16.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 156, + 550, + 456, + 572.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 156, + 572.6666666666666, + 456, + 595.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 156, + 595.3333333333333, + 456, + 617.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 506, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 125, + 637 + ], + "score": 0.88, + "content": "\\it { W _ { i } ^ { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 621, + 129, + 640 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 129, + 623, + 149, + 637 + ], + "score": 0.87, + "content": "W _ { i } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 621, + 170, + 640 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 170, + 624, + 189, + 637 + ], + "score": 0.92, + "content": "W _ { i } ^ { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 621, + 335, + 640 + ], + "score": 1.0, + "content": "are learned parameter matrices and", + "type": "text" + }, + { + "bbox": [ + 335, + 626, + 347, + 636 + ], + "score": 0.88, + "content": "d _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 621, + 432, + 640 + ], + "score": 1.0, + "content": "is the dimensions of", + "type": "text" + }, + { + "bbox": [ + 433, + 626, + 442, + 635 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 621, + 506, + 640 + ], + "score": 1.0, + "content": ". The output of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 636, + 491, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 475, + 651 + ], + "score": 1.0, + "content": "the Language-Guided Multi-Head Attention are the attended image embeddings, denoted as", + "type": "text" + }, + { + "bbox": [ + 475, + 636, + 487, + 650 + ], + "score": 0.9, + "content": "F _ { i } ^ { ' }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 636, + 491, + 651 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "Frame-Scoring Transformer. Finally, it is also important to ensure that we do not include redundant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "information, e.g., several key shots from the same event. To better model interactions across frames", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "and contextualize them w.r.t. each other, we add a Frame-Scoring Transformer that takes image-text", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "representations as input and outputs one score per frame. Based on the Transformer model [38], this", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 374, + 712 + ], + "score": 1.0, + "content": "module assigns relevance scores to the attended image embeddings", + "type": "text" + }, + { + "bbox": [ + 375, + 699, + 386, + 712 + ], + "score": 0.9, + "content": "F _ { i } ^ { ' }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 699, + 425, + 712 + ], + "score": 1.0, + "content": ". 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Given an input video, CLIP-It generates a summary conditioned on either a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "user-defined natural language query or an automatically generated dense video caption. The Language-Guided", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "Attention head fuses the image and language embeddings and the Frame-Scoring Transformer jointly attends to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "all frames to predict their relevance scores. 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Given a video, we extract", + "type": "text" + }, + { + "bbox": [ + 430, + 321, + 441, + 331 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "frames denoted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 118, + 344 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 332, + 129, + 343 + ], + "score": 0.79, + "content": "F _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 331, + 133, + 344 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 187, + 343 + ], + "score": 0.91, + "content": "i \\in [ 1 , \\dots \\bar { N } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 331, + 505, + 344 + ], + "score": 1.0, + "content": ". We formulate the video summarization task as a per-frame binary classification", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 342, + 356 + ], + "score": 1.0, + "content": "problem. We embed the frames using a pretrained network", + "type": "text" + }, + { + "bbox": [ + 342, + 343, + 363, + 355 + ], + "score": 0.91, + "content": "f _ { i m g }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 343, + 506, + 356 + ], + "score": 1.0, + "content": ". If a query is provided (in the form", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 426, + 366 + ], + "score": 1.0, + "content": "of a natural language string), we embed the query using a pretrained network", + "type": "text" + }, + { + "bbox": [ + 426, + 354, + 443, + 365 + ], + "score": 0.9, + "content": "f _ { t x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 353, + 506, + 366 + ], + "score": 1.0, + "content": ". 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In the following,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "we describe the Language-Guided Attention and the Frame-Selection Transformer modules, followed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 452, + 300, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 300, + 465 + ], + "score": 1.0, + "content": "by discussing the visual and language encoders.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 320, + 507, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 468, + 504, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Language-Guided Attention. We use Language-Guided Multi-head Attention to efficiently fuse", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "information across the video and language modalities and to infer long-term dependencies across", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "both. Using a single attention head does not suffice as our goal is to allow all captions to attend to all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "frames in the video. We modify the Multi-Head Attention described in Vaswani et al. 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Based on the Transformer model [38], this", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 374, + 712 + ], + "score": 1.0, + "content": "module assigns relevance scores to the attended image embeddings", + "type": "text" + }, + { + "bbox": [ + 375, + 699, + 386, + 712 + ], + "score": 0.9, + "content": "F _ { i } ^ { ' }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 699, + 425, + 712 + ], + "score": 1.0, + "content": ". 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We experiment with the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 124 + ], + "score": 1.0, + "content": "following networks: GoogleNet [37] (for a fair comparison to prior work), ResNet [8], and CLIP [29].", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 127, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "Text Encoding. We encode the user-defined query or the system generated dense caption using a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 187, + 151 + ], + "score": 1.0, + "content": "pre-trained network", + "type": "text" + }, + { + "bbox": [ + 187, + 138, + 204, + 150 + ], + "score": 0.92, + "content": "f _ { t x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 138, + 506, + 151 + ], + "score": 1.0, + "content": ". In this case we tried the CLIP (ViT and RN101) model. In case of generic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "video summarization, we generate dense video captions for the whole video in order to condition", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "on language and incorporate added semantics into our video summarization pipeline. We use the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 184 + ], + "score": 1.0, + "content": "Bi-Modal Transformer [11] dense video captioning model that generates a multi-sentence description", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "given a video with audio. Since there are multiple sentences in the dense video caption, we first", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 352, + 206 + ], + "score": 1.0, + "content": "embed each sentence of the caption using the text encoder", + "type": "text" + }, + { + "bbox": [ + 352, + 193, + 369, + 204 + ], + "score": 0.9, + "content": "f _ { t x t }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 192, + 506, + 206 + ], + "score": 1.0, + "content": "described above. They are then", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 204, + 357, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 357, + 216 + ], + "score": 1.0, + "content": "concatenated and fused using a multi-layer perceptron (MLP).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 221, + 171, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 172, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 172, + 237 + ], + "score": 1.0, + "content": "3.1 Learning", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 506, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "We employ 3 loss functions (classification, diversity, and reconstruction) to train our model. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "supervised setting uses all 3 and the unsupervised setting uses only diversity and reconstruction", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 137, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 137, + 270 + ], + "score": 1.0, + "content": "losses.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 273, + 474, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 475, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 475, + 288 + ], + "score": 1.0, + "content": "Classification loss. We use a weighted binary cross entropy loss for classifying each frame:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 291, + 435, + 326 + ], + "lines": [ + { + "bbox": [ + 175, + 291, + 435, + 326 + ], + "spans": [ + { + "bbox": [ + 175, + 291, + 435, + 326 + ], + "score": 0.95, + "content": "\\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 } ) ] ,", + "type": "interline_equation", + "image_path": "97e15b9ce3a49ab5a72cb5809875e3e18ddca3b297d27441b6b535f6d5eafee2.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 175, + 291, + 435, + 302.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 175, + 302.6666666666667, + 435, + 314.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 175, + 314.33333333333337, + 435, + 326.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 134, + 348 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 337, + 146, + 348 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 335, + 279, + 348 + ], + "score": 1.0, + "content": "is the ground-truth label of the", + "type": "text" + }, + { + "bbox": [ + 279, + 337, + 284, + 346 + ], + "score": 0.79, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 335, + 342, + 348 + ], + "score": 1.0, + "content": "-th frame and", + "type": "text" + }, + { + "bbox": [ + 342, + 336, + 352, + 346 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "is the total number of frames in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 134, + 344, + 507, + 377 + ], + "spans": [ + { + "bbox": [ + 134, + 349, + 147, + 359 + ], + "score": 0.86, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 344, + 183, + 377 + ], + "score": 1.0, + "content": "weif", + "type": "text" + }, + { + "bbox": [ + 195, + 344, + 285, + 377 + ], + "score": 1.0, + "content": "ht assigned to the class is a background frame", + "type": "text" + }, + { + "bbox": [ + 285, + 349, + 296, + 361 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 344, + 360, + 377 + ], + "score": 1.0, + "content": ", which is set to", + "type": "text" + }, + { + "bbox": [ + 361, + 347, + 410, + 362 + ], + "score": 0.92, + "content": "\\frac { \\# k e y f r a m e s } { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 344, + 420, + 377 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 420, + 349, + 431, + 361 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 344, + 507, + 377 + ], + "score": 1.0, + "content": "is a keyframe and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 361, + 195, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 173, + 376 + ], + "score": 0.86, + "content": "1 - { \\frac { \\# k e y f r a m e s } { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 362, + 195, + 375 + ], + "score": 0.88, + "content": "\\boldsymbol { x } _ { i } ^ { * }", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "For the purpose of training without any supervision, we employ two additional losses that enforce", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 351, + 402 + ], + "score": 1.0, + "content": "diversity in the selected keyframes. 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The reconstruction loss", + "type": "text" + }, + { + "bbox": [ + 295, + 450, + 307, + 461 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 450, + 504, + 462 + ], + "score": 1.0, + "content": "is defined as the mean squared error between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 502, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 502, + 473 + ], + "score": 1.0, + "content": "reconstructed features and the original features corresponding to the selected keyframes, such that:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 477, + 358, + 507 + ], + "lines": [ + { + "bbox": [ + 252, + 477, + 358, + 507 + ], + "spans": [ + { + "bbox": [ + 252, + 477, + 358, + 507 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { r } = \\frac { 1 } { X } \\sum _ { i \\in X } | | \\mathbf { x } _ { i } - \\hat { \\mathbf { y } } _ { i } | | _ { 2 } ,", + "type": "interline_equation", + "image_path": "cfa3e99f4ed6b7899a92854fede6a172f1b755f8d7055df221b74adb4bf92716.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 477, + 358, + 492.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 252, + 492.0, + 358, + 507.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 280, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 282, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 132, + 525 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 513, + 141, + 524 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 512, + 282, + 525 + ], + "score": 1.0, + "content": "denotes the reconstructed features.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "Diversity Loss. 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The reconstruction loss", + "type": "text" + }, + { + "bbox": [ + 295, + 450, + 307, + 461 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 450, + 504, + 462 + ], + "score": 1.0, + "content": "is defined as the mean squared error between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 502, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 502, + 473 + ], + "score": 1.0, + "content": "reconstructed features and the original features corresponding to the selected keyframes, such that:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 450, + 504, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 477, + 358, + 507 + ], + "lines": [ + { + "bbox": [ + 252, + 477, + 358, + 507 + ], + "spans": [ + { + "bbox": [ + 252, + 477, + 358, + 507 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { r } = \\frac { 1 } { X } \\sum _ { i \\in X } | | \\mathbf { x } _ { i } - \\hat { \\mathbf { y } } _ { i } | | _ { 2 } ,", + "type": "interline_equation", + "image_path": "cfa3e99f4ed6b7899a92854fede6a172f1b755f8d7055df221b74adb4bf92716.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 477, + 358, + 492.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 252, + 492.0, + 358, + 507.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 280, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 282, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 132, + 525 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 513, + 141, + 524 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 512, + 282, + 525 + ], + "score": 1.0, + "content": "denotes the reconstructed features.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 512, + 282, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 504, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "Diversity Loss. 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Comparing F1 Scores of our methods with supervised baselines on the SumMe [7] and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 401, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 401, + 92 + ], + "score": 1.0, + "content": "TVSum [36] datasets using Standard, Augment, and Transfer data configurations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 93, + 507, + 336 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 93, + 507, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 336 + ], + "score": 0.985, + "html": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Zhang et al. (SumTransfer) [42]40.941.338.5-=1
Zhang et al. (LSTM) [43]38.642.941.854.759.658.7
Mahasseni et al. (SUM-GANsup) [22]41.743.6-56.361.2=
Rochan et al. (SUM-FCN) [31]47.551.144.156.859.258.2
Rochan et al. (SUM-DeepLab) [31]48.850.245.058.459.157.4
Zhou et al. [47]42.143.942.658.159.858.9
Zhang et al. [44]=44.91=63.91
Fajtl et al. [3]49.751.1-61.462.41
Rochan et al. [30]148.041.656.155.7
Chen et al. (V2TS) [1]-1162.111
He et al. [9]47.2--59.4--
Park et al. (SumGraph) [26]51.452.948.763.965.860.5
GoogleNet+bi-LSTM38.542.440.753.959.658.6
ResNet+bi-LSTM39.444.042.655.061.059.9
CLIP-Image+bi-LSTM41.145.944.956.863.761.6
CLIP-Image+Video Caption+bi-LSTM41.246.145.557.164.362.4
GoogleNet+Transformer51.653.549.464.266.361.3
ResNet+Transformer52.854.950.365.067.562.8
CLIP-Image+Transformer53.555.351.065.568.163.4
CLIP-It: CLIP-Image+Video Caption+Transformer54.256.451.966.369.065.5
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Note that while prior works only use visual cues from the video, our method also allows", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "for video captions as an input feature. For a fair comparison, we include ablations of our method that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 443, + 445, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 445, + 456 + ], + "score": 1.0, + "content": "do not use any language cues and only the visual features used in earlier works [43].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "Datasets. We evaluate our approach on two standard video summarization datasets (TVSum [36]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "and SumMe [7]) and on the generic summaries for UT Egocentric videos [16] provided by the QFVS", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 480, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 507, + 495 + ], + "score": 1.0, + "content": "dataset [33]. TVSum [36] consists of 50 videos pertaining to 10 categories (how to videos, news,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "documentary, etc) with 5 videos from each category, typically 1-5 minutes in length. 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In addition to training on each dataset independently, we follow", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "prior work and augment training data with 39 videos from the YouTube dataset [2] and 50 videos", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "from the Open Video Project (OVP) dataset [24]. YouTube dataset consists of news, sports and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "cartoon videos. OVP dataset consists of multiple different genres including documentary videos.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "These datasets are diverse in nature and come with different types of annotations, frame-level scores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 583 + ], + "score": 1.0, + "content": "for TVSum and shot-level scores for SumMe. They are integrated to create the ground-truth using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "the procedure in [43]. The UT Egocentric dataset consists of 4 videos captured from head-mounted", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "cameras. Each video is about 3-5 hours long, captured in a natural, uncontrolled setting and contains", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "a diverse set of events. The QFVS dataset [33] provides ground-truth generic summaries for these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "4 videos. The summaries were constructed by dividing the video into shots and asking 3 users to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 623, + 470, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 470, + 635 + ], + "score": 1.0, + "content": "select the relevant shots. The final ground-truth is an average of annotations from all users.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 108, + 640, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 653 + ], + "score": 1.0, + "content": "Note. All the datasets - YouTube [2], Open Video Project (OVP) dataset [24], TVSum [36],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "SumMe [7], and QFVS [32] were collected by the creators (cited) and consent for any person-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 661, + 437, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 437, + 676 + ], + "score": 1.0, + "content": "ally identifiable information (PII) was ascertained by the authors where necessary.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "Data configuration for TVSum and SumMe. Following previous works [43, 42, 31, 26], we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "evaluate our approach in three different data settings: Standard, Augment, and Transfer. In the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "Standard setting, the training and test splits are from the same dataset (i.e. either TVSum or SumMe).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 710, + 389, + 724 + ], + "score": 1.0, + "content": "For SumMe we use available splits, and for TVSum we randomly select", + "type": "text" + }, + { + "bbox": [ + 389, + 711, + 409, + 721 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "of the videos for testing", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 93, + 507, + 336 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 71, + 504, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 71, + 505, + 82 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 505, + 82 + ], + "score": 1.0, + "content": "Table 1: Supervised. Comparing F1 Scores of our methods with supervised baselines on the SumMe [7] and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 401, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 401, + 92 + ], + "score": 1.0, + "content": "TVSum [36] datasets using Standard, Augment, and Transfer data configurations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 93, + 507, + 336 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 93, + 507, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 336 + ], + "score": 0.985, + "html": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Zhang et al. (SumTransfer) [42]40.941.338.5-=1
Zhang et al. (LSTM) [43]38.642.941.854.759.658.7
Mahasseni et al. (SUM-GANsup) [22]41.743.6-56.361.2=
Rochan et al. (SUM-FCN) [31]47.551.144.156.859.258.2
Rochan et al. (SUM-DeepLab) [31]48.850.245.058.459.157.4
Zhou et al. [47]42.143.942.658.159.858.9
Zhang et al. [44]=44.91=63.91
Fajtl et al. [3]49.751.1-61.462.41
Rochan et al. [30]148.041.656.155.7
Chen et al. (V2TS) [1]-1162.111
He et al. [9]47.2--59.4--
Park et al. (SumGraph) [26]51.452.948.763.965.860.5
GoogleNet+bi-LSTM38.542.440.753.959.658.6
ResNet+bi-LSTM39.444.042.655.061.059.9
CLIP-Image+bi-LSTM41.145.944.956.863.761.6
CLIP-Image+Video Caption+bi-LSTM41.246.145.557.164.362.4
GoogleNet+Transformer51.653.549.464.266.361.3
ResNet+Transformer52.854.950.365.067.562.8
CLIP-Image+Transformer53.555.351.065.568.163.4
CLIP-It: CLIP-Image+Video Caption+Transformer54.256.451.966.369.065.5
", + "type": "table", + "image_path": "0a457e51185718c2ccebd26e980cbf0b4be695ca09bbc09811abbe800366204f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 105, + 93, + 507, + 174.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 105, + 174.0, + 507, + 255.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 105, + 255.0, + 507, + 336.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "title", + "bbox": [ + 107, + 349, + 192, + 363 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 193, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 193, + 366 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 504, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "In this section, we describe the experimental setup and evaluation of our method on two tasks: generic", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 378, + 357, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 357, + 391 + ], + "score": 1.0, + "content": "video summarization and query-focused video summarization.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 367, + 505, + 391 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 396, + 259, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 261, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 261, + 408 + ], + "score": 1.0, + "content": "4.1 Generic Video Summarization", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "Generic video summarization involves generating a single general-purpose summary to describe the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 423, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 433 + ], + "score": 1.0, + "content": "input video. Note that while prior works only use visual cues from the video, our method also allows", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "for video captions as an input feature. For a fair comparison, we include ablations of our method that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 443, + 445, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 445, + 456 + ], + "score": 1.0, + "content": "do not use any language cues and only the visual features used in earlier works [43].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 410, + 505, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "Datasets. We evaluate our approach on two standard video summarization datasets (TVSum [36]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "and SumMe [7]) and on the generic summaries for UT Egocentric videos [16] provided by the QFVS", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 480, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 507, + 495 + ], + "score": 1.0, + "content": "dataset [33]. TVSum [36] consists of 50 videos pertaining to 10 categories (how to videos, news,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "documentary, etc) with 5 videos from each category, typically 1-5 minutes in length. SumMe [7]", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "consists of 25 videos capturing multiple events such as cooking and sports, and the lengths of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "videos vary from 1 to 6 minutes. In addition to training on each dataset independently, we follow", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "prior work and augment training data with 39 videos from the YouTube dataset [2] and 50 videos", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "from the Open Video Project (OVP) dataset [24]. YouTube dataset consists of news, sports and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "cartoon videos. OVP dataset consists of multiple different genres including documentary videos.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "These datasets are diverse in nature and come with different types of annotations, frame-level scores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 583 + ], + "score": 1.0, + "content": "for TVSum and shot-level scores for SumMe. They are integrated to create the ground-truth using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "the procedure in [43]. The UT Egocentric dataset consists of 4 videos captured from head-mounted", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "cameras. Each video is about 3-5 hours long, captured in a natural, uncontrolled setting and contains", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "a diverse set of events. The QFVS dataset [33] provides ground-truth generic summaries for these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "4 videos. The summaries were constructed by dividing the video into shots and asking 3 users to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 623, + 470, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 470, + 635 + ], + "score": 1.0, + "content": "select the relevant shots. The final ground-truth is an average of annotations from all users.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 459, + 507, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 640, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 653 + ], + "score": 1.0, + "content": "Note. All the datasets - YouTube [2], Open Video Project (OVP) dataset [24], TVSum [36],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "SumMe [7], and QFVS [32] were collected by the creators (cited) and consent for any person-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 661, + 437, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 437, + 676 + ], + "score": 1.0, + "content": "ally identifiable information (PII) was ascertained by the authors where necessary.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 638, + 506, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "Data configuration for TVSum and SumMe. Following previous works [43, 42, 31, 26], we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "evaluate our approach in three different data settings: Standard, Augment, and Transfer. In the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "Standard setting, the training and test splits are from the same dataset (i.e. either TVSum or SumMe).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 710, + 389, + 724 + ], + "score": 1.0, + "content": "For SumMe we use available splits, and for TVSum we randomly select", + "type": "text" + }, + { + "bbox": [ + 389, + 711, + 409, + 721 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "of the videos for testing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 479, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 507, + 496 + ], + "score": 1.0, + "content": "and construct 5 different splits and report an average result on all 5 splits. For the Augment setting,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "the training set from one dataset (e.g., TVSum) is combined with all the data from the remaining", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "three datasets (e.g., SumMe, OVP, and YouTube). This setting yields the best performing models due", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "to the additional training data. The Transfer setting is the most challenging of the three. It involves", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 525, + 420, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 420, + 537 + ], + "score": 1.0, + "content": "training a model on three datasets and evaluating on the fourth unseen dataset.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 677, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 93, + 507, + 276 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 71, + 504, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 505, + 83 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 505, + 83 + ], + "score": 1.0, + "content": "Table 2: Unsupervised. Comparing F1 Scores of our methods with unsupervised baselines on the SumMe [7]", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 416, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 416, + 92 + ], + "score": 1.0, + "content": "and TVSum [36] datasets using Standard, Augment, and Transfer data configurations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 93, + 507, + 276 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 93, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 507, + 276 + ], + "score": 0.982, + "html": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Mahasseni et al. [22]39.143.451.759.5
Yuan et al. [41]41.91=57.61
Rochan et al. (SUM-FCNun sup) [31]41.5139.552.71-
Rochan et al. [30]47.5=41.655.6=55.7
He et al. [9]46.047.044.558.558.957.8
Park et al. (SumGraph) [26]49.852.147.059.361.257.6
GoogleNet+bi-LSTM33.138.036.547.754.952.3
ResNet+bi-LSTM34.540.139.651.056.253.8
CLIP-Image+bi-LSTM35.741.041.452.858.756.0
CLIP-Image+Video Caption+bi-LSTM36.942.442.553.559.457.6
GoogleNet+Transformer50.052.747.659.962.158.4
ResNet+Transformer50.853.949.361.163.059.9
CLIP-Image+Transformer51.253.649.261.964.060.6
CLIP-It: CLIP-Image+Video Caption+Transformer52.554.750.063.065.762.8
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SupervisedVid 1Vid 2Vid 3Vid 4Avg
SubMod [6]49.5151.0364.5235.8250.22
QFVS [33]62.6646.1158.8533.5050.29
CLIP-Image + bi-LSTM ResNet +Transformer65.4356.5568.6340.0657.67
66.9758.3270.1043.3159.67
CLIP-Image + Transformer70.861.6772.4347.4863.11
CLIP-It (Gen. Caption)74.1363.4475.8650.2365.92
CLIP-It (GT Caption)84.9871.2682.5561.4675.06
UnsupervisedVid 1Vid 2Vid 3Vid 4Avg
Quasi [46]53.0653.8049.9122.3144.77
CLIP-Image + Transformer65.4457.2165.1041.6357.35
CLIP-It (Gen. Caption)67.0259.4866.4344.1959.28
CLIP-It (GT Caption)73.9066.8375.4452.3167.12
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Comparing F1 Scores of our methods with unsupervised baselines on the SumMe [7]", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 416, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 416, + 92 + ], + "score": 1.0, + "content": "and TVSum [36] datasets using Standard, Augment, and Transfer data configurations.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 93, + 507, + 276 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 93, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 507, + 276 + ], + "score": 0.982, + "html": "
MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Mahasseni et al. [22]39.143.451.759.5
Yuan et al. [41]41.91=57.61
Rochan et al. (SUM-FCNun sup) [31]41.5139.552.71-
Rochan et al. [30]47.5=41.655.6=55.7
He et al. [9]46.047.044.558.558.957.8
Park et al. (SumGraph) [26]49.852.147.059.361.257.6
GoogleNet+bi-LSTM33.138.036.547.754.952.3
ResNet+bi-LSTM34.540.139.651.056.253.8
CLIP-Image+bi-LSTM35.741.041.452.858.756.0
CLIP-Image+Video Caption+bi-LSTM36.942.442.553.559.457.6
GoogleNet+Transformer50.052.747.659.962.158.4
ResNet+Transformer50.853.949.361.163.059.9
CLIP-Image+Transformer51.253.649.261.964.060.6
CLIP-It: CLIP-Image+Video Caption+Transformer52.554.750.063.065.762.8
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SupervisedVid 1Vid 2Vid 3Vid 4Avg
SubMod [6]49.5151.0364.5235.8250.22
QFVS [33]62.6646.1158.8533.5050.29
CLIP-Image + bi-LSTM ResNet +Transformer65.4356.5568.6340.0657.67
66.9758.3270.1043.3159.67
CLIP-Image + Transformer70.861.6772.4347.4863.11
CLIP-It (Gen. Caption)74.1363.4475.8650.2365.92
CLIP-It (GT Caption)84.9871.2682.5561.4675.06
UnsupervisedVid 1Vid 2Vid 3Vid 4Avg
Quasi [46]53.0653.8049.9122.3144.77
CLIP-Image + Transformer65.4457.2165.1041.6357.35
CLIP-It (Gen. Caption)67.0259.4866.4344.1959.28
CLIP-It (GT Caption)73.9066.8375.4452.3167.12
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MethodF-MeasureRecall
Supervised
Submod-V+Both et al. [27]34.1531.59
CLIP Image + Transformer41.5839.96
CLIP-It: CLIP Image + Gen. Video Caption + Transformer44.7043.28
CLIP-It: CLIP Image + GT Video Caption + Transformer52.1050.76
Unsupervised
CLIP Image + Transformer39.2237.46
CLIP-It: CLIP Image + Gen.Video Caption + Transformer42.1040.65
CLIP-It: CLIP Image + GT Video Caption + Transformer49.9847.91
Table 5: Results on TV Episodes dataset [16]
MethodF-MeasureRecall
Supervised
Submod-V+Sem. Rep. et al. [27]40.9037.02
CLIP Image+ Transformer47.8246.02
CLIP-It: CLIP Image + GT Video Caption + Transformer55.3453.90
Unsupervised
CLIP Image + Transformer45.7744.01
CLIP-It: CLIP Image + GT Video Caption+ Transformer53.4252.50
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As ground truth, we obtain 15 summaries for each video using the same greedy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "n-gram matching and ordered subshot selection procedures as previous work [27]. We follow the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "same procedure as in prior work [40, 27] for creating and evaluating text summaries from video", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "summaries. 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The input is a video", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "of a woman demonstrating how to make a chicken sandwich. The ground truth summary shows", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "the scores computed by averaging the annotations from all users as in [43] and the keyframes that", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "table", + "bbox": [ + 118, + 623, + 491, + 719 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 171, + 610, + 437, + 621 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 172, + 609, + 439, + 622 + ], + "spans": [ + { + "bbox": [ + 172, + 609, + 439, + 622 + ], + "score": 1.0, + "content": "Table 6: Comparing F1 Scores of different methods on the QFVS dataset.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 118, + 623, + 491, + 719 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 118, + 623, + 491, + 719 + ], + "spans": [ + { + "bbox": [ + 118, + 623, + 491, + 719 + ], + "score": 0.982, + "html": "
Vid 1Vid 2Vid 3Vid 4Avg
SeqDPP[5136.5943.6725.2618.1530.92
SH-DPP [32]35.6742.7436.5118.6233.38
QFVS [33]48.6841.6656.4729.9644.19
CLIP-Image + Query + bi-LSTM54.4748.5962.8138.6451.13
ResNet + Query + Transformer55.1951.0364.2639.4752.49
CLIP-It: CLIP-Image + Query + Transformer57.1353.6066.0841.4154.55
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MethodF-MeasureRecall
Supervised
Submod-V+Both et al. [27]34.1531.59
CLIP Image + Transformer41.5839.96
CLIP-It: CLIP Image + Gen. Video Caption + Transformer44.7043.28
CLIP-It: CLIP Image + GT Video Caption + Transformer52.1050.76
Unsupervised
CLIP Image + Transformer39.2237.46
CLIP-It: CLIP Image + Gen.Video Caption + Transformer42.1040.65
CLIP-It: CLIP Image + GT Video Caption + Transformer49.9847.91
Table 5: Results on TV Episodes dataset [16]
MethodF-MeasureRecall
Supervised
Submod-V+Sem. Rep. et al. [27]40.9037.02
CLIP Image+ Transformer47.8246.02
CLIP-It: CLIP Image + GT Video Caption + Transformer55.3453.90
Unsupervised
CLIP Image + Transformer45.7744.01
CLIP-It: CLIP Image + GT Video Caption+ Transformer53.4252.50
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Captions) outperforms both supervised", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "and unsupervised baselines by a large margin on all four videos, and particularly on Video 4, which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 412, + 504, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 481, + 425 + ], + "score": 1.0, + "content": "happens to be the most difficult for all methods. Adding captions helps significantly improve", + "type": "text" + }, + { + "bbox": [ + 481, + 412, + 504, + 423 + ], + "score": 0.8, + "content": "( 2 \\% )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 298, + 435 + ], + "score": 1.0, + "content": "the summaries and outperforms the CLIP Image", + "type": "text" + }, + { + "bbox": [ + 298, + 424, + 307, + 433 + ], + "score": 0.79, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "Transformer baseline. 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Likewise, replacing the Transformer with a bi-LSTM also hurts performance.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 367, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "We include results of our method (1) without captions (2) using generated captions (3) using the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "ground truth captions provided by VideoSet [40] for UT Egocentric and TV Episodes datasets in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "Tables 4 and 5. As ground truth, we obtain 15 summaries for each video using the same greedy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "n-gram matching and ordered subshot selection procedures as previous work [27]. We follow the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "same procedure as in prior work [40, 27] for creating and evaluating text summaries from video", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "summaries. Our method outperforms [27] in both the supervised and unsupervised settings on both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 537, + 144, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 144, + 550 + ], + "score": 1.0, + "content": "datasets.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 471, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "Qualitative Results. We highlight the need to use language, specifically dense video captions, for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "constructing generic video summaries through a qualitative example in Fig. 3. The input is a video", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "of a woman demonstrating how to make a chicken sandwich. The ground truth summary shows", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "the scores computed by averaging the annotations from all users as in [43] and the keyframes that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "received high scores. Next we show the result from the baseline CLIP-Image+Transformer, which", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "uses only visual features and no language input. The predicted scores show that the high scoring", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "frames in the ground truth also receive a high score by our baseline, however a lot of irrelevant frames", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 270, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 286 + ], + "score": 1.0, + "content": "end up receiving a high score too. Thus, when the model finally picks the keyframes it ends up", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "selecting frames where the person is talking or eating the sandwich (shown in red) which do not", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "correspond to the key steps in the video. Adding language guidance via generated captions helps", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "address this problem. The last row shows the captions generated by BMT [11]. The results shown are", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "from our full CLIP-It model. The predicted scores are more similar to ground truth scores and the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 326, + 345, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 345, + 339 + ], + "score": 1.0, + "content": "highest scoring keyframes are the same as the ground truth.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 553, + 505, + 599 + ] + }, + { + "type": "table", + "bbox": [ + 118, + 623, + 491, + 719 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 171, + 610, + 437, + 621 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 172, + 609, + 439, + 622 + ], + "spans": [ + { + "bbox": [ + 172, + 609, + 439, + 622 + ], + "score": 1.0, + "content": "Table 6: Comparing F1 Scores of different methods on the QFVS dataset.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 118, + 623, + 491, + 719 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 118, + 623, + 491, + 719 + ], + "spans": [ + { + "bbox": [ + 118, + 623, + 491, + 719 + ], + "score": 0.982, + "html": "
Vid 1Vid 2Vid 3Vid 4Avg
SeqDPP[5136.5943.6725.2618.1530.92
SH-DPP [32]35.6742.7436.5118.6233.38
QFVS [33]48.6841.6656.4729.9644.19
CLIP-Image + Query + bi-LSTM54.4748.5962.8138.6451.13
ResNet + Query + Transformer55.1951.0364.2639.4752.49
CLIP-It: CLIP-Image + Query + Transformer57.1353.6066.0841.4154.55
", + "type": "table", + "image_path": "bf1ab8fad45ed422cb09bb2d772b5e4e5900b4e6dbf40f63aa8eb4fe634018e7.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 118, + 623, + 491, + 655.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 118, + 655.0, + 491, + 687.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 118, + 687.0, + 491, + 719.0 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 25.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 70, + 506, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 70, + 506, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 70, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 114, + 70, + 506, + 186 + ], + "score": 0.965, + "type": "image", + "image_path": "8598abec91315572d62925e9e2ddce1b0bafa26350dcb4469546767f520beafd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 70, + 506, + 108.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 108.66666666666666, + 506, + 147.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 147.33333333333331, + 506, + 185.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 193, + 505, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Figure 5: Result of our method on the QFVS dataset. The first row shows some frames from the 4 hour long input", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "score": 1.0, + "content": "video. Given the query \"book and chair\", Summary 1 shows some frames selected by our method. Summary 2", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 213, + 265, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 265, + 223 + ], + "score": 1.0, + "content": "shows frames for the query \"Sun and Tree\".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "received high scores. Next we show the result from the baseline CLIP-Image+Transformer, which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "uses only visual features and no language input. The predicted scores show that the high scoring", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "frames in the ground truth also receive a high score by our baseline, however a lot of irrelevant frames", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 270, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 286 + ], + "score": 1.0, + "content": "end up receiving a high score too. Thus, when the model finally picks the keyframes it ends up", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "selecting frames where the person is talking or eating the sandwich (shown in red) which do not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "correspond to the key steps in the video. Adding language guidance via generated captions helps", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "address this problem. The last row shows the captions generated by BMT [11]. The results shown are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "from our full CLIP-It model. The predicted scores are more similar to ground truth scores and the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 326, + 345, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 345, + 339 + ], + "score": 1.0, + "content": "highest scoring keyframes are the same as the ground truth.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "Fig. 4 shows compares summaries generated by our method to the ground truth summary on a cooking", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "video from the SumMe dataset. Predicted scores are the frame-wise scores predicted by CLIP-It and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "GT Scores are the scores computed from user annotations [43]. The top row (True Positives) shows", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "high scoring keyframes which are chosen by both our method and the ground truth, and the green", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "arrows point to the assigned scores. As we see, they are clear, distinct and represent key actions in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "the recipe. The bottom row (True Negatives) shows frames which are assigned a low score (shown", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "by the red arrows) and are not part of the final summaries (both GT and predicted). E.g., the first", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "frame is irrelevant and corresponds to a segment between key steps, while the second frame has poor", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 294, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 294, + 442 + ], + "score": 1.0, + "content": "lighting and its hard to tell what is being done.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 290, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 292, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 292, + 490 + ], + "score": 1.0, + "content": "4.2 Query-Focused Video Summarization", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 504, + 504, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "In Query-Focused Video Summarization, the summarization process is conditioned on an input user", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 513, + 496, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 496, + 528 + ], + "score": 1.0, + "content": "query, thus, multiple summaries can be obtained for the same video using different input queries.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Dataset. We evaluate our method on the QFVS dataset based on the UT Egocentric videos described", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "earlier. The dataset consists of 46 queries for each of the four videos and user-annotated video", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 215, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 215, + 567 + ], + "score": 1.0, + "content": "summaries for each query.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "score": 1.0, + "content": "Quantitative Results. In Tab. 6, we compare F1 scores of our method to 3 baselines, SH-DPP [32],", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "Seq-DPP [5], and QFVS [33]. Following [33], we run four rounds of experiments leaving out one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "video for testing and one for validation, while keeping the remaining two for training. Our full model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 602, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 215, + 613 + ], + "score": 1.0, + "content": "achieves an avg F1 score of", + "type": "text" + }, + { + "bbox": [ + 215, + 602, + 246, + 613 + ], + "score": 0.88, + "content": "5 4 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 603, + 373, + 613 + ], + "score": 1.0, + "content": "outperforming the best baseline", + "type": "text" + }, + { + "bbox": [ + 373, + 602, + 410, + 613 + ], + "score": 0.87, + "content": "( 4 4 . 1 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 603, + 423, + 613 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 424, + 602, + 442, + 613 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 603, + 505, + 613 + ], + "score": 1.0, + "content": ". We would like", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to point out that our method uses more recent image features compared to the baselines. At the same", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "time, when switching from CLIP to ResNet image embedding, we still improve significantly over the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "baselines. We expect the improvement to also hold with weaker features (e.g., as demonstrated with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 234, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 234, + 658 + ], + "score": 1.0, + "content": "GoogleNet results in Tab. 1, 2).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 505, + 728 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "Qualitative Results. Fig. 5 shows a result on the QFVS dataset for UT Egocentric videos. The input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "is an egocentric video shot from a head-mounted camera and spans 4 hours. It consists of multiple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "events from a person’s day, such as reading books, working on the laptop, walking in the streets, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "score": 1.0, + "content": "so on. Summary 1 and 2 show results from our CLIP-It method when the input query is “Book and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 705, + 506, + 718 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 506, + 718 + ], + "score": 1.0, + "content": "chair” and “Sun and tree” respectively. The frames shown are frames assigned high-scores by our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 716, + 465, + 729 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 465, + 729 + ], + "score": 1.0, + "content": "method. As seen, given the same input video, different queries yield different summaries.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 70, + 506, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 70, + 506, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 70, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 114, + 70, + 506, + 186 + ], + "score": 0.965, + "type": "image", + "image_path": "8598abec91315572d62925e9e2ddce1b0bafa26350dcb4469546767f520beafd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 70, + 506, + 108.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 108.66666666666666, + 506, + 147.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 147.33333333333331, + 506, + 185.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 193, + 505, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Figure 5: Result of our method on the QFVS dataset. The first row shows some frames from the 4 hour long input", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 215 + ], + "score": 1.0, + "content": "video. Given the query \"book and chair\", Summary 1 shows some frames selected by our method. Summary 2", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 213, + 265, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 265, + 223 + ], + "score": 1.0, + "content": "shows frames for the query \"Sun and Tree\".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 338 + ], + "lines": [], + "index": 10, + "bbox_fs": [ + 105, + 239, + 506, + 339 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "Fig. 4 shows compares summaries generated by our method to the ground truth summary on a cooking", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "video from the SumMe dataset. Predicted scores are the frame-wise scores predicted by CLIP-It and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "GT Scores are the scores computed from user annotations [43]. The top row (True Positives) shows", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "high scoring keyframes which are chosen by both our method and the ground truth, and the green", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "arrows point to the assigned scores. As we see, they are clear, distinct and represent key actions in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "the recipe. The bottom row (True Negatives) shows frames which are assigned a low score (shown", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "by the red arrows) and are not part of the final summaries (both GT and predicted). E.g., the first", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "frame is irrelevant and corresponds to a segment between key steps, while the second frame has poor", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 294, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 294, + 442 + ], + "score": 1.0, + "content": "lighting and its hard to tell what is being done.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 342, + 506, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 290, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 292, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 292, + 490 + ], + "score": 1.0, + "content": "4.2 Query-Focused Video Summarization", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 504, + 504, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "In Query-Focused Video Summarization, the summarization process is conditioned on an input user", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 513, + 496, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 496, + 528 + ], + "score": 1.0, + "content": "query, thus, multiple summaries can be obtained for the same video using different input queries.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 503, + 505, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Dataset. We evaluate our method on the QFVS dataset based on the UT Egocentric videos described", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "earlier. The dataset consists of 46 queries for each of the four videos and user-annotated video", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 215, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 215, + 567 + ], + "score": 1.0, + "content": "summaries for each query.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 531, + 505, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 507, + 582 + ], + "score": 1.0, + "content": "Quantitative Results. In Tab. 6, we compare F1 scores of our method to 3 baselines, SH-DPP [32],", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "Seq-DPP [5], and QFVS [33]. Following [33], we run four rounds of experiments leaving out one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "video for testing and one for validation, while keeping the remaining two for training. Our full model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 602, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 215, + 613 + ], + "score": 1.0, + "content": "achieves an avg F1 score of", + "type": "text" + }, + { + "bbox": [ + 215, + 602, + 246, + 613 + ], + "score": 0.88, + "content": "5 4 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 603, + 373, + 613 + ], + "score": 1.0, + "content": "outperforming the best baseline", + "type": "text" + }, + { + "bbox": [ + 373, + 602, + 410, + 613 + ], + "score": 0.87, + "content": "( 4 4 . 1 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 603, + 423, + 613 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 424, + 602, + 442, + 613 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 603, + 505, + 613 + ], + "score": 1.0, + "content": ". We would like", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to point out that our method uses more recent image features compared to the baselines. At the same", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "time, when switching from CLIP to ResNet image embedding, we still improve significantly over the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "baselines. We expect the improvement to also hold with weaker features (e.g., as demonstrated with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 646, + 234, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 234, + 658 + ], + "score": 1.0, + "content": "GoogleNet results in Tab. 1, 2).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 568, + 507, + 658 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 505, + 728 + ], + "lines": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "Qualitative Results. Fig. 5 shows a result on the QFVS dataset for UT Egocentric videos. The input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "is an egocentric video shot from a head-mounted camera and spans 4 hours. It consists of multiple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "events from a person’s day, such as reading books, working on the laptop, walking in the streets, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 707 + ], + "score": 1.0, + "content": "so on. Summary 1 and 2 show results from our CLIP-It method when the input query is “Book and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 705, + 506, + 718 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 506, + 718 + ], + "score": 1.0, + "content": "chair” and “Sun and tree” respectively. The frames shown are frames assigned high-scores by our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 716, + 465, + 729 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 465, + 729 + ], + "score": 1.0, + "content": "method. As seen, given the same input video, different queries yield different summaries.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 661, + 506, + 729 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 291, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 293, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 293, + 87 + ], + "score": 1.0, + "content": "5 Discussion and Broader Impacts", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 88, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 106, + 87, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 506, + 101 + ], + "score": 1.0, + "content": "We introduced CLIP-It, a unified language-guided framework for generic and query focused video", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 100, + 504, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 504, + 111 + ], + "score": 1.0, + "content": "summarization. Video summarization is a relevant problem with many use-cases, and our approach", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "provides greater flexibility to the users, allowing them to guide summarization with open-ended natural", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "language queries. We envision potential positive impact from improved user experience if adopted", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "on video platforms such as YouTube. We rely on an off-the-shelf video captioning model [11] and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "a large-scale vision-language model (CLIP [29]) which may have encoded some inappropriate biases", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "that could propagate to our model. Our visual inspection of the obtained summarization results did not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "raise any apparent concerns. However, practitioners who wish to use our approach should be mindful", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "of the sources of bias we have outlined above depending on the specific use case they are addressing.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Acknowledgements. We thank Arun Mallya for very helpful discussions and feedback. We’d also", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "like to thank Huijuan Xu for feedback on the draft. 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We rely on an off-the-shelf video captioning model [11] and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "a large-scale vision-language model (CLIP [29]) which may have encoded some inappropriate biases", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "that could propagate to our model. Our visual inspection of the obtained summarization results did not", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "raise any apparent concerns. However, practitioners who wish to use our approach should be mindful", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "of the sources of bias we have outlined above depending on the specific use case they are addressing.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 87, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "Acknowledgements. We thank Arun Mallya for very helpful discussions and feedback. We’d also", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "like to thank Huijuan Xu for feedback on the draft. 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MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Zhang et al. (SumTransfer) [42]40.941.338.5-=1
Zhang et al. (LSTM) [43]38.642.941.854.759.658.7
Mahasseni et al. (SUM-GANsup) [22]41.743.6-56.361.2=
Rochan et al. (SUM-FCN) [31]47.551.144.156.859.258.2
Rochan et al. (SUM-DeepLab) [31]48.850.245.058.459.157.4
Zhou et al. [47]42.143.942.658.159.858.9
Zhang et al. [44]=44.91=63.91
Fajtl et al. [3]49.751.1-61.462.41
Rochan et al. [30]148.041.656.155.7
Chen et al. (V2TS) [1]-1162.111
He et al. [9]47.2--59.4--
Park et al. (SumGraph) [26]51.452.948.763.965.860.5
GoogleNet+bi-LSTM38.542.440.753.959.658.6
ResNet+bi-LSTM39.444.042.655.061.059.9
CLIP-Image+bi-LSTM41.145.944.956.863.761.6
CLIP-Image+Video Caption+bi-LSTM41.246.145.557.164.362.4
GoogleNet+Transformer51.653.549.464.266.361.3
ResNet+Transformer52.854.950.365.067.562.8
CLIP-Image+Transformer53.555.351.065.568.163.4
CLIP-It: CLIP-Image+Video Caption+Transformer54.256.451.966.369.065.5
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MethodSumMeTVSum
Standard AugmentTransfer Standard AugmentTransfer
Mahasseni et al. [22]39.143.451.759.5
Yuan et al. [41]41.91=57.61
Rochan et al. (SUM-FCNun sup) [31]41.5139.552.71-
Rochan et al. [30]47.5=41.655.6=55.7
He et al. [9]46.047.044.558.558.957.8
Park et al. (SumGraph) [26]49.852.147.059.361.257.6
GoogleNet+bi-LSTM33.138.036.547.754.952.3
ResNet+bi-LSTM34.540.139.651.056.253.8
CLIP-Image+bi-LSTM35.741.041.452.858.756.0
CLIP-Image+Video Caption+bi-LSTM36.942.442.553.559.457.6
GoogleNet+Transformer50.052.747.659.962.158.4
ResNet+Transformer50.853.949.361.163.059.9
CLIP-Image+Transformer51.253.649.261.964.060.6
CLIP-It: CLIP-Image+Video Caption+Transformer52.554.750.063.065.762.8
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SupervisedVid 1Vid 2Vid 3Vid 4Avg
SubMod [6]49.5151.0364.5235.8250.22
QFVS [33]62.6646.1158.8533.5050.29
CLIP-Image + bi-LSTM ResNet +Transformer65.4356.5568.6340.0657.67
66.9758.3270.1043.3159.67
CLIP-Image + Transformer70.861.6772.4347.4863.11
CLIP-It (Gen. Caption)74.1363.4475.8650.2365.92
CLIP-It (GT Caption)84.9871.2682.5561.4675.06
UnsupervisedVid 1Vid 2Vid 3Vid 4Avg
Quasi [46]53.0653.8049.9122.3144.77
CLIP-Image + Transformer65.4457.2165.1041.6357.35
CLIP-It (Gen. Caption)67.0259.4866.4344.1959.28
CLIP-It (GT Caption)73.9066.8375.4452.3167.12
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Vid 1Vid 2Vid 3Vid 4Avg
SeqDPP[5136.5943.6725.2618.1530.92
SH-DPP [32]35.6742.7436.5118.6233.38
QFVS [33]48.6841.6656.4729.9644.19
CLIP-Image + Query + bi-LSTM54.4748.5962.8138.6451.13
ResNet + Query + Transformer55.1951.0364.2639.4752.49
CLIP-It: CLIP-Image + Query + Transformer57.1353.6066.0841.4154.55
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MethodF-MeasureRecall
Supervised
Submod-V+Both et al. [27]34.1531.59
CLIP Image + Transformer41.5839.96
CLIP-It: CLIP Image + Gen. Video Caption + Transformer44.7043.28
CLIP-It: CLIP Image + GT Video Caption + Transformer52.1050.76
Unsupervised
CLIP Image + Transformer39.2237.46
CLIP-It: CLIP Image + Gen.Video Caption + Transformer42.1040.65
CLIP-It: CLIP Image + GT Video Caption + Transformer49.9847.91
Table 5: Results on TV Episodes dataset [16]
MethodF-MeasureRecall
Supervised
Submod-V+Sem. Rep. et al. [27]40.9037.02
CLIP Image+ Transformer47.8246.02
CLIP-It: CLIP Image + GT Video Caption + Transformer55.3453.90
Unsupervised
CLIP Image + Transformer45.7744.01
CLIP-It: CLIP Image + GT Video Caption+ Transformer53.4252.50
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0000000000000000000000000000000000000000..19faf49b793cbc29d35ff7b8c47bbe3cc4324785 --- /dev/null +++ b/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. + +
ExperimentPN-MaxPN-AvgPN-MixPN-AttPN++DGCNNPointConvKPConv
Clustering art.56%/53%98%/99%97%/96%94%/93%100%/98%100%/99%100%/99%100%/99%
FPS vs.uniform50%/50%50.3%/50%51.5%/50%50.1%/50%100%/97%100%/96%100%/98%100%/99%
ESS0.030.490.460.430.950.950.970.99
+ +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. + +
DataFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
Uniformre-sampling0.11530.09260.81410.11040.0014570.6972.16
Farthest point sampling0.17000.15581.88330.10640.0013767.7469.36
Biased sampling2.86310.35249.67190.24690.0014523.1271.28
+ +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. + +
DatasetGeneratorPoolingFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
ChairFCFCMaxMix1.5710.1840.2110.2097.0300.10170.0016423.5672.75
2.1240.06740.0019673.6474.96
FCDeformDeformDeformTreeGANTreeGANGraph-CNNAttention0.6350.6724.9710.11560.0016068.9270.54
MaxMix0.9130.5340.2680.3735.6020.09080.06950.002010.0020068.576.2972.6175.11
2.836
AttentionMaxMixMax0.6960.7552.9870.11410.0016068.7769.36
1.4420.2931.0340.6547.8084.0320.09620.07040.001910.0021124,7474.8248.9073.4978.7963.18
0.334
0.9817.4940.08120.00191
PointFlowOracle1.7820.0881.4050.0933.2560.8140.13350.05940.001900.0016571.7679.5671.7379.69
Multi-CatFCFCFCMaxMix1.5530.3546.9810.08420.0015335.1664.16
0.2550.2852.5500.06560.001840.0013473.572.1972.1669.63
Attention0.4140.4534.2340.1188
DeformDeformDeformTreeGANTreeGANMaxMix1.0720.6333.8450.07990.0017962.564.5
0.6140.3492.4510.06700.0019170.8368.8369.60
Attention0.6160.7202.5310.11130.0014172.04
MaxMix1.7140.3880.4376.3420.10930.0015825.3372.6667.071.0
0.4204.3000.06990.00184
Oracle0.1310.1770.6730.060120.0012477.1477.57
+ +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. + +
DiscriminatorFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
PointNet-Max3.8282.45619.2790.10390.002872656
PointNet-Avg19.64019.60439.7930.11540.006423212
PointNet-Mix1.8352.20813.1970.08980.003315051
PointNet-Attention1.3501.64212.3590.16220.002715248
PointNet++15.08114.18733.9580.11550.005031937
DGCNN10.6299.93914.8820.09320.003964648
KPConv41.1549.9657.320.26950.0113562.5
PointConv27.6927.0546.820.28320.0072138
+ +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 + +Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. 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In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 206–215, 2018. \ No newline at end of file diff --git a/parse/train/784_F-WCW46/784_F-WCW46_content_list.json b/parse/train/784_F-WCW46/784_F-WCW46_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d2f4265cadf50c0b840a2149c67e8cd7991ca430 --- /dev/null +++ b/parse/train/784_F-WCW46/784_F-WCW46_content_list.json @@ -0,0 +1,1179 @@ +[ + { + "type": "text", + "text": "RETHINKING SAMPLING IN 3D POINT CLOUD GENERATIVE ADVERSARIAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 725, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 169, + 398, + 195 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 232, + 544, + 247 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 233, + 268, + 766, + 462 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 232, + 453, + 571, + 465 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 503, + 336, + 518 + ], + "page_idx": 0 + }, + { + "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": [ + 173, + 530, + 825, + 625 + ], + "page_idx": 0 + }, + { + "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": [ + 173, + 642, + 823, + 827 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 103, + 825, + 262 + ], + "page_idx": 1 + }, + { + "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? ", + "bbox": [ + 178, + 270, + 823, + 309 + ], + "page_idx": 1 + }, + { + "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). ", + "bbox": [ + 173, + 315, + 825, + 515 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 521, + 825, + 614 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 619, + 825, + 832 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 POINT CLOUD GAN LANDSCAPE ", + "text_level": 1, + "bbox": [ + 176, + 864, + 482, + 881 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 900, + 825, + 928 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 POINT CLOUD GAN GENERATORS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 454, + 117 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 122, + 825, + 256 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 262, + 825, + 356 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 POINT CLOUD GAN DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 174, + 367, + 482, + 381 + ], + "page_idx": 2 + }, + { + "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 $F = \\overline { { g } } ( \\{ \\dot { h ( p _ { 1 } ) } , h ( p _ { 2 } ) , \\cdots , h ( \\overline { { p _ { N } } } ) \\} )$ . ", + "bbox": [ + 174, + 386, + 825, + 469 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 173, + 473, + 825, + 606 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 612, + 825, + 693 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 699, + 825, + 838 + ], + "page_idx": 2 + }, + { + "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). ", + "bbox": [ + 174, + 843, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 131 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 POINT CLOUD GAN EVALUATION METRICS ", + "text_level": 1, + "bbox": [ + 174, + 147, + 521, + 161 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 173, + 166, + 825, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 SAMPLING SPECTRUM OF DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 174, + 354, + 571, + 371 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 380, + 825, + 433 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 SAMPLING SENSITIVITY OF DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 178, + 443, + 531, + 457 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 460, + 825, + 607 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 613, + 820, + 654 + ], + "page_idx": 3 + }, + { + "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). ", + "bbox": [ + 174, + 660, + 825, + 714 + ], + "page_idx": 3 + }, + { + "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)). ", + "bbox": [ + 173, + 719, + 821, + 747 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 SAMPLING SENSITIVITY EXAMINATION RESULTS ", + "text_level": 1, + "bbox": [ + 173, + 757, + 558, + 771 + ], + "page_idx": 3 + }, + { + "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). ", + "bbox": [ + 173, + 775, + 826, + 829 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 830, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/3ae82dce49479ace6c8954ee595b6c8228b7e2314af5e5c672162bd1b4dfc6b9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ExperimentPN-MaxPN-AvgPN-MixPN-AttPN++DGCNNPointConvKPConv
Clustering art.56%/53%98%/99%97%/96%94%/93%100%/98%100%/99%100%/99%100%/99%
FPS vs.uniform50%/50%50.3%/50%51.5%/50%50.1%/50%100%/97%100%/96%100%/98%100%/99%
ESS0.030.490.460.430.950.950.970.99
", + "bbox": [ + 173, + 102, + 841, + 148 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 165, + 826, + 205 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 217, + 825, + 337 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 343, + 825, + 397 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 404, + 825, + 457 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 463, + 825, + 531 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 536, + 825, + 604 + ], + "page_idx": 4 + }, + { + "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: ", + "bbox": [ + 173, + 619, + 825, + 728 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 232, + 729, + 759, + 744 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 748, + 825, + 776 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 SAMPLING SPECTRUM OF EVALUATION METRICS ", + "text_level": 1, + "bbox": [ + 174, + 786, + 617, + 803 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 809, + 825, + 890 + ], + "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. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/2ae5f2cf923fea9d318098171de62c1544d26af427878d0d5240f4db7813e788.jpg", + "table_caption": [ + "Table 2: Examining sampling sensitivity of evaluation metrics. " + ], + "table_footnote": [], + "table_body": "
DataFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
Uniformre-sampling0.11530.09260.81410.11040.0014570.6972.16
Farthest point sampling0.17000.15581.88330.10640.0013767.7469.36
Biased sampling2.86310.35249.67190.24690.0014523.1271.28
", + "bbox": [ + 202, + 102, + 792, + 148 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 193, + 825, + 258 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 265, + 825, + 345 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 352, + 825, + 433 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 438, + 825, + 479 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 494, + 326, + 510 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 520, + 825, + 613 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 SETTING AND DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 626, + 387, + 640 + ], + "page_idx": 5 + }, + { + "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). ", + "bbox": [ + 174, + 645, + 825, + 712 + ], + "page_idx": 5 + }, + { + "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). ", + "bbox": [ + 174, + 718, + 825, + 811 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 EVALUATING POINTNET-BASED DISCRIMINATORS WITH VARIOUS GENERATORS ", + "text_level": 1, + "bbox": [ + 173, + 824, + 769, + 838 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 843, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "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": "
DatasetGeneratorPoolingFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
ChairFCFCMaxMix1.5710.1840.2110.2097.0300.10170.0016423.5672.75
2.1240.06740.0019673.6474.96
FCDeformDeformDeformTreeGANTreeGANGraph-CNNAttention0.6350.6724.9710.11560.0016068.9270.54
MaxMix0.9130.5340.2680.3735.6020.09080.06950.002010.0020068.576.2972.6175.11
2.836
AttentionMaxMixMax0.6960.7552.9870.11410.0016068.7769.36
1.4420.2931.0340.6547.8084.0320.09620.07040.001910.0021124,7474.8248.9073.4978.7963.18
0.334
0.9817.4940.08120.00191
PointFlowOracle1.7820.0881.4050.0933.2560.8140.13350.05940.001900.0016571.7679.5671.7379.69
Multi-CatFCFCFCMaxMix1.5530.3546.9810.08420.0015335.1664.16
0.2550.2852.5500.06560.001840.0013473.572.1972.1669.63
Attention0.4140.4534.2340.1188
DeformDeformDeformTreeGANTreeGANMaxMix1.0720.6333.8450.07990.0017962.564.5
0.6140.3492.4510.06700.0019170.8368.8369.60
Attention0.6160.7202.5310.11130.0014172.04
MaxMix1.7140.3880.4376.3420.10930.0015825.3372.6667.071.0
0.4204.3000.06990.00184
Oracle0.1310.1770.6730.060120.0012477.1477.57
", + "bbox": [ + 173, + 102, + 828, + 327 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 414, + 821, + 441 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 448, + 825, + 527 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 534, + 825, + 588 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 DIAGNOSING FAILURES FOR VARIOUS DISCRIMINATORS ", + "text_level": 1, + "bbox": [ + 174, + 599, + 606, + 613 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 618, + 825, + 672 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 679, + 825, + 799 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 808, + 825, + 876 + ], + "page_idx": 6 + }, + { + "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 ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/9ab46f0bd251aa0ef4577907186cc5ad8e8d73246239da39a1cce07f60003cd8.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 174, + 99, + 825, + 433 + ], + "page_idx": 7 + }, + { + "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": [ + 186, + 493, + 810, + 621 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a6502f96127bea14acbd72cf43811ba4be0dad6184ce827043f8ab3e9573dc5e.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DiscriminatorFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
PointNet-Max3.8282.45619.2790.10390.002872656
PointNet-Avg19.64019.60439.7930.11540.006423212
PointNet-Mix1.8352.20813.1970.08980.003315051
PointNet-Attention1.3501.64212.3590.16220.002715248
PointNet++15.08114.18733.9580.11550.005031937
DGCNN10.6299.93914.8820.09320.003964648
KPConv41.1549.9657.320.26950.0113562.5
PointConv27.6927.0546.820.28320.0072138
", + "bbox": [ + 207, + 700, + 789, + 795 + ], + "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. ", + "bbox": [ + 173, + 806, + 826, + 848 + ], + "page_idx": 7 + }, + { + "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. ", + "bbox": [ + 173, + 857, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 131 + ], + "page_idx": 8 + }, + { + "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, + 349, + 808, + 366 + ], + "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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We point out that, given that recent research has been", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 338, + 471, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 471, + 349 + ], + "score": 1.0, + "content": "focused on the generator design, the discriminator design needs more attention.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 348, + 471, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 348, + 471, + 360 + ], + "score": 1.0, + "content": "Our work provides both suggestions and tools for building future discriminators.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 357, + 351, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 351, + 370 + ], + "score": 1.0, + "content": "We will release the code to facilitate future research.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 212, + 471, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 359, + 350, + 369 + ], + "lines": [ + { + "bbox": [ + 142, + 359, + 350, + 369 + ], + "spans": [], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 142, + 359, + 350, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 399, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 208, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 208, + 415 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "Point cloud, as the most common form of 3D sensor data, has been widely used in a variety of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "3D vision applications due to its compact yet expressive nature and its amenability to geometric", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "manipulations. It is natural to consider how to generate point cloud through deep learning approaches,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "which has been a popular research topic recently. The previous research efforts in the community have", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "been mainly devoted to conditional generation of point clouds with 3D supervision. The condition", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "could either be images (Fan et al., 2017; Groueix et al., 2018; Park et al., 2019) or partial point", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 275, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 275, + 496 + ], + "score": 1.0, + "content": "clouds (Li et al., 2019; Yang et al., 2018).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 421, + 506, + 496 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 509, + 504, + 655 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 509, + 504, + 655 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 509, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 504, + 655 + ], + "score": 0.97, + "type": "image", + "image_path": "c2faa45a97279dcce0b62c1f7a939ffd1317175ba945bb4c981516043c089985.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 106, + 509, + 504, + 557.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 106, + 557.6666666666666, + 504, + 606.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 106, + 606.3333333333333, + 504, + 654.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 670, + 505, + 744 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 683 + ], + "score": 1.0, + "content": "Figure 1: We visualize the behavior of different discriminators when judging four different chair point", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "clouds. (a) and (b) are generated results, (c) and (d) are point clouds sampled using FPS and uniform", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "score": 1.0, + "content": "sampling from a real chair surface. When training the discriminators on data like (d), different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "discriminators make distinct decisions on the point cloud realness forming a sampling spectrum", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 711, + 506, + 726 + ], + "spans": [ + { + "bbox": [ + 104, + 711, + 506, + 726 + ], + "score": 1.0, + "content": "ranging from sampling-insensitive, sampling-aware to sampling-oversensitive. We advocate sampling-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 723, + 505, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 723, + 505, + 735 + ], + "score": 1.0, + "content": "aware discriminators in the middle of the spectrum, which provide good guidance for fixing geometric", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 733, + 498, + 745 + ], + "spans": [ + { + "bbox": [ + 106, + 733, + 498, + 745 + ], + "score": 1.0, + "content": "flaws (a) and big sample artifacts (b) and tolerate subtle sampling differences between (c) and (d).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + } + ], + "index": 32.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Generating 3D point clouds with GANs in an unsupervised manner is an important but less explored", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "problem. 3D point cloud GAN learns to transform a random latent code into a 3D surface point cloud", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 116 + ], + "score": 1.0, + "content": "by playing an adversarial game. Its development is still in an early stage compared with 2D image", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "GANs. While existing works such as (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 122, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 122, + 506, + 138 + ], + "score": 1.0, + "content": "have developed a variety of generators, they all use PointNet (Qi et al., 2017a) with max pooling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "(PointNet-Max) as their discriminator. PointNet, which is essentially a pointwise MLP followed by a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 507, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 507, + 158 + ], + "score": 1.0, + "content": "global pooling operation, is too limited in capturing shape details for a successful GAN. However,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "advanced networks, e.g. PointNet++(Qi et al., 2017b), DGCNN(Wang et al., 2019), KPConv (Thomas", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "et al., 2019), PointConv (Wu et al., 2019), which leverage relative positions between points and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "hierarchical feature extraction, may not help. From our empirical study, we find they all fail to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "a functioning discriminator. Understanding their failure mechanism and improving discriminator", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 263, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 263, + 210 + ], + "score": 1.0, + "content": "design are hence important and urgent.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 109, + 214, + 504, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "To design a better discriminator, we first need to answer the following question: what should the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 107, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "discriminator examine for improving the generation quality? Or, even more fundamentally, what", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 235, + 325, + 246 + ], + "spans": [ + { + "bbox": [ + 107, + 235, + 325, + 246 + ], + "score": 1.0, + "content": "does it mean by the quality of generated point clouds?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Since a shape point cloud are the points sampled from an object surface, its quality should be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "evaluated from two perspectives: the depicted surface geometry and the point sampling. Arguably,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "geometry plays a decisive role and should be the main focus of a discriminator. However, when the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "generated point clouds have a good shape, there is still a full spectrum on how much a discriminator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "cares about the sampling patterns. We introduce the concept of Sampling Spectrum to depict the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "sampling sensitivity of discriminators, as illustrated in Figure 1. A sampling-insensitive discriminator", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "(e.g. PointNet-Max) may ignore the point density variations as long as it perceives a good overall", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 322, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 507, + 338 + ], + "score": 1.0, + "content": "shape. Such a discriminator could identify big geometric flaws as shown in Figure 1 (a), but turns a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "blind eye to highly non-uniform density distribution, e.g. point clusters in Figure 1 (b). On the other", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 357, + 357 + ], + "score": 1.0, + "content": "extreme, a sampling-oversensitive discriminator (e.g. PointNet", + "type": "text" + }, + { + "bbox": [ + 358, + 345, + 369, + 354 + ], + "score": 0.31, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 344, + 506, + 357 + ], + "score": 1.0, + "content": ", DGCNN, KPConv) can even tell", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "the subtle difference in sampling patterns, e.g. between furthest point sampling (FPS) in Figure 1 (c)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "and uniform sampling in Figure 1 (d), and hence can be very narrow-minded about what a real point", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "cloud should look like. A sampling-aware discriminator, e.g. PointNet-Mix/Attention, which lies", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "in the middle of the spectrum, is able to identify density-related artifacts such as point clusters in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "Figure 1 (b), while not being too sensitive to different sampling patterns, such as Figure 1 (c) and (d).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Resembling the sampling spectrum of discriminators, we also examine the existing point-cloud GAN", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "evaluation metrics from the perspective of sampling sensitivity and propose several perceptual metrics", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "forming a Sampling Spectrum of evaluation metrics. Understanding how the metrics weigh between", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "score": 1.0, + "content": "sampling and geometry is a prerequisite for evaluating point cloud GANs. Many of the existing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "sampling-insensitive metrics only evaluate the geometry factor of the generated point clouds shapes,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "which are blind to the obvious point clustering artifacts and uneven point density. We propose novel", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 475, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 507, + 487 + ], + "score": 1.0, + "content": "sampling-sensitive metrics to further complete the spectrum of point-cloud GAN evaluation metrics.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Guided by the proposed sampling spectrum of discriminators and evaluation metrics, experiments", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "show that different discriminators in the spectrum could provide very different suggestions to improve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 513, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 506, + 524 + ], + "score": 1.0, + "content": "a generator according to its sampling sensitivity. Sampling-insensitive discriminators, e.g. PointNet-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "Max, is unaware of point density variations and hence its generated point cloud inevitably suffer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 448, + 547 + ], + "score": 1.0, + "content": "from clustering artifacts, while sampling-oversensitive discriminators, e.g. PointNe", + "type": "text" + }, + { + "bbox": [ + 449, + 534, + 461, + 543 + ], + "score": 0.51, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 533, + 506, + 547 + ], + "score": 1.0, + "content": ", DGCNN,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "KPConv, PointConv, simply fail to function as the discriminators and can generate much degraded", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "point cloud shapes. We design a diagnostic “no generator” experiment to factor out the impact from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "generators and reveal that the gradients of sampling-oversensitive discriminators prioritize adjusting", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "sampling patterns over producing better shape geometry. Picking a middle-point on the sampling", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "spectrum, we discover a simple yet effective sampling-aware discriminator, PointNet-Mix, and find", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "that it can supervise both shape generation and point density uniformity. It improves all existing", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "generators by a large margin on sampling-related metrics. Surprisingly, we find that even the most", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "naive fully-connected generator, coupled with PointNet-Mix, simply beats all the start-of-the-art point", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "cloud GANs. This discovery conveys an important message to the community: instead of focusing on", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "the generator design, people should invest more time into discriminator and seek for more powerful", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 648, + 235, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 235, + 661 + ], + "score": 1.0, + "content": "sampling-aware discriminators.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 295, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 298, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 298, + 700 + ], + "score": 1.0, + "content": "2 POINT CLOUD GAN LANDSCAPE", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 713, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 105, + 713, + 507, + 727 + ], + "spans": [ + { + "bbox": [ + 105, + 713, + 507, + 727 + ], + "score": 1.0, + "content": "In this section, we review the current state of point cloud GAN covering the generators, the discrimi-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 724, + 368, + 736 + ], + "spans": [ + { + "bbox": [ + 106, + 724, + 368, + 736 + ], + "score": 1.0, + "content": "nators, and the evaluation metrics we are examining in this work.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Generating 3D point clouds with GANs in an unsupervised manner is an important but less explored", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "problem. 3D point cloud GAN learns to transform a random latent code into a 3D surface point cloud", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 116 + ], + "score": 1.0, + "content": "by playing an adversarial game. Its development is still in an early stage compared with 2D image", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "GANs. While existing works such as (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 122, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 122, + 506, + 138 + ], + "score": 1.0, + "content": "have developed a variety of generators, they all use PointNet (Qi et al., 2017a) with max pooling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "(PointNet-Max) as their discriminator. PointNet, which is essentially a pointwise MLP followed by a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 507, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 507, + 158 + ], + "score": 1.0, + "content": "global pooling operation, is too limited in capturing shape details for a successful GAN. However,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "advanced networks, e.g. PointNet++(Qi et al., 2017b), DGCNN(Wang et al., 2019), KPConv (Thomas", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "et al., 2019), PointConv (Wu et al., 2019), which leverage relative positions between points and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "hierarchical feature extraction, may not help. From our empirical study, we find they all fail to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "a functioning discriminator. Understanding their failure mechanism and improving discriminator", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 263, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 263, + 210 + ], + "score": 1.0, + "content": "design are hence important and urgent.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 82, + 507, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 214, + 504, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "To design a better discriminator, we first need to answer the following question: what should the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 107, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "discriminator examine for improving the generation quality? Or, even more fundamentally, what", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 235, + 325, + 246 + ], + "spans": [ + { + "bbox": [ + 107, + 235, + 325, + 246 + ], + "score": 1.0, + "content": "does it mean by the quality of generated point clouds?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 213, + 505, + 246 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Since a shape point cloud are the points sampled from an object surface, its quality should be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "evaluated from two perspectives: the depicted surface geometry and the point sampling. Arguably,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "geometry plays a decisive role and should be the main focus of a discriminator. However, when the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "generated point clouds have a good shape, there is still a full spectrum on how much a discriminator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "cares about the sampling patterns. We introduce the concept of Sampling Spectrum to depict the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "sampling sensitivity of discriminators, as illustrated in Figure 1. A sampling-insensitive discriminator", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "(e.g. PointNet-Max) may ignore the point density variations as long as it perceives a good overall", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 322, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 507, + 338 + ], + "score": 1.0, + "content": "shape. Such a discriminator could identify big geometric flaws as shown in Figure 1 (a), but turns a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "blind eye to highly non-uniform density distribution, e.g. point clusters in Figure 1 (b). On the other", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 357, + 357 + ], + "score": 1.0, + "content": "extreme, a sampling-oversensitive discriminator (e.g. PointNet", + "type": "text" + }, + { + "bbox": [ + 358, + 345, + 369, + 354 + ], + "score": 0.31, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 344, + 506, + 357 + ], + "score": 1.0, + "content": ", DGCNN, KPConv) can even tell", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "the subtle difference in sampling patterns, e.g. between furthest point sampling (FPS) in Figure 1 (c)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "and uniform sampling in Figure 1 (d), and hence can be very narrow-minded about what a real point", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 389 + ], + "score": 1.0, + "content": "cloud should look like. A sampling-aware discriminator, e.g. PointNet-Mix/Attention, which lies", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "in the middle of the spectrum, is able to identify density-related artifacts such as point clusters in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "Figure 1 (b), while not being too sensitive to different sampling patterns, such as Figure 1 (c) and (d).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 250, + 507, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Resembling the sampling spectrum of discriminators, we also examine the existing point-cloud GAN", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "evaluation metrics from the perspective of sampling sensitivity and propose several perceptual metrics", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "forming a Sampling Spectrum of evaluation metrics. Understanding how the metrics weigh between", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "score": 1.0, + "content": "sampling and geometry is a prerequisite for evaluating point cloud GANs. Many of the existing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "sampling-insensitive metrics only evaluate the geometry factor of the generated point clouds shapes,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "which are blind to the obvious point clustering artifacts and uneven point density. We propose novel", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 475, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 507, + 487 + ], + "score": 1.0, + "content": "sampling-sensitive metrics to further complete the spectrum of point-cloud GAN evaluation metrics.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 413, + 507, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Guided by the proposed sampling spectrum of discriminators and evaluation metrics, experiments", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "show that different discriminators in the spectrum could provide very different suggestions to improve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 513, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 506, + 524 + ], + "score": 1.0, + "content": "a generator according to its sampling sensitivity. Sampling-insensitive discriminators, e.g. PointNet-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "Max, is unaware of point density variations and hence its generated point cloud inevitably suffer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 448, + 547 + ], + "score": 1.0, + "content": "from clustering artifacts, while sampling-oversensitive discriminators, e.g. PointNe", + "type": "text" + }, + { + "bbox": [ + 449, + 534, + 461, + 543 + ], + "score": 0.51, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 533, + 506, + 547 + ], + "score": 1.0, + "content": ", DGCNN,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "KPConv, PointConv, simply fail to function as the discriminators and can generate much degraded", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "point cloud shapes. We design a diagnostic “no generator” experiment to factor out the impact from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "generators and reveal that the gradients of sampling-oversensitive discriminators prioritize adjusting", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "sampling patterns over producing better shape geometry. Picking a middle-point on the sampling", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "spectrum, we discover a simple yet effective sampling-aware discriminator, PointNet-Mix, and find", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "that it can supervise both shape generation and point density uniformity. It improves all existing", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "generators by a large margin on sampling-related metrics. Surprisingly, we find that even the most", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "naive fully-connected generator, coupled with PointNet-Mix, simply beats all the start-of-the-art point", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "cloud GANs. This discovery conveys an important message to the community: instead of focusing on", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "the generator design, people should invest more time into discriminator and seek for more powerful", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 648, + 235, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 235, + 661 + ], + "score": 1.0, + "content": "sampling-aware discriminators.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 491, + 506, + 661 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 295, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 298, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 298, + 700 + ], + "score": 1.0, + "content": "2 POINT CLOUD GAN LANDSCAPE", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 713, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 105, + 713, + 507, + 727 + ], + "spans": [ + { + "bbox": [ + 105, + 713, + 507, + 727 + ], + "score": 1.0, + "content": "In this section, we review the current state of point cloud GAN covering the generators, the discrimi-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 724, + 368, + 736 + ], + "spans": [ + { + "bbox": [ + 106, + 724, + 368, + 736 + ], + "score": 1.0, + "content": "nators, and the evaluation metrics we are examining in this work.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 713, + 507, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 278, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 280, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 280, + 96 + ], + "score": 1.0, + "content": "2.1 POINT CLOUD GAN GENERATORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 110 + ], + "score": 1.0, + "content": "Recent point cloud GAN works primarily focus on the generator design. The generator takes input", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 103, + 105, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 103, + 105, + 171, + 123 + ], + "score": 1.0, + "content": "a random noise", + "type": "text" + }, + { + "bbox": [ + 172, + 110, + 223, + 121 + ], + "score": 0.93, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 105, + 330, + 123 + ], + "score": 1.0, + "content": "and outputs a point cloud", + "type": "text" + }, + { + "bbox": [ + 330, + 108, + 373, + 120 + ], + "score": 0.9, + "content": "\\boldsymbol { p } \\in \\mathbb { R } ^ { N \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 105, + 506, + 123 + ], + "score": 1.0, + "content": ". The existing generators can be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 119, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 506, + 131 + ], + "score": 1.0, + "content": "categorized into two classes: fully-connected (FC) generators and graph convolutional generators.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "The first point cloud GAN, r-GAN (Achlioptas et al., 2018), simply uses an FC network as its", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "score": 1.0, + "content": "generator. GraphCNN-GAN (Valsesia et al., 2018) and TreeGAN (Shu et al., 2019) are the rest two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "published works in this field that use graph convolutional generators. The two methods are very", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "similar in principle. The main difference lies in how they build graphs. GraphCNN-GAN builds", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 151, + 184 + ], + "score": 1.0, + "content": "a dynamic", + "type": "text" + }, + { + "bbox": [ + 151, + 173, + 158, + 181 + ], + "score": 0.84, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "-nn graph based upon feature space distance while TreeGAN enforces a tree structure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "throughout its sequential graph expansion and the messages can only be passed from ancestors", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 236, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 236, + 204 + ], + "score": 1.0, + "content": "vertices to descendants vertices.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "Deformation-based decoders (Groueix et al., 2018; Yang et al., 2018) are widely used in the point", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "cloud auto-encoder networks for 3D shape reconstruction. The decoders leverage Multiple-layer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "Perceptrons (MLP) to deform template surfaces into shape surfaces taking as inputs the concatenation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "of template point coordinates and the latent shape feature vectors. Though the decoders can truly act", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "as generators for point cloud GANs, they have not yet been used in unconditioned point cloud GAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "score": 1.0, + "content": "literature. Recently, Mo et al. (Mo et al., 2020) use deformation-based decoder as part generators for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 268, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 268, + 283 + ], + "score": 1.0, + "content": "structure-conditioned point cloud GAN.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 291, + 295, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 297, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 297, + 303 + ], + "score": 1.0, + "content": "2.2 POINT CLOUD GAN DISCRIMINATORS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 506, + 317 + ], + "score": 1.0, + "content": "All existing works on unconditioned point cloud GANs use PointNet with max-pooling (PointNet-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 314, + 507, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 314, + 408, + 330 + ], + "score": 1.0, + "content": "Max) as their discriminators. The discriminator takes input a point cloud", + "type": "text" + }, + { + "bbox": [ + 409, + 317, + 453, + 329 + ], + "score": 0.91, + "content": "p \\in \\mathbb { R } ^ { N \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 314, + 507, + 330 + ], + "score": 1.0, + "content": "and outputs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 303, + 340 + ], + "score": 1.0, + "content": "a score from 0 to 1. PointNet learns a function", + "type": "text" + }, + { + "bbox": [ + 303, + 328, + 310, + 338 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 326, + 399, + 340 + ], + "score": 1.0, + "content": "that maps each point", + "type": "text" + }, + { + "bbox": [ + 400, + 329, + 410, + 339 + ], + "score": 0.85, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "in the point cloud to a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 177, + 352 + ], + "score": 1.0, + "content": "per-point feature", + "type": "text" + }, + { + "bbox": [ + 178, + 338, + 226, + 351 + ], + "score": 0.93, + "content": "h ( p _ { i } ) \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 338, + 457, + 352 + ], + "score": 1.0, + "content": "and then extracts a permutation-invariant global feature", + "type": "text" + }, + { + "bbox": [ + 457, + 338, + 491, + 349 + ], + "score": 0.92, + "content": "F \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 416, + 362 + ], + "score": 1.0, + "content": "pooling the per-point features across all points using a symmetric function", + "type": "text" + }, + { + "bbox": [ + 417, + 352, + 423, + 361 + ], + "score": 0.7, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 349, + 506, + 362 + ], + "score": 1.0, + "content": ", which can be max", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 360, + 448, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 299, + 373 + ], + "score": 1.0, + "content": "pooling, average pooling, etc. Namely, we have", + "type": "text" + }, + { + "bbox": [ + 299, + 360, + 443, + 372 + ], + "score": 0.92, + "content": "F = \\overline { { g } } ( \\{ \\dot { h ( p _ { 1 } ) } , h ( p _ { 2 } ) , \\cdots , h ( \\overline { { p _ { N } } } ) \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 360, + 448, + 373 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 233, + 388 + ], + "score": 1.0, + "content": "PointNet-Max/Avg. Though", + "type": "text" + }, + { + "bbox": [ + 233, + 378, + 240, + 387 + ], + "score": 0.75, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "can be any symmetric function, most existing works use max-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "pooling. In PointNet, the authors show that max-pooling outperforms average-pooling on 3D shape", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 367, + 410 + ], + "score": 1.0, + "content": "classification tasks. They further show that the global feature", + "type": "text" + }, + { + "bbox": [ + 367, + 397, + 376, + 406 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 395, + 506, + 410 + ], + "score": 1.0, + "content": "obtained from max-pooling is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 393, + 420 + ], + "score": 1.0, + "content": "determined by only a sparse subset of the points, namely critical points", + "type": "text" + }, + { + "bbox": [ + 393, + 407, + 405, + 418 + ], + "score": 0.88, + "content": "\\mathcal { C } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ", bringing PointNet-Max", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "with robustness against small data perturbation and corruption. However, this property may limit its", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "discriminative power on telling the density variations and classifying different sampling patterns. To", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "investigate how different aggregation operations affect the sampling sensitivity and generation quality,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 345, + 461 + ], + "score": 1.0, + "content": "we study the two common choices of the symmetric function", + "type": "text" + }, + { + "bbox": [ + 346, + 451, + 352, + 460 + ], + "score": 0.75, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 449, + 505, + 461 + ], + "score": 1.0, + "content": ", max-pooling, and average-pooling, in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "this paper. We will show in Sec. 3 that using different aggregation operations makes huge differences", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 469, + 415, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 415, + 482 + ], + "score": 1.0, + "content": "when adapting PointNet-based networks as point cloud GAN discriminators.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "PointNet-Mix. By simply concatenating the max-pooling feature and the average pool feature,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 473, + 509 + ], + "score": 1.0, + "content": "we obtain another permutation-invariant feature. We name this PointNet-Mix. Formally,", + "type": "text" + }, + { + "bbox": [ + 473, + 496, + 505, + 507 + ], + "score": 0.89, + "content": "F _ { \\mathrm { m i x } } =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 204, + 520 + ], + "score": 0.79, + "content": "[ \\operatorname* { m a x } \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 505, + 207, + 520 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 207, + 506, + 334, + 519 + ], + "score": 0.89, + "content": "\\arg \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\} ] \\in \\mathbb { R } ^ { 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 505, + 506, + 520 + ], + "score": 1.0, + "content": ". The mix-pooling operation is a special", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 244, + 530 + ], + "score": 1.0, + "content": "choice of the symmetric function", + "type": "text" + }, + { + "bbox": [ + 245, + 520, + 251, + 529 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 518, + 504, + 530 + ], + "score": 1.0, + "content": ". We will show in our experiments that PointNet-Mix, though", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "simple, surprisingly improves the performance for most point cloud GANs by a large margin on", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 538, + 207, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 207, + 551 + ], + "score": 1.0, + "content": "sampling-aware metrics.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "PointNet-Attention. A recent point cloud upsampling work (Li et al., 2019) incorporates a self-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "attention module into PointNet for its discriminator and shows improved point density in the upsam-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "pled point cloud. We denote this discriminator as PointNet-Attention. 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A weighted features is then obtained through", + "type": "text" + }, + { + "bbox": [ + 344, + 611, + 465, + 623 + ], + "score": 0.89, + "content": "w ( p _ { i } ) \\ = \\ h ( p _ { i } ) \\ + \\ W ^ { T } k ( p _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 607, + 507, + 626 + ], + "score": 1.0, + "content": ". 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GraphCNN-GAN (Valsesia et al., 2018) and TreeGAN (Shu et al., 2019) are the rest two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "published works in this field that use graph convolutional generators. The two methods are very", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "similar in principle. The main difference lies in how they build graphs. 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The decoders leverage Multiple-layer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "Perceptrons (MLP) to deform template surfaces into shape surfaces taking as inputs the concatenation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "of template point coordinates and the latent shape feature vectors. 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The discriminator takes input a point cloud", + "type": "text" + }, + { + "bbox": [ + 409, + 317, + 453, + 329 + ], + "score": 0.91, + "content": "p \\in \\mathbb { R } ^ { N \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 314, + 507, + 330 + ], + "score": 1.0, + "content": "and outputs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 303, + 340 + ], + "score": 1.0, + "content": "a score from 0 to 1. PointNet learns a function", + "type": "text" + }, + { + "bbox": [ + 303, + 328, + 310, + 338 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 326, + 399, + 340 + ], + "score": 1.0, + "content": "that maps each point", + "type": "text" + }, + { + "bbox": [ + 400, + 329, + 410, + 339 + ], + "score": 0.85, + "content": "p _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "in the point cloud to a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 177, + 352 + ], + "score": 1.0, + "content": "per-point feature", + "type": "text" + }, + { + "bbox": [ + 178, + 338, + 226, + 351 + ], + "score": 0.93, + "content": "h ( p _ { i } ) \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 338, + 457, + 352 + ], + "score": 1.0, + "content": "and then extracts a permutation-invariant global feature", + "type": "text" + }, + { + "bbox": [ + 457, + 338, + 491, + 349 + ], + "score": 0.92, + "content": "F \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 416, + 362 + ], + "score": 1.0, + "content": "pooling the per-point features across all points using a symmetric function", + "type": "text" + }, + { + "bbox": [ + 417, + 352, + 423, + 361 + ], + "score": 0.7, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 349, + 506, + 362 + ], + "score": 1.0, + "content": ", which can be max", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 360, + 448, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 299, + 373 + ], + "score": 1.0, + "content": "pooling, average pooling, etc. Namely, we have", + "type": "text" + }, + { + "bbox": [ + 299, + 360, + 443, + 372 + ], + "score": 0.92, + "content": "F = \\overline { { g } } ( \\{ \\dot { h ( p _ { 1 } ) } , h ( p _ { 2 } ) , \\cdots , h ( \\overline { { p _ { N } } } ) \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 360, + 448, + 373 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 306, + 507, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 233, + 388 + ], + "score": 1.0, + "content": "PointNet-Max/Avg. Though", + "type": "text" + }, + { + "bbox": [ + 233, + 378, + 240, + 387 + ], + "score": 0.75, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "can be any symmetric function, most existing works use max-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "pooling. In PointNet, the authors show that max-pooling outperforms average-pooling on 3D shape", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 367, + 410 + ], + "score": 1.0, + "content": "classification tasks. They further show that the global feature", + "type": "text" + }, + { + "bbox": [ + 367, + 397, + 376, + 406 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 395, + 506, + 410 + ], + "score": 1.0, + "content": "obtained from max-pooling is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 393, + 420 + ], + "score": 1.0, + "content": "determined by only a sparse subset of the points, namely critical points", + "type": "text" + }, + { + "bbox": [ + 393, + 407, + 405, + 418 + ], + "score": 0.88, + "content": "\\mathcal { C } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ", bringing PointNet-Max", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "with robustness against small data perturbation and corruption. However, this property may limit its", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "discriminative power on telling the density variations and classifying different sampling patterns. To", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "investigate how different aggregation operations affect the sampling sensitivity and generation quality,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 345, + 461 + ], + "score": 1.0, + "content": "we study the two common choices of the symmetric function", + "type": "text" + }, + { + "bbox": [ + 346, + 451, + 352, + 460 + ], + "score": 0.75, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 449, + 505, + 461 + ], + "score": 1.0, + "content": ", max-pooling, and average-pooling, in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "this paper. We will show in Sec. 3 that using different aggregation operations makes huge differences", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 469, + 415, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 415, + 482 + ], + "score": 1.0, + "content": "when adapting PointNet-based networks as point cloud GAN discriminators.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 375, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "PointNet-Mix. By simply concatenating the max-pooling feature and the average pool feature,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 473, + 509 + ], + "score": 1.0, + "content": "we obtain another permutation-invariant feature. We name this PointNet-Mix. Formally,", + "type": "text" + }, + { + "bbox": [ + 473, + 496, + 505, + 507 + ], + "score": 0.89, + "content": "F _ { \\mathrm { m i x } } =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 204, + 520 + ], + "score": 0.79, + "content": "[ \\operatorname* { m a x } \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 505, + 207, + 520 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 207, + 506, + 334, + 519 + ], + "score": 0.89, + "content": "\\arg \\{ h ( p _ { 1 } ) , . . . , h ( p _ { N } ) \\} ] \\in \\mathbb { R } ^ { 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 505, + 506, + 520 + ], + "score": 1.0, + "content": ". The mix-pooling operation is a special", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 518, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 244, + 530 + ], + "score": 1.0, + "content": "choice of the symmetric function", + "type": "text" + }, + { + "bbox": [ + 245, + 520, + 251, + 529 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 518, + 504, + 530 + ], + "score": 1.0, + "content": ". We will show in our experiments that PointNet-Mix, though", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "simple, surprisingly improves the performance for most point cloud GANs by a large margin on", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 538, + 207, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 207, + 551 + ], + "score": 1.0, + "content": "sampling-aware metrics.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 484, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "PointNet-Attention. A recent point cloud upsampling work (Li et al., 2019) incorporates a self-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "attention module into PointNet for its discriminator and shows improved point density in the upsam-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "pled point cloud. We denote this discriminator as PointNet-Attention. The self-attention module learns", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 261, + 599 + ], + "score": 1.0, + "content": "three separate MLPs to transform each", + "type": "text" + }, + { + "bbox": [ + 261, + 586, + 284, + 598 + ], + "score": 0.91, + "content": "h ( p _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 586, + 303, + 599 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 304, + 586, + 401, + 598 + ], + "score": 0.88, + "content": "f ( p _ { i } ) , l ( p _ { i } ) , k ( p _ { i } ) \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "correspondingly. Then an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 597, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 264, + 612 + ], + "score": 1.0, + "content": "attention weight matrix is formed by", + "type": "text" + }, + { + "bbox": [ + 264, + 599, + 290, + 609 + ], + "score": 0.46, + "content": "W =", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 597, + 330, + 612 + ], + "score": 1.0, + "content": "SoftMax", + "type": "text" + }, + { + "bbox": [ + 330, + 597, + 504, + 611 + ], + "score": 0.88, + "content": "\\left( [ f ( p _ { 1 } ) , . . . , f ( p _ { N } ) ] ^ { T } [ ( l ( p _ { 1 } ) , . . . , l ( p _ { N } ) ) ] \\right) \\ \\in", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 607, + 507, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 136, + 621 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { N \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 607, + 344, + 626 + ], + "score": 1.0, + "content": ". A weighted features is then obtained through", + "type": "text" + }, + { + "bbox": [ + 344, + 611, + 465, + 623 + ], + "score": 0.89, + "content": "w ( p _ { i } ) \\ = \\ h ( p _ { i } ) \\ + \\ W ^ { T } k ( p _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 607, + 507, + 626 + ], + "score": 1.0, + "content": ". The fi-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "nal aggregated features of PointNet-Attention is a max-pooling of the weighted feature, namely", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 631, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 250, + 644 + ], + "score": 0.89, + "content": "F _ { \\mathrm { a t t e n t i o n } } = \\operatorname* { m a x } \\{ w ( p _ { 1 } ) , . . . , w ( p _ { N } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 631, + 507, + 646 + ], + "score": 1.0, + "content": ". Note that PointNet-Attention allows the per-point features to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "communicate with each other according to their similarity, which is more sensitive to sampling", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 652, + 418, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 418, + 666 + ], + "score": 1.0, + "content": "patterns than PointNet-Max. We investigate this strategy in our paper as well.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 553, + 507, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "Discriminators beyond PointNet. Recently, there have been many works (Qi et al., 2017a; Li", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "et al., 2018; Wang et al., 2019; Hermosilla et al., 2018; Thomas et al., 2019) extending PointNet to", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "more advanced 3D deep learning architectures on point clouds. They improve PointNet by extracting", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "more local or hierarchical geometric features via point cloud convolutions or point cloud graph", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "learning. Though proven to be effective on shape classification and segmentation tasks, no published", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "work examines adapting them as point cloud GAN discriminators. In this paper, we investigate four", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 322, + 97 + ], + "score": 1.0, + "content": "exemplar beyond-PointNet discriminators: PointNet", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 322, + 84, + 334, + 92 + ], + "score": 0.37, + "content": "^ { + + }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "(Qi et al., 2017b), DGCNN (Wang et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 379, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 379, + 105 + ], + "score": 1.0, + "content": "2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 53.5, + "bbox_fs": [ + 104, + 668, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 322, + 97 + ], + "score": 1.0, + "content": "exemplar beyond-PointNet discriminators: PointNet", + "type": "text" + }, + { + "bbox": [ + 322, + 84, + 334, + 92 + ], + "score": 0.37, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "(Qi et al., 2017b), DGCNN (Wang et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 379, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 379, + 105 + ], + "score": 1.0, + "content": "2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 117, + 319, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 320, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 320, + 131 + ], + "score": 1.0, + "content": "2.3 POINT CLOUD GAN EVALUATION METRICS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Achlioptas et al. (Achlioptas et al., 2018) introduces two distance metrics in the Euclidean space", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "for evaluating point cloud GAN: the coverage scores (COV) computing the fraction of the point", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 146, + 165 + ], + "score": 1.0, + "content": "clouds in", + "type": "text" + }, + { + "bbox": [ + 146, + 154, + 155, + 163 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 152, + 306, + 165 + ], + "score": 1.0, + "content": "that are closed to the point clouds in", + "type": "text" + }, + { + "bbox": [ + 306, + 154, + 315, + 163 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 152, + 506, + 165 + ], + "score": 1.0, + "content": "using either Earth Mover’s distance (EMD) or", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "Chamfer distance (CD), and the Minimum matching distance (MMD) scores measuring the fidelity", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 118, + 186 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 174, + 127, + 184 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 174, + 191, + 186 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 191, + 175, + 201, + 184 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "using either EMD or CD. In the field of 2D image GAN, it is common to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "use perceptual metrics, such as the Frechét distance (Heusel et al., 2017) between real and fake", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "score": 1.0, + "content": "Gaussian measures in the feature spaces, for evaluating the generated image results. Formally,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 204, + 508, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 204, + 175, + 222 + ], + "score": 1.0, + "content": "Frechet Distance", + "type": "text" + }, + { + "bbox": [ + 176, + 206, + 361, + 221 + ], + "score": 0.86, + "content": "= | | \\mu _ { r } - \\mu _ { g } | | ^ { 2 } + \\operatorname { T r } ( \\Sigma _ { r } + \\Sigma _ { g } - 2 \\left( \\Sigma _ { r } \\Sigma _ { g } \\right) ^ { 1 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 204, + 391, + 222 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 392, + 210, + 399, + 220 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 204, + 416, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 208, + 425, + 218 + ], + "score": 0.8, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 204, + 508, + 222 + ], + "score": 1.0, + "content": "are the mean vector", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "and the covariance matrix of the features calculated from either real or fake data distribution, and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 117, + 239 + ], + "score": 0.26, + "content": "\\mathrm { T r }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "is the matrix trace. For point clouds, Shu et al. (Shu et al., 2019) proposes Frechét Point Cloud", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "Distance (FPD), which uses the features extracted from a pre-trained PointNet-Max model. In this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "paper, we position the existing metrics on a sampling spectrum according to their sampling sensitivity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 382, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 382, + 273 + ], + "score": 1.0, + "content": "and propose novel sampling-aware metrics to augment the spectrum.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 350, + 294 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 351, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 351, + 297 + ], + "score": 1.0, + "content": "3 SAMPLING SPECTRUM OF DISCRIMINATORS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "While recent works propose many advanced generator improvements for point cloud GANs, we find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "that designing a good discriminator is of equal importance, if not more. In this section, we introduce", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "the sampling spectrum of discriminators, on which we thoroughly examine the positions of different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 413, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 413, + 344 + ], + "score": 1.0, + "content": "discriminators that explain their behaviors when training point cloud GANs.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 109, + 351, + 325, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "score": 1.0, + "content": "3.1 SAMPLING SENSITIVITY OF DISCRIMINATORS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "The sampling sensitivity of a discriminator depicts how much it responds to a change in the point", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "density or sampling pattern of an input point cloud. We find it extremely hard to quantitatively", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "measure this sensitivity given the difficulty of measuring changes in the sampling patterns. Naively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "using Euclidean metrics (e.g. CD or EMD) to measure the distance between two sampled point sets is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "not a solution, since given the same distance budget, the discriminator’s responses can be dramatically", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "different depending on how the sampled points move. Instead, we can set landmarks in the continuous", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "spectrum of sampling sensitivity by examining the discriminative power of the discriminators against", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "different sampling patterns under a series of experiments from easy to hard. Specifically, we design", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 504, + 461 + ], + "score": 1.0, + "content": "two experiments to test whether a discriminator could tell clustering artifacts in point clouds and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "whether it could distinguish between FPS and uniform sampling patterns. Accordingly, we divide the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 474, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 474, + 482 + ], + "score": 1.0, + "content": "spectrum into three regimes: sampling-insensitive/-aware/-oversensitive, as shown in Fig.1.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 502, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "A sampling-insensitive discriminator does not respond to local point density changes if the overall", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "shape remains roughly the same. This kind of discriminators can’t tell clustering artifacts, i.e. Fig.1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 421, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 421, + 519 + ], + "score": 1.0, + "content": "(b), and thus may cause the non-uniform density in the generated point clouds.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "A sampling-aware discriminator can notice the significant point non-uniformity and is hence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "capable of supervising the generator to enforce a similar sampling distribution to the training data,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "while being ignorant to subtle changes when the sampling is already uniform in an intermediate scale.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "Such discriminator will judge Fig.1 (b) as fake but can’t tell the difference between Fig.1 (c) and (d).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 503, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "A sampling-oversensitive discriminator can tell very subtle difference in the sampling patterns, e.g.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 581, + 503, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 503, + 594 + ], + "score": 1.0, + "content": "between FPS and uniform sampling, even if the underlying shapes are the same (Fig.1 (c) and (d)).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 106, + 600, + 342, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 341, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 341, + 612 + ], + "score": 1.0, + "content": "3.2 SAMPLING SENSITIVITY EXAMINATION RESULTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "score": 1.0, + "content": "We design diagnostic experiments to categorize point cloud GAN discriminators on the sampling", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "score": 1.0, + "content": "spectrum: sampling-insensitive (PointNet-Max), sampling-aware (PointNet-Avg, PointNet-Mix,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 633, + 507, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 415, + 649 + ], + "score": 1.0, + "content": "PointNet-Attention (Li et al., 2019)), and sampling-oversensitive (PointNet+", + "type": "text" + }, + { + "bbox": [ + 415, + 637, + 423, + 645 + ], + "score": 0.31, + "content": "^ { - + }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 633, + 507, + 649 + ], + "score": 1.0, + "content": ", DGCNN, KPConv,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 157, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 157, + 659 + ], + "score": 1.0, + "content": "PointConv).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "The Discriminating Power against Clustering Artifacts. To examine whether the discriminators", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "can tell clustering artifacts or not, we create a diagnostic classification dataset using the models from", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 507, + 692 + ], + "score": 1.0, + "content": "ShapeNet (Chang et al., 2015). Taking 100 chairs, we uniformly sample 2048 points from each shape,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "forming a set of real point clouds. To form a fake set of point clouds, for each chair, we first uniformly", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "sample 1024 points, and then densely sample another 1024 points around a random position on the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "chair within a 0.1 radius. The real/fake point clouds are used as the labeled training dataset. We", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 721, + 442, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 442, + 732 + ], + "score": 1.0, + "content": "repeat the same process to generate a test dataset using a different set of 100 chairs.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 104 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 105 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 117, + 319, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 320, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 320, + 131 + ], + "score": 1.0, + "content": "2.3 POINT CLOUD GAN EVALUATION METRICS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Achlioptas et al. (Achlioptas et al., 2018) introduces two distance metrics in the Euclidean space", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "for evaluating point cloud GAN: the coverage scores (COV) computing the fraction of the point", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 152, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 146, + 165 + ], + "score": 1.0, + "content": "clouds in", + "type": "text" + }, + { + "bbox": [ + 146, + 154, + 155, + 163 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 152, + 306, + 165 + ], + "score": 1.0, + "content": "that are closed to the point clouds in", + "type": "text" + }, + { + "bbox": [ + 306, + 154, + 315, + 163 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 152, + 506, + 165 + ], + "score": 1.0, + "content": "using either Earth Mover’s distance (EMD) or", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "Chamfer distance (CD), and the Minimum matching distance (MMD) scores measuring the fidelity", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 118, + 186 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 174, + 127, + 184 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 174, + 191, + 186 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 191, + 175, + 201, + 184 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "using either EMD or CD. In the field of 2D image GAN, it is common to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "use perceptual metrics, such as the Frechét distance (Heusel et al., 2017) between real and fake", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "score": 1.0, + "content": "Gaussian measures in the feature spaces, for evaluating the generated image results. 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For point clouds, Shu et al. (Shu et al., 2019) proposes Frechét Point Cloud", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "Distance (FPD), which uses the features extracted from a pre-trained PointNet-Max model. In this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "paper, we position the existing metrics on a sampling spectrum according to their sampling sensitivity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 382, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 382, + 273 + ], + "score": 1.0, + "content": "and propose novel sampling-aware metrics to augment the spectrum.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 131, + 508, + 273 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 350, + 294 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 351, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 351, + 297 + ], + "score": 1.0, + "content": "3 SAMPLING SPECTRUM OF DISCRIMINATORS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "While recent works propose many advanced generator improvements for point cloud GANs, we find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "that designing a good discriminator is of equal importance, if not more. In this section, we introduce", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "the sampling spectrum of discriminators, on which we thoroughly examine the positions of different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 413, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 413, + 344 + ], + "score": 1.0, + "content": "discriminators that explain their behaviors when training point cloud GANs.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 300, + 505, + 344 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 351, + 325, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 327, + 363 + ], + "score": 1.0, + "content": "3.1 SAMPLING SENSITIVITY OF DISCRIMINATORS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "The sampling sensitivity of a discriminator depicts how much it responds to a change in the point", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "density or sampling pattern of an input point cloud. We find it extremely hard to quantitatively", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "measure this sensitivity given the difficulty of measuring changes in the sampling patterns. Naively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "using Euclidean metrics (e.g. CD or EMD) to measure the distance between two sampled point sets is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "not a solution, since given the same distance budget, the discriminator’s responses can be dramatically", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "different depending on how the sampled points move. Instead, we can set landmarks in the continuous", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "spectrum of sampling sensitivity by examining the discriminative power of the discriminators against", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "different sampling patterns under a series of experiments from easy to hard. Specifically, we design", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 449, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 504, + 461 + ], + "score": 1.0, + "content": "two experiments to test whether a discriminator could tell clustering artifacts in point clouds and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "whether it could distinguish between FPS and uniform sampling patterns. Accordingly, we divide the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 474, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 474, + 482 + ], + "score": 1.0, + "content": "spectrum into three regimes: sampling-insensitive/-aware/-oversensitive, as shown in Fig.1.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 366, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 502, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "A sampling-insensitive discriminator does not respond to local point density changes if the overall", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "shape remains roughly the same. This kind of discriminators can’t tell clustering artifacts, i.e. Fig.1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 421, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 421, + 519 + ], + "score": 1.0, + "content": "(b), and thus may cause the non-uniform density in the generated point clouds.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 486, + 505, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "A sampling-aware discriminator can notice the significant point non-uniformity and is hence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "capable of supervising the generator to enforce a similar sampling distribution to the training data,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "while being ignorant to subtle changes when the sampling is already uniform in an intermediate scale.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "Such discriminator will judge Fig.1 (b) as fake but can’t tell the difference between Fig.1 (c) and (d).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 523, + 507, + 567 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 570, + 503, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "A sampling-oversensitive discriminator can tell very subtle difference in the sampling patterns, e.g.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 581, + 503, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 503, + 594 + ], + "score": 1.0, + "content": "between FPS and uniform sampling, even if the underlying shapes are the same (Fig.1 (c) and (d)).", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 569, + 505, + 594 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 600, + 342, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 341, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 341, + 612 + ], + "score": 1.0, + "content": "3.2 SAMPLING SENSITIVITY EXAMINATION RESULTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 506, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "score": 1.0, + "content": "We design diagnostic experiments to categorize point cloud GAN discriminators on the sampling", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 638 + ], + "score": 1.0, + "content": "spectrum: sampling-insensitive (PointNet-Max), sampling-aware (PointNet-Avg, PointNet-Mix,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 633, + 507, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 415, + 649 + ], + "score": 1.0, + "content": "PointNet-Attention (Li et al., 2019)), and sampling-oversensitive (PointNet+", + "type": "text" + }, + { + "bbox": [ + 415, + 637, + 423, + 645 + ], + "score": 0.31, + "content": "^ { - + }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 633, + 507, + 649 + ], + "score": 1.0, + "content": ", DGCNN, KPConv,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 157, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 157, + 659 + ], + "score": 1.0, + "content": "PointConv).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 614, + 507, + 659 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "The Discriminating Power against Clustering Artifacts. To examine whether the discriminators", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "can tell clustering artifacts or not, we create a diagnostic classification dataset using the models from", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 507, + 692 + ], + "score": 1.0, + "content": "ShapeNet (Chang et al., 2015). Taking 100 chairs, we uniformly sample 2048 points from each shape,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "forming a set of real point clouds. To form a fake set of point clouds, for each chair, we first uniformly", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "sample 1024 points, and then densely sample another 1024 points around a random position on the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "chair within a 0.1 radius. The real/fake point clouds are used as the labeled training dataset. We", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 721, + 442, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 442, + 732 + ], + "score": 1.0, + "content": "repeat the same process to generate a test dataset using a different set of 100 chairs.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 658, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 81, + 515, + 118 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 81, + 515, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 515, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 515, + 118 + ], + "score": 0.972, + "html": "
ExperimentPN-MaxPN-AvgPN-MixPN-AttPN++DGCNNPointConvKPConv
Clustering art.56%/53%98%/99%97%/96%94%/93%100%/98%100%/99%100%/99%100%/99%
FPS vs.uniform50%/50%50.3%/50%51.5%/50%50.1%/50%100%/97%100%/96%100%/98%100%/99%
ESS0.030.490.460.430.950.950.970.99
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In each cell", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 490, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 490, + 165 + ], + "score": 1.0, + "content": "of the top two rows, the left shows the training accuracy while the right shows the test accuracy.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "We supervisely train each discriminator to classify the real/fake point clouds. We train them 200", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "score": 1.0, + "content": "epochs until convergence. The training and test accuracies are shown in the first row of Table 1.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "Despite the huge density variation and the remarkable clustering artifacts, PointNet-Max is just barely", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "better than a random guess while the rest of the discriminators are very successful in telling the fake", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "from the real. This indicates that PointNet-Max is sampling-insensitive. Note that the discriminating", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "power of a network towards certain artifacts is maximized under such supervised training scheme. A", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "network will fail to identify such artifacts in an adversarial training scheme if it fails in a supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "training scheme. We will see in Sec. 5.2, point clouds generated by GAN using PointNet-Max as the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 360, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 360, + 268 + ], + "score": 1.0, + "content": "discriminator indeed suffer from non-uniform density artifacts.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "Our insight why PointNet-Avg/Mix can tell the artifacts but PointNet-Max fails is that the average", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "pooling feature computes the center of the mass of the points in feature space and is hence aware to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "certain global non-uniform density distributions. PointNet-Attention leverages a learnable weighted", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 336, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 336, + 317 + ], + "score": 1.0, + "content": "averaging and is hence capable to identify the difference.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "Distinguishing between FPS and Uniform Sampling. We construct another diagnostic dataset", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "with real and fake data which are the same in their shapes but only differ in their sampling patterns.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "Specifically, we perform uniform sampling to generate real data and use FPS to generate fake data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 350, + 457, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 457, + 364 + ], + "score": 1.0, + "content": "from 100 chairs. Figure 1 (c) and (d) illustrate the different sampling pattern outcomes.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 491, + 380 + ], + "score": 1.0, + "content": "We present the training and test accuracies in the second row of Table 1 and show that PointNet", + "type": "text" + }, + { + "bbox": [ + 491, + 369, + 503, + 378 + ], + "score": 0.46, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "DGCNN, KPConv, and PointConv can perfectly distinguish the subtle difference in sampling patterns", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "while the rest discriminators make no progress even on the training set. The experiment indicates", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "that PointNet-Avg/Mix/Attention are sampling-aware while PointNet++, DGCNN, KPConv, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 262, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 262, + 423 + ], + "score": 1.0, + "content": "PointConv are sampling-oversensitive.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 213, + 439 + ], + "score": 1.0, + "content": "We believe, for PointNet+", + "type": "text" + }, + { + "bbox": [ + 213, + 428, + 219, + 436 + ], + "score": 0.37, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 424, + 505, + 439 + ], + "score": 1.0, + "content": ", DGCNN, KPConv, and PointConv, their capability of distinguishing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "FPS from uniform sampling owes to their usage of relative point positions or edge information,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "which are highly sensitive to any change in sampling. We will show in Sec. 5.3 that their remarkable", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "discriminating power on sampling patterns actually leads to the failures as functioning discriminators", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 216, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 216, + 480 + ], + "score": 1.0, + "content": "to train point cloud GANs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "Empirical Definition of Sampling Sensitivity Given the difficulty of measuring the change in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "sampling pattern, we would like to provide an empirical definition of sampling sensitivity of one", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "discriminator according to its performance on the two experiments above: when the discriminator", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "fails in telling clustering artifacts and telling the difference between FPS and uniform sampling", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "pattern, its sampling sensitivity should be closed to 0; when the discriminator can tell clustering", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 557 + ], + "score": 1.0, + "content": "artifacts but can’t distinguish between FPS and uniform sampling, we want to assign it a sampling", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 553, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 504, + 567 + ], + "score": 1.0, + "content": "sensitivity closed to 0.5 ; when the discriminator can do both, we want to assign a sampling sensitivity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 466, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 396, + 578 + ], + "score": 1.0, + "content": "closed to 1 to it. Based on this, we define empirical sampling sensitivity", + "type": "text" + }, + { + "bbox": [ + 396, + 567, + 450, + 577 + ], + "score": 0.93, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 562, + 466, + 578 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 142, + 578, + 465, + 590 + ], + "lines": [ + { + "bbox": [ + 145, + 576, + 466, + 593 + ], + "spans": [ + { + "bbox": [ + 145, + 576, + 279, + 593 + ], + "score": 1.0, + "content": "ESS = Acc(Tell clustering art.)", + "type": "text" + }, + { + "bbox": [ + 279, + 579, + 289, + 590 + ], + "score": 0.3, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 576, + 445, + 593 + ], + "score": 1.0, + "content": "Acc(Tell FPS from uniform sampling)", + "type": "text" + }, + { + "bbox": [ + 446, + 581, + 461, + 590 + ], + "score": 0.66, + "content": "^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 576, + 466, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 593, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "where Acc denotes the test accuracy of an experiment. See Table 1 for results. Using this metric, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 332, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 332, + 615 + ], + "score": 1.0, + "content": "would form a continuous sampling sensitivity spectrum.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 378, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 378, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 378, + 638 + ], + "score": 1.0, + "content": "4 SAMPLING SPECTRUM OF EVALUATION METRICS", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "Similar to the discriminator design, it is very important to understand how different evaluation metrics", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "weigh the differences in sampling patterns against geometry quality. Thus, we introduce the sampling", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "score": 1.0, + "content": "spectrum of evaluation metrics, which exactly resembles the sampling spectrum of discriminators", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "introduced in Sec. 3. On the spectrum, we have sampling-insensitive metrics that measure only", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "the shape geometry and are ignorant of the sampling patterns, and sampling-sensitive metrics that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 695, + 232, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 232, + 705 + ], + "score": 1.0, + "content": "measure both at the same time.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "For perceptual metrics, i.e. Frechét distances in feature spaces, the sampling sensitivity of the metric", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "purely depends on the sampling sensitivity of its feature extractor. 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ExperimentPN-MaxPN-AvgPN-MixPN-AttPN++DGCNNPointConvKPConv
Clustering art.56%/53%98%/99%97%/96%94%/93%100%/98%100%/99%100%/99%100%/99%
FPS vs.uniform50%/50%50.3%/50%51.5%/50%50.1%/50%100%/97%100%/96%100%/98%100%/99%
ESS0.030.490.460.430.950.950.970.99
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In each cell", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 490, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 490, + 165 + ], + "score": 1.0, + "content": "of the top two rows, the left shows the training accuracy while the right shows the test accuracy.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 130, + 505, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "We supervisely train each discriminator to classify the real/fake point clouds. We train them 200", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 507, + 195 + ], + "score": 1.0, + "content": "epochs until convergence. The training and test accuracies are shown in the first row of Table 1.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "Despite the huge density variation and the remarkable clustering artifacts, PointNet-Max is just barely", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "better than a random guess while the rest of the discriminators are very successful in telling the fake", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "from the real. This indicates that PointNet-Max is sampling-insensitive. Note that the discriminating", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "power of a network towards certain artifacts is maximized under such supervised training scheme. A", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "network will fail to identify such artifacts in an adversarial training scheme if it fails in a supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "training scheme. We will see in Sec. 5.2, point clouds generated by GAN using PointNet-Max as the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 360, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 360, + 268 + ], + "score": 1.0, + "content": "discriminator indeed suffer from non-uniform density artifacts.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 172, + 507, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "Our insight why PointNet-Avg/Mix can tell the artifacts but PointNet-Max fails is that the average", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "pooling feature computes the center of the mass of the points in feature space and is hence aware to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "certain global non-uniform density distributions. PointNet-Attention leverages a learnable weighted", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 336, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 336, + 317 + ], + "score": 1.0, + "content": "averaging and is hence capable to identify the difference.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 271, + 505, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "Distinguishing between FPS and Uniform Sampling. We construct another diagnostic dataset", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "with real and fake data which are the same in their shapes but only differ in their sampling patterns.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "Specifically, we perform uniform sampling to generate real data and use FPS to generate fake data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 350, + 457, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 457, + 364 + ], + "score": 1.0, + "content": "from 100 chairs. Figure 1 (c) and (d) illustrate the different sampling pattern outcomes.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 319, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 491, + 380 + ], + "score": 1.0, + "content": "We present the training and test accuracies in the second row of Table 1 and show that PointNet", + "type": "text" + }, + { + "bbox": [ + 491, + 369, + 503, + 378 + ], + "score": 0.46, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "DGCNN, KPConv, and PointConv can perfectly distinguish the subtle difference in sampling patterns", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "while the rest discriminators make no progress even on the training set. The experiment indicates", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "that PointNet-Avg/Mix/Attention are sampling-aware while PointNet++, DGCNN, KPConv, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 262, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 262, + 423 + ], + "score": 1.0, + "content": "PointConv are sampling-oversensitive.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 367, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 213, + 439 + ], + "score": 1.0, + "content": "We believe, for PointNet+", + "type": "text" + }, + { + "bbox": [ + 213, + 428, + 219, + 436 + ], + "score": 0.37, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 424, + 505, + 439 + ], + "score": 1.0, + "content": ", DGCNN, KPConv, and PointConv, their capability of distinguishing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "FPS from uniform sampling owes to their usage of relative point positions or edge information,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "which are highly sensitive to any change in sampling. We will show in Sec. 5.3 that their remarkable", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "discriminating power on sampling patterns actually leads to the failures as functioning discriminators", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 216, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 216, + 480 + ], + "score": 1.0, + "content": "to train point cloud GANs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 424, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "Empirical Definition of Sampling Sensitivity Given the difficulty of measuring the change in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "sampling pattern, we would like to provide an empirical definition of sampling sensitivity of one", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "discriminator according to its performance on the two experiments above: when the discriminator", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "fails in telling clustering artifacts and telling the difference between FPS and uniform sampling", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "pattern, its sampling sensitivity should be closed to 0; when the discriminator can tell clustering", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 557 + ], + "score": 1.0, + "content": "artifacts but can’t distinguish between FPS and uniform sampling, we want to assign it a sampling", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 553, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 504, + 567 + ], + "score": 1.0, + "content": "sensitivity closed to 0.5 ; when the discriminator can do both, we want to assign a sampling sensitivity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 466, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 396, + 578 + ], + "score": 1.0, + "content": "closed to 1 to it. Based on this, we define empirical sampling sensitivity", + "type": "text" + }, + { + "bbox": [ + 396, + 567, + 450, + 577 + ], + "score": 0.93, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 562, + 466, + 578 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 490, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 578, + 465, + 590 + ], + "lines": [ + { + "bbox": [ + 145, + 576, + 466, + 593 + ], + "spans": [ + { + "bbox": [ + 145, + 576, + 279, + 593 + ], + "score": 1.0, + "content": "ESS = Acc(Tell clustering art.)", + "type": "text" + }, + { + "bbox": [ + 279, + 579, + 289, + 590 + ], + "score": 0.3, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 576, + 445, + 593 + ], + "score": 1.0, + "content": "Acc(Tell FPS from uniform sampling)", + "type": "text" + }, + { + "bbox": [ + 446, + 581, + 461, + 590 + ], + "score": 0.66, + "content": "^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 576, + 466, + 593 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "where Acc denotes the test accuracy of an experiment. See Table 1 for results. Using this metric, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 332, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 332, + 615 + ], + "score": 1.0, + "content": "would form a continuous sampling sensitivity spectrum.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 145, + 576, + 466, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 593, + 505, + 615 + ], + "lines": [], + "index": 42.5, + "bbox_fs": [ + 105, + 591, + 505, + 615 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 378, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 378, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 378, + 638 + ], + "score": 1.0, + "content": "4 SAMPLING SPECTRUM OF EVALUATION METRICS", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "Similar to the discriminator design, it is very important to understand how different evaluation metrics", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "weigh the differences in sampling patterns against geometry quality. Thus, we introduce the sampling", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "score": 1.0, + "content": "spectrum of evaluation metrics, which exactly resembles the sampling spectrum of discriminators", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "introduced in Sec. 3. On the spectrum, we have sampling-insensitive metrics that measure only", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "the shape geometry and are ignorant of the sampling patterns, and sampling-sensitive metrics that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 695, + 232, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 232, + 705 + ], + "score": 1.0, + "content": "measure both at the same time.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 641, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "For perceptual metrics, i.e. Frechét distances in feature spaces, the sampling sensitivity of the metric", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "purely depends on the sampling sensitivity of its feature extractor. Frechét distance measured in", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 151, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 167 + ], + "score": 1.0, + "content": "different feature spaces may respond very differently to changes in point density and sampling", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 164, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 175 + ], + "score": 1.0, + "content": "patterns. In this work, we examine three Frechét distance metrics that extract features using PointNet-", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "Max, PointNet-Mix, and DGCNN, respectively. We denote them as Frechét PointNet-Max Distance", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "(FPD-Max), Frechét PointNet-Mix Distance (FPD-Mix), and Frechét DGCNN Distance (FGD). We", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "pretrain all the three feature extraction networks on ModelNet40 (Wu et al., 2015) shape classification.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 710, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 124, + 81, + 485, + 118 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 124, + 81, + 485, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 81, + 485, + 118 + ], + "spans": [ + { + "bbox": [ + 124, + 81, + 485, + 118 + ], + "score": 0.97, + "html": "
DataFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
Uniformre-sampling0.11530.09260.81410.11040.0014570.6972.16
Farthest point sampling0.17000.15581.88330.10640.0013767.7469.36
Biased sampling2.86310.35249.67190.24690.0014523.1271.28
", + "type": "table", + "image_path": "2ae5f2cf923fea9d318098171de62c1544d26af427878d0d5240f4db7813e788.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 81, + 485, + 93.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 93.33333333333333, + 485, + 105.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 105.66666666666666, + 485, + 117.99999999999999 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 179, + 127, + 430, + 139 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 180, + 127, + 431, + 140 + ], + "spans": [ + { + "bbox": [ + 180, + 127, + 431, + 140 + ], + "score": 1.0, + "content": "Table 2: Examining sampling sensitivity of evaluation metrics.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 153, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 167 + ], + "score": 1.0, + "content": "different feature spaces may respond very differently to changes in point density and sampling", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 164, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 175 + ], + "score": 1.0, + "content": "patterns. In this work, we examine three Frechét distance metrics that extract features using PointNet-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "Max, PointNet-Mix, and DGCNN, respectively. We denote them as Frechét PointNet-Max Distance", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "(FPD-Max), Frechét PointNet-Mix Distance (FPD-Mix), and Frechét DGCNN Distance (FGD). We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "pretrain all the three feature extraction networks on ModelNet40 (Wu et al., 2015) shape classification.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "To examine the sampling sensitivity of FPD-Mix/Max and FGD, we create several copies of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "score": 1.0, + "content": "training split of our ShapeNet chair dataset (see Sec.5.1), each of which uses a different sampling", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "strategy to obtain the shape point clouds. The reference one used as the ground truth is using uniform", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "sampling. Then we consider 1) uniform sampling with a different random seed; 2) FPS; and 3)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "biased sampling with clustered artifacts (as described in Sec. 3.2). We use all the available metrics to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 435, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 435, + 275 + ], + "score": 1.0, + "content": "evaluate their distances to the ground truth data. The results are shown in Table 2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "We observe that the Frechét distance metrics share the same sampling sensitivity of their correspond-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "ing discriminators. For example, since PointNet-Max is sampling-insensitive, FPD-Max remains", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 507, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 507, + 313 + ], + "score": 1.0, + "content": "very low even on biased sampling data, hence FPD-Max serves as a perceptual geometry metric,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "which is ignorant to sampling patterns. Similarly, we find that FPD-Mix is sampling-aware since it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "clearly detects the biased sampling patterns while not being able to distinguish the uniform sampling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 497, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 497, + 345 + ], + "score": 1.0, + "content": "and FPS, while FGD is sampling-oversensitive in that it can tell apart FPS and uniform sampling.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "score": 1.0, + "content": "For Euclidean distance metrics, results in Table 2 show that COV-EMD and MMD-EMD are sampling-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "aware, which is intuitively reasonable since EMD enforces a one-to-one matching and is aware of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 390, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 390, + 382 + ], + "score": 1.0, + "content": "point density, while COV-CD and MMD-CD are sampling-insensitive.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 200, + 404 + ], + "lines": [ + { + "bbox": [ + 104, + 390, + 201, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 201, + 407 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "Aware of both the sampling spectrums, we conduct experiments to further evaluate the performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "of point cloud GANs under various evaluation metrics. We show that the point cloud GANs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 447 + ], + "score": 1.0, + "content": "using sampling-insensitive discriminators may produce point clustering artifacts, while sampling-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "oversensitive discriminators fail to supervise point cloud GAN training at all. We further devise a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "diagnostic \"no-generator\" experiment that factors out the generators to better illustrate our discoveries", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "score": 1.0, + "content": "on discriminators. More interesting, we find that the simple PointNet-Mix paired of any generator,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 460, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 460, + 488 + ], + "score": 1.0, + "content": "even with the most naive fully-connected one, achieves the state-of-the-art performance.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 237, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 239, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 239, + 509 + ], + "score": 1.0, + "content": "5.1 SETTING AND DATASETS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "We provide a thorough comparison of all the discriminators investigated in Sec.3 combining with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 522, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 507, + 534 + ], + "score": 1.0, + "content": "all the available generators in the published literature, including the FC generator proposed in r-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "GAN (Achlioptas et al., 2018), and graph convolutional generators used in TreeGAN (Shu et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 556 + ], + "score": 1.0, + "content": "2019). We also add a deformation-based generator into the comparison given its popularity for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "score": 1.0, + "content": "supervised point cloud reconstruction (Groueix et al., 2018; Yang et al., 2018).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "We use two datasets to evaluate the GANs. One is a single-category dataset containing point clouds", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "sampled from all 6,778 chair meshes in ShapeNet (Chang et al., 2015). The other is a multi-category", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "dataset combining shapes from airplane, car, chair, rifle, sofa, table, vessel categories in ShapeNet.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "The multi-category dataset contains 34,313 shapes in total. We uniformly sample 2048 points", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 344, + 623 + ], + "score": 1.0, + "content": "from each shape to form the two datasets. We follow the", + "type": "text" + }, + { + "bbox": [ + 344, + 611, + 402, + 622 + ], + "score": 0.91, + "content": "8 5 \\% / 5 \\% / 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "train/validation/test split", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "in (Achlioptas et al., 2018) and use WGAN-gp (Gulrajani et al., 2017) for the GAN training, similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 433, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 433, + 644 + ], + "score": 1.0, + "content": "to previous works (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 106, + 653, + 471, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 473, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 473, + 666 + ], + "score": 1.0, + "content": "5.2 EVALUATING POINTNET-BASED DISCRIMINATORS WITH VARIOUS GENERATORS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "We report the performance for point cloud GANs that combine PointNet-Max/Min/Attention discrim-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "inators and FC/Deform/TreeGAN/Graph-CNN generators in Table 3. We observe that GANs using", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "PointNet-Mix as the discriminator outperform the ones using PointNet-Max/Attention across all differ-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "ent generators. On sampling-aware/sensitive metrics (i.e. FPD-Mix, FGD, MMD-EMD, COV-EMD),", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "PointNet-Mix is always significantly better than PointNet-Max and is better than PointNet-Attention", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "mainly on FPD-Mix and FGD. Regarding geometry quality evaluated using sampling-insensitive", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 124, + 81, + 485, + 118 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 124, + 81, + 485, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 81, + 485, + 118 + ], + "spans": [ + { + "bbox": [ + 124, + 81, + 485, + 118 + ], + "score": 0.97, + "html": "
DataFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
Uniformre-sampling0.11530.09260.81410.11040.0014570.6972.16
Farthest point sampling0.17000.15581.88330.10640.0013767.7469.36
Biased sampling2.86310.35249.67190.24690.0014523.1271.28
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The reference one used as the ground truth is using uniform", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "sampling. Then we consider 1) uniform sampling with a different random seed; 2) FPS; and 3)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "biased sampling with clustered artifacts (as described in Sec. 3.2). We use all the available metrics to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 435, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 435, + 275 + ], + "score": 1.0, + "content": "evaluate their distances to the ground truth data. The results are shown in Table 2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 210, + 506, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "We observe that the Frechét distance metrics share the same sampling sensitivity of their correspond-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "ing discriminators. For example, since PointNet-Max is sampling-insensitive, FPD-Max remains", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 507, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 507, + 313 + ], + "score": 1.0, + "content": "very low even on biased sampling data, hence FPD-Max serves as a perceptual geometry metric,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "which is ignorant to sampling patterns. Similarly, we find that FPD-Mix is sampling-aware since it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "clearly detects the biased sampling patterns while not being able to distinguish the uniform sampling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 497, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 497, + 345 + ], + "score": 1.0, + "content": "and FPS, while FGD is sampling-oversensitive in that it can tell apart FPS and uniform sampling.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 279, + 507, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 361 + ], + "score": 1.0, + "content": "For Euclidean distance metrics, results in Table 2 show that COV-EMD and MMD-EMD are sampling-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "aware, which is intuitively reasonable since EMD enforces a one-to-one matching and is aware of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 390, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 390, + 382 + ], + "score": 1.0, + "content": "point density, while COV-CD and MMD-CD are sampling-insensitive.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 346, + 506, + 382 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 200, + 404 + ], + "lines": [ + { + "bbox": [ + 104, + 390, + 201, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 201, + 407 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "Aware of both the sampling spectrums, we conduct experiments to further evaluate the performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "of point cloud GANs under various evaluation metrics. We show that the point cloud GANs", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 507, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 447 + ], + "score": 1.0, + "content": "using sampling-insensitive discriminators may produce point clustering artifacts, while sampling-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "oversensitive discriminators fail to supervise point cloud GAN training at all. We further devise a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "diagnostic \"no-generator\" experiment that factors out the generators to better illustrate our discoveries", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 507, + 478 + ], + "score": 1.0, + "content": "on discriminators. More interesting, we find that the simple PointNet-Mix paired of any generator,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 460, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 460, + 488 + ], + "score": 1.0, + "content": "even with the most naive fully-connected one, achieves the state-of-the-art performance.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 412, + 507, + 488 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 496, + 237, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 239, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 239, + 509 + ], + "score": 1.0, + "content": "5.1 SETTING AND DATASETS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "We provide a thorough comparison of all the discriminators investigated in Sec.3 combining with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 522, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 507, + 534 + ], + "score": 1.0, + "content": "all the available generators in the published literature, including the FC generator proposed in r-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "GAN (Achlioptas et al., 2018), and graph convolutional generators used in TreeGAN (Shu et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 556 + ], + "score": 1.0, + "content": "2019). We also add a deformation-based generator into the comparison given its popularity for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "score": 1.0, + "content": "supervised point cloud reconstruction (Groueix et al., 2018; Yang et al., 2018).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 511, + 507, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "We use two datasets to evaluate the GANs. One is a single-category dataset containing point clouds", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "sampled from all 6,778 chair meshes in ShapeNet (Chang et al., 2015). The other is a multi-category", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "dataset combining shapes from airplane, car, chair, rifle, sofa, table, vessel categories in ShapeNet.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "The multi-category dataset contains 34,313 shapes in total. We uniformly sample 2048 points", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 344, + 623 + ], + "score": 1.0, + "content": "from each shape to form the two datasets. We follow the", + "type": "text" + }, + { + "bbox": [ + 344, + 611, + 402, + 622 + ], + "score": 0.91, + "content": "8 5 \\% / 5 \\% / 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "train/validation/test split", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "in (Achlioptas et al., 2018) and use WGAN-gp (Gulrajani et al., 2017) for the GAN training, similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 433, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 433, + 644 + ], + "score": 1.0, + "content": "to previous works (Achlioptas et al., 2018; Valsesia et al., 2018; Shu et al., 2019).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 569, + 506, + 644 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 653, + 471, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 473, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 473, + 666 + ], + "score": 1.0, + "content": "5.2 EVALUATING POINTNET-BASED DISCRIMINATORS WITH VARIOUS GENERATORS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "We report the performance for point cloud GANs that combine PointNet-Max/Min/Attention discrim-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "inators and FC/Deform/TreeGAN/Graph-CNN generators in Table 3. We observe that GANs using", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "PointNet-Mix as the discriminator outperform the ones using PointNet-Max/Attention across all differ-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "ent generators. On sampling-aware/sensitive metrics (i.e. FPD-Mix, FGD, MMD-EMD, COV-EMD),", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "PointNet-Mix is always significantly better than PointNet-Max and is better than PointNet-Attention", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "mainly on FPD-Mix and FGD. Regarding geometry quality evaluated using sampling-insensitive", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "metrics (i.e. FPD-Max, MMD-CD, COV-CD), PointNet-Mix is always significantly better than", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 338, + 480, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 480, + 352 + ], + "score": 1.0, + "content": "PointNet-Attention on FPD-Max and COV-CD while remaining on a par with PointNet-Max.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 669, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 81, + 507, + 259 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 81, + 507, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 507, + 259 + ], + "score": 0.974, + "html": "
DatasetGeneratorPoolingFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
ChairFCFCMaxMix1.5710.1840.2110.2097.0300.10170.0016423.5672.75
2.1240.06740.0019673.6474.96
FCDeformDeformDeformTreeGANTreeGANGraph-CNNAttention0.6350.6724.9710.11560.0016068.9270.54
MaxMix0.9130.5340.2680.3735.6020.09080.06950.002010.0020068.576.2972.6175.11
2.836
AttentionMaxMixMax0.6960.7552.9870.11410.0016068.7769.36
1.4420.2931.0340.6547.8084.0320.09620.07040.001910.0021124,7474.8248.9073.4978.7963.18
0.334
0.9817.4940.08120.00191
PointFlowOracle1.7820.0881.4050.0933.2560.8140.13350.05940.001900.0016571.7679.5671.7379.69
Multi-CatFCFCFCMaxMix1.5530.3546.9810.08420.0015335.1664.16
0.2550.2852.5500.06560.001840.0013473.572.1972.1669.63
Attention0.4140.4534.2340.1188
DeformDeformDeformTreeGANTreeGANMaxMix1.0720.6333.8450.07990.0017962.564.5
0.6140.3492.4510.06700.0019170.8368.8369.60
Attention0.6160.7202.5310.11130.0014172.04
MaxMix1.7140.3880.4376.3420.10930.0015825.3372.6667.071.0
0.4204.3000.06990.00184
Oracle0.1310.1770.6730.060120.0012477.1477.57
", + "type": "table", + "image_path": "bb0b6d2514c2aef60eea70ee2a670c45a56e495ce19debe24e184b17ef906f14.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 81, + 507, + 140.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 140.33333333333334, + 507, + 199.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 199.66666666666669, + 507, + 259.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 263, + 507, + 306 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "Table 3: Evaluating PointNet-based discriminators with different generators. We observe that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "PointNet-Mix can significantly improve all sampling-aware/sensitive metric, including FPD-Mix,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 284, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 506, + 296 + ], + "score": 1.0, + "content": "FGD, MMD-EMD, COV-EMD for all the generators. When paired with FC generator, PointNet-Mix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "score": 1.0, + "content": "achieves the best performance. We include PointFlow(Yang et al., 2019) and Oracle for a comparison.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 503, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 341 + ], + "score": 1.0, + "content": "metrics (i.e. FPD-Max, MMD-CD, COV-CD), PointNet-Mix is always significantly better than", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 338, + 480, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 480, + 352 + ], + "score": 1.0, + "content": "PointNet-Attention on FPD-Max and COV-CD while remaining on a par with PointNet-Max.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "In Figure 2, we present the generated point clouds for all the experiments with color-coding for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "the local point density. We see that the generators trained using PointNet-Max usually suffer from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "non-uniform density, except for the deformation generator. On the chair class, points are usually", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "clustering around the seat area while being sparse at the back. On the contrary, PointNet-Mix enforces", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "a globally uniform point density and hence greatly improves the visual quality of the generated point", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 406, + 289, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 289, + 420 + ], + "score": 1.0, + "content": "clouds. PointNet-Attention is in the between.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "With PointNet-Mix, we observe that the most naive FC generator works the best outperforming the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "previously state-of-the-state generators on almost of metrics. This showcases the importance of being", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "sampling-aware but not sampling-insensitive/oversensitive as a discriminator. It also suggests that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 455, + 446, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 446, + 467 + ], + "score": 1.0, + "content": "future works may focus on designing more powerful sampling-aware discriminators.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 371, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 372, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 372, + 488 + ], + "score": 1.0, + "content": "5.3 DIAGNOSING FAILURES FOR VARIOUS DISCRIMINATORS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 490, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "Observing that the discriminator forms the bottleneck of the point cloud GAN, it is important to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 326, + 514 + ], + "score": 1.0, + "content": "examine whether advanced networks, e.g. PointNet+", + "type": "text" + }, + { + "bbox": [ + 326, + 502, + 333, + 511 + ], + "score": 0.46, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "(Qi et al., 2017b), DGCNN (Wang et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019) can outperform PointNet-based", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 521, + 164, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 164, + 534 + ], + "score": 1.0, + "content": "discriminator.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Surprisingly, we fail to observe any of them ever to be successful to train a GAN in our extensive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 547, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 507, + 561 + ], + "score": 1.0, + "content": "experiments, in which we perform a systematic search, including varying generator architecture,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "changing hyperparameters (e.g. the number of layers and parameters in discriminator, learning rate),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "whether use batch norm, etc. During GAN training, we observe the four advanced discriminator", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "behave very similarly: they are very discriminative in the sense that the gaps between the real and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "fake scores remain significantly large, but the quality of the generated point clouds stays very bad.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "The contradictory behaviors seem unreasonable at the first glance: how can a discriminator be very", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "discriminative but teach nothing to the generator? Thus, we design a diagnostic \"no generator\"", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 622, + 295, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 295, + 634 + ], + "score": 1.0, + "content": "experiment to examine the underlying reasons.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "No Generator Experiments. During GAN training, the gradients from discriminator output scores", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "back-propagate to the generated point clouds and then further back-propagate to the weights of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "generator to update the generator. We design a diagnostic experiment that removes the generator and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "examines whether the discriminator gradients are informative enough for supervising point cloud", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 138, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 138, + 695 + ], + "score": 1.0, + "content": "GANs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 187, + 710 + ], + "score": 1.0, + "content": "Concretely, we take", + "type": "text" + }, + { + "bbox": [ + 188, + 700, + 200, + 708 + ], + "score": 0.82, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 698, + 401, + 710 + ], + "score": 1.0, + "content": "real chairs as real data and we randomly initialize", + "type": "text" + }, + { + "bbox": [ + 402, + 701, + 413, + 708 + ], + "score": 0.84, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 698, + 506, + 710 + ], + "score": 1.0, + "content": "learnable point clouds", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 152, + 722 + ], + "score": 0.9, + "content": "p \\in \\mathbb { R } ^ { N \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 708, + 263, + 722 + ], + "score": 1.0, + "content": "with i.i.d Gaussian noises", + "type": "text" + }, + { + "bbox": [ + 263, + 712, + 303, + 722 + ], + "score": 0.93, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 708, + 470, + 722 + ], + "score": 1.0, + "content": ". For each training iteration, we sample", + "type": "text" + }, + { + "bbox": [ + 471, + 712, + 479, + 720 + ], + "score": 0.79, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "point", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 339, + 732 + ], + "score": 1.0, + "content": "clouds from the real dataset as the real data and sample", + "type": "text" + }, + { + "bbox": [ + 339, + 723, + 347, + 730 + ], + "score": 0.89, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "point clouds from the learnable point", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 81, + 507, + 259 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 81, + 507, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 507, + 259 + ], + "score": 0.974, + "html": "
DatasetGeneratorPoolingFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
ChairFCFCMaxMix1.5710.1840.2110.2097.0300.10170.0016423.5672.75
2.1240.06740.0019673.6474.96
FCDeformDeformDeformTreeGANTreeGANGraph-CNNAttention0.6350.6724.9710.11560.0016068.9270.54
MaxMix0.9130.5340.2680.3735.6020.09080.06950.002010.0020068.576.2972.6175.11
2.836
AttentionMaxMixMax0.6960.7552.9870.11410.0016068.7769.36
1.4420.2931.0340.6547.8084.0320.09620.07040.001910.0021124,7474.8248.9073.4978.7963.18
0.334
0.9817.4940.08120.00191
PointFlowOracle1.7820.0881.4050.0933.2560.8140.13350.05940.001900.0016571.7679.5671.7379.69
Multi-CatFCFCFCMaxMix1.5530.3546.9810.08420.0015335.1664.16
0.2550.2852.5500.06560.001840.0013473.572.1972.1669.63
Attention0.4140.4534.2340.1188
DeformDeformDeformTreeGANTreeGANMaxMix1.0720.6333.8450.07990.0017962.564.5
0.6140.3492.4510.06700.0019170.8368.8369.60
Attention0.6160.7202.5310.11130.0014172.04
MaxMix1.7140.3880.4376.3420.10930.0015825.3372.6667.071.0
0.4204.3000.06990.00184
Oracle0.1310.1770.6730.060120.0012477.1477.57
", + "type": "table", + "image_path": "bb0b6d2514c2aef60eea70ee2a670c45a56e495ce19debe24e184b17ef906f14.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 81, + 507, + 140.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 140.33333333333334, + 507, + 199.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 199.66666666666669, + 507, + 259.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 263, + 507, + 306 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "Table 3: Evaluating PointNet-based discriminators with different generators. We observe that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "PointNet-Mix can significantly improve all sampling-aware/sensitive metric, including FPD-Mix,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 284, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 506, + 296 + ], + "score": 1.0, + "content": "FGD, MMD-EMD, COV-EMD for all the generators. When paired with FC generator, PointNet-Mix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "score": 1.0, + "content": "achieves the best performance. We include PointFlow(Yang et al., 2019) and Oracle for a comparison.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 503, + 350 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 327, + 505, + 352 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "In Figure 2, we present the generated point clouds for all the experiments with color-coding for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "the local point density. We see that the generators trained using PointNet-Max usually suffer from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "non-uniform density, except for the deformation generator. On the chair class, points are usually", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "clustering around the seat area while being sparse at the back. On the contrary, PointNet-Mix enforces", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "a globally uniform point density and hence greatly improves the visual quality of the generated point", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 406, + 289, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 289, + 420 + ], + "score": 1.0, + "content": "clouds. PointNet-Attention is in the between.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 355, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "With PointNet-Mix, we observe that the most naive FC generator works the best outperforming the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "previously state-of-the-state generators on almost of metrics. This showcases the importance of being", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "sampling-aware but not sampling-insensitive/oversensitive as a discriminator. It also suggests that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 455, + 446, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 446, + 467 + ], + "score": 1.0, + "content": "future works may focus on designing more powerful sampling-aware discriminators.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 423, + 506, + 467 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 371, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 372, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 372, + 488 + ], + "score": 1.0, + "content": "5.3 DIAGNOSING FAILURES FOR VARIOUS DISCRIMINATORS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 490, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "Observing that the discriminator forms the bottleneck of the point cloud GAN, it is important to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 326, + 514 + ], + "score": 1.0, + "content": "examine whether advanced networks, e.g. PointNet+", + "type": "text" + }, + { + "bbox": [ + 326, + 502, + 333, + 511 + ], + "score": 0.46, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "(Qi et al., 2017b), DGCNN (Wang et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "2019), KPConv (Thomas et al., 2019), PointConv (Wu et al., 2019) can outperform PointNet-based", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 521, + 164, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 164, + 534 + ], + "score": 1.0, + "content": "discriminator.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 490, + 506, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Surprisingly, we fail to observe any of them ever to be successful to train a GAN in our extensive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 547, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 507, + 561 + ], + "score": 1.0, + "content": "experiments, in which we perform a systematic search, including varying generator architecture,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "changing hyperparameters (e.g. the number of layers and parameters in discriminator, learning rate),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "whether use batch norm, etc. During GAN training, we observe the four advanced discriminator", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "behave very similarly: they are very discriminative in the sense that the gaps between the real and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "fake scores remain significantly large, but the quality of the generated point clouds stays very bad.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "The contradictory behaviors seem unreasonable at the first glance: how can a discriminator be very", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "discriminative but teach nothing to the generator? Thus, we design a diagnostic \"no generator\"", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 622, + 295, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 295, + 634 + ], + "score": 1.0, + "content": "experiment to examine the underlying reasons.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 537, + 507, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "No Generator Experiments. During GAN training, the gradients from discriminator output scores", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "back-propagate to the generated point clouds and then further back-propagate to the weights of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "generator to update the generator. We design a diagnostic experiment that removes the generator and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "examines whether the discriminator gradients are informative enough for supervising point cloud", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 138, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 138, + 695 + ], + "score": 1.0, + "content": "GANs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 640, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 187, + 710 + ], + "score": 1.0, + "content": "Concretely, we take", + "type": "text" + }, + { + "bbox": [ + 188, + 700, + 200, + 708 + ], + "score": 0.82, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 698, + 401, + 710 + ], + "score": 1.0, + "content": "real chairs as real data and we randomly initialize", + "type": "text" + }, + { + "bbox": [ + 402, + 701, + 413, + 708 + ], + "score": 0.84, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 698, + 506, + 710 + ], + "score": 1.0, + "content": "learnable point clouds", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 152, + 722 + ], + "score": 0.9, + "content": "p \\in \\mathbb { R } ^ { N \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 708, + 263, + 722 + ], + "score": 1.0, + "content": "with i.i.d Gaussian noises", + "type": "text" + }, + { + "bbox": [ + 263, + 712, + 303, + 722 + ], + "score": 0.93, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 708, + 470, + 722 + ], + "score": 1.0, + "content": ". For each training iteration, we sample", + "type": "text" + }, + { + "bbox": [ + 471, + 712, + 479, + 720 + ], + "score": 0.79, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "point", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 339, + 732 + ], + "score": 1.0, + "content": "clouds from the real dataset as the real data and sample", + "type": "text" + }, + { + "bbox": [ + 339, + 723, + 347, + 730 + ], + "score": 0.89, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "point clouds from the learnable point", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 505, + 343 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 505, + 343 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 79, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 505, + 343 + ], + "score": 0.934, + "type": "image", + "image_path": "9ab46f0bd251aa0ef4577907186cc5ad8e8d73246239da39a1cce07f60003cd8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 505, + 167.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 167.0, + 505, + 255.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 255.0, + 505, + 343.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 114, + 391, + 496, + 492 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 106, + 350, + 505, + 391 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "Figure 2: Visualization of point clouds generated by different methods. We show generated point", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "clouds in few exemplar shapes for a fair and easier comparison regarding their sampling quality and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "geometry quality. 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On the left, we show exemplar", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "point clouds generated by different discriminators. On the right, we plot the training curves for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 178, + 544 + ], + "score": 1.0, + "content": "experiments. 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DiscriminatorFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
PointNet-Max3.8282.45619.2790.10390.002872656
PointNet-Avg19.64019.60439.7930.11540.006423212
PointNet-Mix1.8352.20813.1970.08980.003315051
PointNet-Attention1.3501.64212.3590.16220.002715248
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DGCNN10.6299.93914.8820.09320.003964648
KPConv41.1549.9657.320.26950.0113562.5
PointConv27.6927.0546.820.28320.0072138
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DiscriminatorFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
PointNet-Max3.8282.45619.2790.10390.002872656
PointNet-Avg19.64019.60439.7930.11540.006423212
PointNet-Mix1.8352.20813.1970.08980.003315051
PointNet-Attention1.3501.64212.3590.16220.002715248
PointNet++15.08114.18733.9580.11550.005031937
DGCNN10.6299.93914.8820.09320.003964648
KPConv41.1549.9657.320.26950.0113562.5
PointConv27.6927.0546.820.28320.0072138
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Figure 3 (right) presents the training curves of the Wasserstein", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 166 + ], + "score": 1.0, + "content": "estimates, which essentially describe the gap between the real and fake scores. The large score", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 161, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 161, + 180, + 176 + ], + "score": 1.0, + "content": "gaps for PointNet", + "type": "text" + }, + { + "bbox": [ + 180, + 164, + 192, + 173 + ], + "score": 0.39, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 161, + 506, + 176 + ], + "score": 1.0, + "content": ", DGCNN and KPConv throughout the training indicate the two models are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "very discriminative in telling apart the fake samples from the real data. 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Note that both of them leverage relative point", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "positions/edge information during their feature extraction, which leads to a huge amount of gradients", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 218 + ], + "score": 1.0, + "content": "flowing along the surface direction focusing on changing sampling patterns instead of supervising", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "shape geometry. 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Note that training such \"no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 245, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 507, + 259 + ], + "score": 1.0, + "content": "generator\" experiment shares a similar flavor to the SGD sampling in introspective CNN(Jin et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 254, + 136, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 136, + 270 + ], + "score": 1.0, + "content": "2017).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 106, + 277, + 495, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 497, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 497, + 291 + ], + "score": 1.0, + "content": "6 CONCLUSION AND SUGGESTIONS FOR FUTURE DISCRIMINATOR DESIGN", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "In this work, we study the importance of sampling for 3D point cloud GAN design. We propose the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "sampling spectrum of discriminators and evaluation metrics that provide insights on the behaviors of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "point cloud GANs. We propose several empirical experiments for identifying the sampling sensitivity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "of a discriminator or an evaluation metric. 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For future discriminators, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 399, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 506, + 415 + ], + "score": 1.0, + "content": "suggest they should be more discriminative in shape and aware but not oversensitive to sampling", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "patterns. For the sanity check of any novel discriminators, one can leverage our proposed \"no", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 200, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 200, + 433 + ], + "score": 1.0, + "content": "generator\" experiment.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 104 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 506, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 123 + ], + "score": 1.0, + "content": "Results and Analysis. Table 4 summarizes the results and Figure 3 (left) shows exemplar generated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 133 + ], + "score": 1.0, + "content": "point clouds. 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Figure 3 (right) presents the training curves of the Wasserstein", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 166 + ], + "score": 1.0, + "content": "estimates, which essentially describe the gap between the real and fake scores. The large score", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 161, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 161, + 180, + 176 + ], + "score": 1.0, + "content": "gaps for PointNet", + "type": "text" + }, + { + "bbox": [ + 180, + 164, + 192, + 173 + ], + "score": 0.39, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 161, + 506, + 176 + ], + "score": 1.0, + "content": ", DGCNN and KPConv throughout the training indicate the two models are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "very discriminative in telling apart the fake samples from the real data. 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For the sanity check of any novel discriminators, one can leverage our proposed \"no", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 200, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 200, + 433 + ], + "score": 1.0, + "content": "generator\" experiment.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 295, + 506, + 433 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 506, + 131 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 111 + ], + "score": 1.0, + "content": "Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. 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ExperimentPN-MaxPN-AvgPN-MixPN-AttPN++DGCNNPointConvKPConv
Clustering art.56%/53%98%/99%97%/96%94%/93%100%/98%100%/99%100%/99%100%/99%
FPS vs.uniform50%/50%50.3%/50%51.5%/50%50.1%/50%100%/97%100%/96%100%/98%100%/99%
ESS0.030.490.460.430.950.950.970.99
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DatasetGeneratorPoolingFPD-Mix↓FPD-Max↓FGD↓MMD-E↓MMD-C↓COV-E↑COV-C↑
ChairFCFCMaxMix1.5710.1840.2110.2097.0300.10170.0016423.5672.75
2.1240.06740.0019673.6474.96
FCDeformDeformDeformTreeGANTreeGANGraph-CNNAttention0.6350.6724.9710.11560.0016068.9270.54
MaxMix0.9130.5340.2680.3735.6020.09080.06950.002010.0020068.576.2972.6175.11
2.836
AttentionMaxMixMax0.6960.7552.9870.11410.0016068.7769.36
1.4420.2931.0340.6547.8084.0320.09620.07040.001910.0021124,7474.8248.9073.4978.7963.18
0.334
0.9817.4940.08120.00191
PointFlowOracle1.7820.0881.4050.0933.2560.8140.13350.05940.001900.0016571.7679.5671.7379.69
Multi-CatFCFCFCMaxMix1.5530.3546.9810.08420.0015335.1664.16
0.2550.2852.5500.06560.001840.0013473.572.1972.1669.63
Attention0.4140.4534.2340.1188
DeformDeformDeformTreeGANTreeGANMaxMix1.0720.6333.8450.07990.0017962.564.5
0.6140.3492.4510.06700.0019170.8368.8369.60
Attention0.6160.7202.5310.11130.0014172.04
MaxMix1.7140.3880.4376.3420.10930.0015825.3372.6667.071.0
0.4204.3000.06990.00184
Oracle0.1310.1770.6730.060120.0012477.1477.57
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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. + +
MethodsessionsThe gap with cRT
123456789
cRT[25]*65.1863.8960.2057.2353.7150.3948.7747.2945.28=
Joint-training*65.1861.4557.3653.6850.8447.3344.7942.6240.08-5.20
Baseline65.1861.6758.6155.1151.8649.4347.6045.6443.83-1.45
iCaRL[40]*66.5257.2654.2750.6247.3344.9943.1441.1639.49-5.79
Rebalance [20]*66.6661.4257.2953.0248.8545.6843.0640.5638.35-6.93
FSLL [34]*65.1856.2454.5551.6149.1147.2745.3543.9542.22-3.08
iCaRL[40]64.1053.2841.6934.1327.9325.0620.4115.4813.73-31.55
Rebalance [20]64.1053.0543.9636.9731.6126.7321.2316.7813.54-31.74
TOPIC [47]64.1055.8847.0745.1640.1136.3833.9631.5529.37-15.91
FSLL [34]64.1055.8551.7148.5945.3443.2541.5239.8138.16-7.12
FSLL+SS [34]66.7655.5252.2049.1746.2344.6443.0741.2039.57-5.71
F2M64.7162.0559.0155.5852.5549.9648.0846.2844.67-0.61
+ +Table 2: Classification accuracy on miniImageNet for 5-way 5-shot incremental learning. ∗ indicates our re-implementation. + +
MethodsessionsThe gap with cRT
123456789
cRT[25]*67.3064.1560.5957.3254.2251.4348.9246.7844.85=
Joint-training*67.3062.3457.7954.0850.9347.6544.6442.6140.29-4.56
Baseline67.3063.1859.6256.3353.2850.5047.9645.8543.88-0.97
iCaRL [40]*67.3559.9155.6452.6049.4346.7344.1342.1740.29-4.56
Rebalance [20]*67.9163.1158.7554.8350.6847.1143.8841.1938.72-6.13
FSLL [34]*67.3059.8157.2654.5752.0549.4246.9544.9442.87-1.11
iCaRL[40]61.3146.3242.9437.6330.4924.0020.8918.8017.21-27.64
Rebalance [20]61.3147.8039.3131.9125.6821.3518.6717.2414.17-30.68
TOPIC [47]61.3150.0945.1741.1637.4835.5232.1929.4624.42-20.43
FSLL [34]66.4861.7558.1654.1651.1048.5346.5444.2042.28-2.57
FSLL+SS[34]68.8563.1459.2455.2352.2449.6547.7445.2343.92-0.93
IDLVQ-C[8]64.7759.8755.9352.6249.8847.5544.8343.1441.84-3.01
F2M67.2863.8060.3857.0654.0851.3948.8246.5844.65-0.20
+ +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. + +
MethodsessionsThe gap with cRT
1234567891011
cRT[25]*80.8378.5176.1273.9371.4668.9667.7366.7564.2262.5361.08
Joint-training*80.8377.5774.1170.7568.5265.9764.5862.2260.1858.4956.78-4.30
Baseline80.8777.1574.4672.2669.4767.1865.6263.6861.3059.7258.12-2.96
iCaRL [40]*79.5867.6364.1761.8058.1055.5153.3450.8948.6247.3445.60-15.48
Rebalance [20]*80.9470.3262.9657.1951.0646.7044.0340.1536.7534.8832.09-28.99
FSLL [34]*80.8377.3872.3771.8467.5165.3063.7561.1659.0558.0355.82-5.26
iCaRL [40]68.6852.6548.6144.1636.6229.5227.8326.2624.0123.8921.16-39.92
Rebalance [20]68.6857.1244.2128.7826.7125.6624.6221.5220.1220.0619.87-41.21
TOPIC[47]68.6862.4954.8149.9945.2541.4038.3535.3632.2228.3126.28-34.80
FSLL [34]72.7769.3365.5162.6661.1058.6557.7857.2655.5955.3954.21-6.87
FSLL+SS [34]75.6371.8168.1664.3262.6160.1058.8258.7056.4556.4155.82-5.26
IDLVQ-C [8]77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81-3.27
F2M81.0778.1675.5772.8970.8668.1767.0165.2663.3661.7660.26-0.82
+ +Table 4: Comparison of the flatness of the local minima found by the Baseline and our F2M. + +
MethodIndicator IVariance g2
Training SetTesting SetTraining SetTesting Set
Baseline0.29930.45820.14510.2395
F2M0.05060.08000.02960.0334
+ +# 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. + +
FMPFPCPNsessionsPD↓
123456789
广65.1860.8353.1343.5723.7510.7608.2607.2406.4558.73
√65.1859.4856.7752.9950.0947.8045.9244.2042.5522.63
√√64.7159.5453.0345.0941.6839.0438.6437.1936.0128.70
√√64.5561.2758.3354.8251.6049.2247.4845.7844.0820.47
√√√64.7161.7558.8055.3352.2749.7547.7246.0144.4320.28
√√√64.7161.9958.9955.5852.5549.9648.0846.2844.6720.04
+ +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. + +
SessionThe hyperparameter b
0.00250.0050.010.020.040.08
Session 1(60 bases classes)64.8564.6764.8164.7163.3062.25
Session 9 (All 100 classes)44.1644.5444.5844.6743.7543.04
Session 9 (60 base classes)59.5859.6959.7359.4458.3857.21
Session 9 (40 new classes)21.0321.8121.8622.5221.8021.77
+ +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. + +References +[1] Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. In Proceedings of the European Conference on Computer Vision (ECCV), pages 139–154, 2018. +[2] Carlo Baldassi, Fabrizio Pittorino, and Riccardo Zecchina. Shaping the learning landscape in neural networks around wide flat minima. Proc. Natl. 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[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] \ No newline at end of file diff --git a/parse/train/ALvt7nXa2q/ALvt7nXa2q_content_list.json b/parse/train/ALvt7nXa2q/ALvt7nXa2q_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a45bea3471eaec9d485970cd49b30d932999cee7 --- /dev/null +++ b/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": [ + 184, + 122, + 815, + 172 + ], + "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, + 767, + 542 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 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": [ + 174, + 601, + 825, + 696 + ], + "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": [ + 174, + 704, + 825, + 787 + ], + "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": [ + 174, + 792, + 825, + 863 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 147 + ], + "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": [ + 173, + 154, + 825, + 291 + ], + "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": [ + 217, + 304, + 825, + 454 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 473, + 321, + 489 + ], + "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": [ + 173, + 503, + 825, + 628 + ], + "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": [ + 174, + 635, + 825, + 760 + ], + "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": [ + 174, + 766, + 825, + 877 + ], + "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": [ + 173, + 883, + 821, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Severity of Catastrophic Forgetting in Incremental Few-Shot Learning ", + "text_level": 1, + "bbox": [ + 173, + 171, + 794, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Problem Statement ", + "text_level": 1, + "bbox": [ + 174, + 205, + 346, + 220 + ], + "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, + 825, + 404 + ], + "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": [ + 174, + 409, + 825, + 506 + ], + "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, + 555, + 823, + 583 + ], + "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": [ + 174, + 589, + 823, + 632 + ], + "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, + 825, + 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, + 723, + 741 + ], + "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, + 823, + 795 + ], + "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, + 800, + 825, + 911 + ], + "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, + 88, + 816, + 234 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 303, + 823, + 332 + ], + "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": [ + 217, + 342, + 826, + 433 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Overcoming Catastrophic Forgetting in IFL by Finding Flat Minima ", + "text_level": 1, + "bbox": [ + 173, + 457, + 777, + 476 + ], + "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": [ + 173, + 488, + 825, + 531 + ], + "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, + 536, + 826, + 621 + ], + "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": [ + 313, + 627, + 681, + 662 + ], + "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": [ + 173, + 670, + 825, + 727 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 Searching for Flat Local Minima in the Base Training Stage ", + "text_level": 1, + "bbox": [ + 174, + 742, + 629, + 758 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A formal definition of $b$ -flat local minima is given as follows. ", + "bbox": [ + 173, + 768, + 575, + 784 + ], + "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": [ + 173, + 787, + 826, + 818 + ], + "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 $\\mathbf { b } _ { i } = b$ . ", + "bbox": [ + 217, + 827, + 697, + 844 + ], + "page_idx": 3 + }, + { + "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": [ + 212, + 851, + 823, + 881 + ], + "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": [ + 179, + 90, + 810, + 239 + ], + "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": [ + 173, + 387, + 825, + 472 + ], + "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": [ + 174, + 477, + 825, + 534 + ], + "page_idx": 4 + }, + { + "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, + 722, + 575 + ], + "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": [ + 178, + 577, + 825, + 618 + ], + "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, + 642, + 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": [ + 173, + 703, + 825, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 Incremental Few-shot Learning within the Flat Region ", + "text_level": 1, + "bbox": [ + 173, + 815, + 593, + 832 + ], + "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": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1: F2M ", + "text_level": 1, + "bbox": [ + 174, + 94, + 302, + 108 + ], + "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": [ + 171, + 113, + 802, + 431 + ], + "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, + 823, + 484 + ], + "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": [ + 173, + 530, + 826, + 669 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 Convergence Analysis ", + "text_level": 1, + "bbox": [ + 174, + 684, + 366, + 699 + ], + "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": [ + 173, + 709, + 825, + 792 + ], + "page_idx": 5 + }, + { + "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": [ + 173, + 797, + 823, + 828 + ], + "page_idx": 5 + }, + { + "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, + 766, + 875 + ], + "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": [ + 173, + 882, + 823, + 912 + ], + "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": [ + 171, + 90, + 823, + 119 + ], + "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": [ + 173, + 156, + 826, + 185 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Assumption 4.2. The expected loss function satisfies the following conditions: ", + "bbox": [ + 176, + 189, + 692, + 204 + ], + "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, + 214, + 816, + 232 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• Condition 2: For all $k \\in \\mathbb N$ and $j \\in [ 1 , M ]$ , ", + "bbox": [ + 218, + 238, + 517, + 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, + 655, + 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, + 784, + 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": [ + 173, + 344, + 825, + 489 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Assumption 4.3. The learning rates satisfy: ", + "bbox": [ + 174, + 492, + 467, + 507 + ], + "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": [ + 401, + 513, + 593, + 555 + ], + "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": [ + 171, + 568, + 825, + 597 + ], + "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, + 826, + 630 + ], + "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, + 584, + 660 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This theorem establishes the convergence of our algorithm. The proof is provided in Appendix A.1. ", + "bbox": [ + 171, + 671, + 823, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Experiments ", + "text_level": 1, + "bbox": [ + 173, + 705, + 313, + 723 + ], + "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": [ + 173, + 737, + 825, + 766 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 781, + 352, + 796 + ], + "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": [ + 173, + 806, + 825, + 877 + ], + "page_idx": 6 + }, + { + "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, + 911 + ], + "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": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*65.1863.8960.2057.2353.7150.3948.7747.2945.28=
Joint-training*65.1861.4557.3653.6850.8447.3344.7942.6240.08-5.20
Baseline65.1861.6758.6155.1151.8649.4347.6045.6443.83-1.45
iCaRL[40]*66.5257.2654.2750.6247.3344.9943.1441.1639.49-5.79
Rebalance [20]*66.6661.4257.2953.0248.8545.6843.0640.5638.35-6.93
FSLL [34]*65.1856.2454.5551.6149.1147.2745.3543.9542.22-3.08
iCaRL[40]64.1053.2841.6934.1327.9325.0620.4115.4813.73-31.55
Rebalance [20]64.1053.0543.9636.9731.6126.7321.2316.7813.54-31.74
TOPIC [47]64.1055.8847.0745.1640.1136.3833.9631.5529.37-15.91
FSLL [34]64.1055.8551.7148.5945.3443.2541.5239.8138.16-7.12
FSLL+SS [34]66.7655.5252.2049.1746.2344.6443.0741.2039.57-5.71
F2M64.7162.0559.0155.5852.5549.9648.0846.2844.67-0.61
", + "bbox": [ + 209, + 125, + 790, + 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": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*67.3064.1560.5957.3254.2251.4348.9246.7844.85=
Joint-training*67.3062.3457.7954.0850.9347.6544.6442.6140.29-4.56
Baseline67.3063.1859.6256.3353.2850.5047.9645.8543.88-0.97
iCaRL [40]*67.3559.9155.6452.6049.4346.7344.1342.1740.29-4.56
Rebalance [20]*67.9163.1158.7554.8350.6847.1143.8841.1938.72-6.13
FSLL [34]*67.3059.8157.2654.5752.0549.4246.9544.9442.87-1.11
iCaRL[40]61.3146.3242.9437.6330.4924.0020.8918.8017.21-27.64
Rebalance [20]61.3147.8039.3131.9125.6821.3518.6717.2414.17-30.68
TOPIC [47]61.3150.0945.1741.1637.4835.5232.1929.4624.42-20.43
FSLL [34]66.4861.7558.1654.1651.1048.5346.5444.2042.28-2.57
FSLL+SS[34]68.8563.1459.2455.2352.2449.6547.7445.2343.92-0.93
IDLVQ-C[8]64.7759.8755.9352.6249.8847.5544.8343.1441.84-3.01
F2M67.2863.8060.3857.0654.0851.3948.8246.5844.65-0.20
", + "bbox": [ + 209, + 382, + 790, + 578 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 627, + 825, + 752 + ], + "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": "
MethodsessionsThe gap with cRT
1234567891011
cRT[25]*80.8378.5176.1273.9371.4668.9667.7366.7564.2262.5361.08
Joint-training*80.8377.5774.1170.7568.5265.9764.5862.2260.1858.4956.78-4.30
Baseline80.8777.1574.4672.2669.4767.1865.6263.6861.3059.7258.12-2.96
iCaRL [40]*79.5867.6364.1761.8058.1055.5153.3450.8948.6247.3445.60-15.48
Rebalance [20]*80.9470.3262.9657.1951.0646.7044.0340.1536.7534.8832.09-28.99
FSLL [34]*80.8377.3872.3771.8467.5165.3063.7561.1659.0558.0355.82-5.26
iCaRL [40]68.6852.6548.6144.1636.6229.5227.8326.2624.0123.8921.16-39.92
Rebalance [20]68.6857.1244.2128.7826.7125.6624.6221.5220.1220.0619.87-41.21
TOPIC[47]68.6862.4954.8149.9945.2541.4038.3535.3632.2228.3126.28-34.80
FSLL [34]72.7769.3365.5162.6661.1058.6557.7857.2655.5955.3954.21-6.87
FSLL+SS [34]75.6371.8168.1664.3262.6160.1058.8258.7056.4556.4155.82-5.26
IDLVQ-C [8]77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81-3.27
F2M81.0778.1675.5772.8970.8668.1767.0165.2663.3661.7660.26-0.82
", + "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": "
MethodIndicator IVariance g2
Training SetTesting SetTraining SetTesting Set
Baseline0.29930.45820.14510.2395
F2M0.05060.08000.02960.0334
", + "bbox": [ + 272, + 349, + 723, + 421 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2 Comparison with the State-of-the-Art ", + "text_level": 1, + "bbox": [ + 174, + 449, + 475, + 463 + ], + "page_idx": 8 + }, + { + "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": [ + 173, + 474, + 826, + 641 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 Ablation Study and Analysis ", + "text_level": 1, + "bbox": [ + 174, + 659, + 411, + 674 + ], + "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, + 825, + 787 + ], + "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, + 825, + 876 + ], + "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, + 911 + ], + "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": "
FMPFPCPNsessionsPD↓
123456789
广65.1860.8353.1343.5723.7510.7608.2607.2406.4558.73
√65.1859.4856.7752.9950.0947.8045.9244.2042.5522.63
√√64.7159.5453.0345.0941.6839.0438.6437.1936.0128.70
√√64.5561.2758.3354.8251.6049.2247.4845.7844.0820.47
√√√64.7161.7558.8055.3352.2749.7547.7246.0144.4320.28
√√√64.7161.9958.9955.5852.5549.9648.0846.2844.6720.04
", + "bbox": [ + 202, + 111, + 795, + 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": "
SessionThe hyperparameter b
0.00250.0050.010.020.040.08
Session 1(60 bases classes)64.8564.6764.8164.7163.3062.25
Session 9 (All 100 classes)44.1644.5444.5844.6743.7543.04
Session 9 (60 base classes)59.5859.6959.7359.4458.3857.21
Session 9 (40 new classes)21.0321.8121.8622.5221.8021.77
", + "bbox": [ + 240, + 272, + 756, + 371 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 396, + 823, + 439 + ], + "page_idx": 9 + }, + { + "type": "text", + "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 $0 . 6 \\%$ . ", + "bbox": [ + 174, + 444, + 823, + 513 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 520, + 825, + 563 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 569, + 825, + 611 + ], + "page_idx": 9 + }, + { + "type": "text", + "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 $b$ . ", + "bbox": [ + 174, + 616, + 825, + 714 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 733, + 299, + 750 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 763, + 825, + 904 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 88, + 553, + 107 + ], + "page_idx": 10 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 121, + 826, + 150 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References \n[1] Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. In Proceedings of the European Conference on Computer Vision (ECCV), pages 139–154, 2018. \n[2] Carlo Baldassi, Fabrizio Pittorino, and Riccardo Zecchina. Shaping the learning landscape in neural networks around wide flat minima. Proc. Natl. Acad. Sci. 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", + "bbox": [ + 169, + 73, + 826, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 828, + 731 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 174, + 755, + 254, + 771 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 782, + 339, + 796 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See Section 6. \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 801, + 825, + 893 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 897, + 493, + 911 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4.3. \n(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1. ", + "bbox": [ + 236, + 90, + 823, + 122 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 127, + 393, + 141 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The link to the source code is provided in the abstract. \n(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. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Appendix A.3. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 5.1 ", + "bbox": [ + 238, + 145, + 825, + 277 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 212, + 281, + 825, + 296 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We have cited [29, 49, 50]. \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 300, + 825, + 435 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 439, + 705, + 454 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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Our analysis further suggests that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 343, + 471, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 471, + 355 + ], + "score": 1.0, + "content": "to prevent catastrophic forgetting, actions need to be taken in the primitive stage –", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 354, + 470, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 366 + ], + "score": 1.0, + "content": "the training of base classes instead of later few-shot learning sessions. 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In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 387, + 471, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 387, + 471, + 399 + ], + "score": 1.0, + "content": "this way, the model can efficiently learn new classes while preserving the old ones.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 397, + 470, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 470, + 410 + ], + "score": 1.0, + "content": "Comprehensive experimental results demonstrate that our approach outperforms", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 408, + 471, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 471, + 421 + ], + "score": 1.0, + "content": "all prior state-of-the-art methods and is very close to the approximate upper bound.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 419, + 450, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 419, + 450, + 432 + ], + "score": 1.0, + "content": "The source code is available at https://github.com/moukamisama/F2M.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 288, + 471, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 191, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 192, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 192, + 466 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "Why study incremental few-shot learning? Incremental learning enables a model to continually", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "learn new concepts from new data without forgetting previously learned knowledge. Rooted from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "real-world applications, this topic has attracted a significant amount of interest in recent years [5, 30,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "31, 40, 26]. Incremental learning assumes sufficient training data is provided for new classes, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "is impractical in many application scenarios, especially when the new classes are rare categories", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "which are costly or difficult to collect. This motivates the study of incremental few-shot learning, a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 470, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 470, + 554 + ], + "score": 1.0, + "content": "more difficult paradigm that aims to continually learn new tasks with only a few examples.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 475, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "Challenges. The major challenge for incremental learning is catastrophic forgetting [14, 28, 35],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "which refers to the drastic performance drop on previous tasks after learning new tasks. This", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "phenomenon is caused by the inaccessibility to previous data while learning on new data. Catastrophic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "forgetting presents a bigger challenge for incremental few-shot learning. Due to the small amount of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "training data in new tasks, the model tends to severely overfit on new classes while quickly forgetting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 613, + 307, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 307, + 625 + ], + "score": 1.0, + "content": "old classes, resulting in catastrophic performance.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 557, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "Current research. The study of incremental few-shot learning has just started [47, 41, 60, 9, 8,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "34, 59]. Current works mainly borrow ideas from research in incremental learning to overcome the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "forgetting problem, by enforcing strong constraints on model parameters to penalize the changes", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 676 + ], + "score": 1.0, + "content": "of parameters [34, 28, 56], or by saving a small amount of exemplars from old classes and adding", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "constraints on the exemplars to avoid forgetting [40, 20, 4]. However, in our empirical study, we find", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "that an intransigent model that only trains on base classes and does not tune on new tasks consistently", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "outperforms state-of-the-art methods, including a joint-training method [47] that uses all encountered", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "data for training and hence suffers from severe data imbalance. This observation motivates us to", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 307, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 307, + 117 + ], + "score": 1.0, + "content": "address this harsh problem from a different angle.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 629, + 506, + 685 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "that an intransigent model that only trains on base classes and does not tune on new tasks consistently", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "outperforms state-of-the-art methods, including a joint-training method [47] that uses all encountered", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "data for training and hence suffers from severe data imbalance. This observation motivates us to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 307, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 307, + 117 + ], + "score": 1.0, + "content": "address this harsh problem from a different angle.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "Our solution. Unlike existing solutions that try to overcome the catastrophic forgetting problem", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "during the process of learning new tasks, we adopt a different approach by considering this issue", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "during the training of base classes. Specifically, we propose to search for flat local minima of the base", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "training objective function. For any parameter vector in the flat region around the minima, the loss is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "small, and the base classes are supposed to be well separated. The flat local minima can be found", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "by adding random noise to the model parameters for multiple times and jointly optimizing multiple", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "loss functions. During the following incremental few-shot learning stage, we fine-tune the model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "parameters within the flat region, which can be achieved by clamping the parameters after updating", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "them on few-shot tasks. In this way, the model can efficiently learn new classes while preserving the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 347, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 347, + 232 + ], + "score": 1.0, + "content": "old ones. Our key contributions are summarized as follows:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 133, + 241, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 132, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 132, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "• We conduct a comprehensive empirical study on existing incremental few-shot learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "methods and discover that a simple baseline model that only trains on base classes outper-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "forms state-of-the-art methods, which demonstrates the severity of catastrophic forgetting.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 136, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 136, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "• We propose a novel approach for incremental few-shot learning by addressing the catas-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "trophic forgetting problem in the primitive stage. Through finding the flat minima region", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "during training on base classes and fine-tuning within the region while learning on new", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 311, + 447, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 447, + 325 + ], + "score": 1.0, + "content": "tasks, our model can overcome catastrophic forgetting and avoid overfitting.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 135, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 135, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "• Comprehensive experimental results on CIFAR-100, miniImageNet, and CUB-200-2011", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "show that our approach outperforms all state-of-the-art methods and achieves performance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 348, + 344, + 361 + ], + "spans": [ + { + "bbox": [ + 142, + 348, + 344, + 361 + ], + "score": 1.0, + "content": "that is very close to the approximate upper bound.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 375, + 197, + 388 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 198, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 198, + 390 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "Few-shot learning aims to learn to generalize to new categories with a few labeled samples in each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "class. Current few-shot methods mainly include optimization-based methods [12, 23, 32, 39, 45,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "46, 55] and metric-based methods [13, 19, 44, 49, 52, 58, 57, 53]. Optimization-based methods", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "can achieve fast adaptation to new tasks with limited samples by learning a specific optimization", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "algorithm. Metric-based approaches exploit different distance metrics such as L2 distance [44],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "cosine similarity [49], and DeepEMD [58] in the learned metric/embedding space to measure the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "similarity between samples. Recently, Tian et al. [48] find that standard supervised training can learn", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "a good metric space for unseen classes, which echoes with our observation on the proposed baseline", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 173, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 173, + 498 + ], + "score": 1.0, + "content": "model in Sec. 3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "Incremental learning focuses on the challenging problem of continually learning to recognize", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "new classes in new coming data without forgetting old classes [6, 7, 10, 51]. Previous research", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "mainly includes multi-class incremental learning [4, 38, 22, 33, 54, 51] and multi-task incremental", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "learning [21, 31, 42]. To overcome the catastrophic forgetting problem, some attempts propose to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "impose strong constraints on model parameters by penalizing the changes of parameters [28, 1].", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 557, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 572 + ], + "score": 1.0, + "content": "Other attempts try to enforce constraints on the exemplars of old classes by restricting the output", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "logits [40] or penalizing the changes of embedding angles [20]. In this work, our empirical study", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 578, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 594 + ], + "score": 1.0, + "content": "shows that imposing strong constraints on the arriving new classes may not be a promising way to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 461, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 461, + 604 + ], + "score": 1.0, + "content": "tackle incremental few-shot learning, due to the scarcity of training data for new classes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 507, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 507, + 620 + ], + "score": 1.0, + "content": "Incremental few-shot learning [47, 41, 60, 9, 8] aims to incrementally learn from very few samples.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "TOPCI [47] proposes a neural gas network to learn and preserve the topology of the feature manifold", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "formed by different classes. FSLL [34] only selects few model parameters for incremental learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 638, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 507, + 654 + ], + "score": 1.0, + "content": "and ensures the parameters are close to the optimal ones. To overcome catastrophic forgetting,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "score": 1.0, + "content": "IDLVQC [8] imposes constraints on the saved exemplars of each class by restricting the embedding", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "drift, and Zhang et al. [59] propose to fix the embedding network for incremental learning. Similar to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "the finding of Zhang et al., we also discover that an intransigent model that simply does not adapt to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 684, + 334, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 334, + 695 + ], + "score": 1.0, + "content": "new tasks can outperform prior state-of-the-art methods.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "Robust optimization. It has been found that flat local minima leads to better generalization capabili-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "ties than sharp minima in the sense that a flat minimizer is more robust when the test loss is shifted", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 505, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "Our solution. Unlike existing solutions that try to overcome the catastrophic forgetting problem", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "during the process of learning new tasks, we adopt a different approach by considering this issue", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "during the training of base classes. Specifically, we propose to search for flat local minima of the base", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "training objective function. For any parameter vector in the flat region around the minima, the loss is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "small, and the base classes are supposed to be well separated. The flat local minima can be found", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "by adding random noise to the model parameters for multiple times and jointly optimizing multiple", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "loss functions. During the following incremental few-shot learning stage, we fine-tune the model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "parameters within the flat region, which can be achieved by clamping the parameters after updating", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "them on few-shot tasks. In this way, the model can efficiently learn new classes while preserving the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 347, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 347, + 232 + ], + "score": 1.0, + "content": "old ones. Our key contributions are summarized as follows:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 121, + 506, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 241, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 132, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 132, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "• We conduct a comprehensive empirical study on existing incremental few-shot learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "methods and discover that a simple baseline model that only trains on base classes outper-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "forms state-of-the-art methods, which demonstrates the severity of catastrophic forgetting.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 136, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 136, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "• We propose a novel approach for incremental few-shot learning by addressing the catas-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "trophic forgetting problem in the primitive stage. Through finding the flat minima region", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "during training on base classes and fine-tuning within the region while learning on new", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 311, + 447, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 447, + 325 + ], + "score": 1.0, + "content": "tasks, our model can overcome catastrophic forgetting and avoid overfitting.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 135, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 135, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "• Comprehensive experimental results on CIFAR-100, miniImageNet, and CUB-200-2011", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "show that our approach outperforms all state-of-the-art methods and achieves performance", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 348, + 344, + 361 + ], + "spans": [ + { + "bbox": [ + 142, + 348, + 344, + 361 + ], + "score": 1.0, + "content": "that is very close to the approximate upper bound.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 132, + 240, + 506, + 361 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 375, + 197, + 388 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 198, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 198, + 390 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "Few-shot learning aims to learn to generalize to new categories with a few labeled samples in each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "class. Current few-shot methods mainly include optimization-based methods [12, 23, 32, 39, 45,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "46, 55] and metric-based methods [13, 19, 44, 49, 52, 58, 57, 53]. Optimization-based methods", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "can achieve fast adaptation to new tasks with limited samples by learning a specific optimization", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "algorithm. Metric-based approaches exploit different distance metrics such as L2 distance [44],", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "cosine similarity [49], and DeepEMD [58] in the learned metric/embedding space to measure the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "similarity between samples. Recently, Tian et al. [48] find that standard supervised training can learn", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "a good metric space for unseen classes, which echoes with our observation on the proposed baseline", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 173, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 173, + 498 + ], + "score": 1.0, + "content": "model in Sec. 3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 400, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "Incremental learning focuses on the challenging problem of continually learning to recognize", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "new classes in new coming data without forgetting old classes [6, 7, 10, 51]. Previous research", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "mainly includes multi-class incremental learning [4, 38, 22, 33, 54, 51] and multi-task incremental", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "learning [21, 31, 42]. To overcome the catastrophic forgetting problem, some attempts propose to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "impose strong constraints on model parameters by penalizing the changes of parameters [28, 1].", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 557, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 572 + ], + "score": 1.0, + "content": "Other attempts try to enforce constraints on the exemplars of old classes by restricting the output", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "logits [40] or penalizing the changes of embedding angles [20]. In this work, our empirical study", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 578, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 594 + ], + "score": 1.0, + "content": "shows that imposing strong constraints on the arriving new classes may not be a promising way to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 461, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 461, + 604 + ], + "score": 1.0, + "content": "tackle incremental few-shot learning, due to the scarcity of training data for new classes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 504, + 506, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 507, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 507, + 620 + ], + "score": 1.0, + "content": "Incremental few-shot learning [47, 41, 60, 9, 8] aims to incrementally learn from very few samples.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "TOPCI [47] proposes a neural gas network to learn and preserve the topology of the feature manifold", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "formed by different classes. FSLL [34] only selects few model parameters for incremental learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 638, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 507, + 654 + ], + "score": 1.0, + "content": "and ensures the parameters are close to the optimal ones. To overcome catastrophic forgetting,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 664 + ], + "score": 1.0, + "content": "IDLVQC [8] imposes constraints on the saved exemplars of each class by restricting the embedding", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "drift, and Zhang et al. [59] propose to fix the embedding network for incremental learning. Similar to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "the finding of Zhang et al., we also discover that an intransigent model that simply does not adapt to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 684, + 334, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 334, + 695 + ], + "score": 1.0, + "content": "new tasks can outperform prior state-of-the-art methods.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 606, + 507, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "Robust optimization. It has been found that flat local minima leads to better generalization capabili-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "ties than sharp minima in the sense that a flat minimizer is more robust when the test loss is shifted", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "due to random perturbations [18, 17, 24]. A substantial body of methods [2, 37, 11, 15] have been", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "proposed to optimize neural networks towards flat local minima. In this paper, we show that for", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "incremental few-shot learning, finding flat minima in the base session and tuning the model within", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 416, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 416, + 119 + ], + "score": 1.0, + "content": "the flat region on new tasks can significantly mitigate catastrophic forgetting.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 51.5, + "bbox_fs": [ + 106, + 700, + 505, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "due to random perturbations [18, 17, 24]. A substantial body of methods [2, 37, 11, 15] have been", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "proposed to optimize neural networks towards flat local minima. In this paper, we show that for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "incremental few-shot learning, finding flat minima in the base session and tuning the model within", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 416, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 416, + 119 + ], + "score": 1.0, + "content": "the flat region on new tasks can significantly mitigate catastrophic forgetting.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 106, + 136, + 486, + 150 + ], + "lines": [ + { + "bbox": [ + 104, + 135, + 487, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 135, + 487, + 153 + ], + "score": 1.0, + "content": "3 Severity of Catastrophic Forgetting in Incremental Few-Shot Learning", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 163, + 212, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 214, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 214, + 176 + ], + "score": 1.0, + "content": "3.1 Problem Statement", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Incremental few-shot learning (IFL) aims to continually learn to recognize new classes with only few", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 281, + 210 + ], + "score": 1.0, + "content": "examples. Similar to incremental learning", + "type": "text" + }, + { + "bbox": [ + 281, + 196, + 298, + 207 + ], + "score": 0.52, + "content": "\\left( \\operatorname { I L } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 195, + 505, + 210 + ], + "score": 1.0, + "content": ", an IFL model is trained by a sequence of training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 204, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 204, + 143, + 222 + ], + "score": 1.0, + "content": "sessions", + "type": "text" + }, + { + "bbox": [ + 143, + 207, + 201, + 218 + ], + "score": 0.91, + "content": "\\{ \\mathcal { D } ^ { 1 } , \\cdots , \\mathcal { D } ^ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 204, + 233, + 222 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 233, + 207, + 325, + 219 + ], + "score": 0.92, + "content": "\\mathcal { D } ^ { t } = \\{ z _ { i } = ( x _ { i } ^ { t } , y _ { i } ^ { t } ) \\} _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 204, + 448, + 222 + ], + "score": 1.0, + "content": "is the training data of session", + "type": "text" + }, + { + "bbox": [ + 448, + 208, + 453, + 217 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 204, + 471, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 207, + 482, + 219 + ], + "score": 0.89, + "content": "\\ v x _ { i } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 204, + 506, + 222 + ], + "score": 1.0, + "content": "is an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 177, + 231 + ], + "score": 1.0, + "content": "example of class", + "type": "text" + }, + { + "bbox": [ + 177, + 218, + 209, + 230 + ], + "score": 0.91, + "content": "y _ { i } ^ { t } \\in \\mathcal { C } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 217, + 308, + 231 + ], + "score": 1.0, + "content": "(the class set of session", + "type": "text" + }, + { + "bbox": [ + 309, + 220, + 313, + 228 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 217, + 421, + 231 + ], + "score": 1.0, + "content": "). 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Similar to", + "type": "text" + }, + { + "bbox": [ + 394, + 262, + 405, + 272 + ], + "score": 0.52, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 260, + 505, + 274 + ], + "score": 1.0, + "content": ", in each training session", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 111, + 282 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 272, + 334, + 285 + ], + "score": 1.0, + "content": "of IFL, the model has only access to the training data", + "type": "text" + }, + { + "bbox": [ + 334, + 272, + 347, + 282 + ], + "score": 0.88, + "content": "\\mathcal { D } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "and possibly a small amount of saved", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 356, + 295 + ], + "score": 1.0, + "content": "exemplars from previous sessions. 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IFL is undoubtedly a more challenging problem than", + "type": "text" + }, + { + "bbox": [ + 430, + 325, + 442, + 335 + ], + "score": 0.51, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "due to the data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "scarcity setting. IL suffers from catastrophic forgetting, a well-known phenomenon and long-standing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "issue, which refers to the drastic drop in test performance on previous (old) classes, caused by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "inaccessibility of old data in the current training session. Unfortunately, catastrophic forgetting is an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "even bigger issue for IFL, because data scarcity makes it difficult to adapt well to new tasks and learn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "new concepts, while the adaptation process could easily lead to the forgetting of base classes. In the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 430, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 430, + 403 + ], + "score": 1.0, + "content": "following, we illustrate this point by evaluating a simple baseline model for IFL.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 418, + 270, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 417, + 273, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 273, + 431 + ], + "score": 1.0, + "content": "3.2 A Simple Baseline Model for IFL", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "We consider an intransigent model that simply does not adapt to new tasks. Particularly, the model", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 489, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 280, + 463 + ], + "score": 1.0, + "content": "only needs to be trained in the base session", + "type": "text" + }, + { + "bbox": [ + 281, + 451, + 294, + 461 + ], + "score": 0.88, + "content": "\\bar { \\mathcal { D } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 451, + 489, + 463 + ], + "score": 1.0, + "content": "and is directly used for inference in all sessions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 148, + 480 + ], + "score": 1.0, + "content": "Training", + "type": "text" + }, + { + "bbox": [ + 148, + 468, + 174, + 478 + ], + "score": 0.85, + "content": "( t = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 465, + 297, + 480 + ], + "score": 1.0, + "content": "). 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The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 174, + 502 + ], + "score": 1.0, + "content": "feature extractor", + "type": "text" + }, + { + "bbox": [ + 174, + 489, + 181, + 501 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 488, + 319, + 502 + ], + "score": 1.0, + "content": "is fixed for the following sessions", + "type": "text" + }, + { + "bbox": [ + 319, + 489, + 344, + 500 + ], + "score": 0.85, + "content": "\\left( t \\geq 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 488, + 505, + 502 + ], + "score": 1.0, + "content": ") without any fine-tuning on new classes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "Inference (test). 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Note that we save the prototypes of all classes in", + "type": "text" + }, + { + "bbox": [ + 376, + 618, + 387, + 628 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 617, + 468, + 630 + ], + "score": 1.0, + "content": "for later evaluation.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "The baseline model outperforms state-of-the-art IFL and IL methods. 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The performance is evaluated on miniImageNet, CIFAR-100, and CUB-200. We tune the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "methods re-implemented by us to the best performance. For the other methods, we use the results", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "reported in the original papers. The experimental details are provided in Sec. 5. As shown in Fig. 1,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "the baseline model consistently outperforms all the compared methods including the joint-training", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 506, + 119 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 106, + 136, + 486, + 150 + ], + "lines": [ + { + "bbox": [ + 104, + 135, + 487, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 135, + 487, + 153 + ], + "score": 1.0, + "content": "3 Severity of Catastrophic Forgetting in Incremental Few-Shot Learning", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 163, + 212, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 214, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 214, + 176 + ], + "score": 1.0, + "content": "3.1 Problem Statement", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Incremental few-shot learning (IFL) aims to continually learn to recognize new classes with only few", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 281, + 210 + ], + "score": 1.0, + "content": "examples. 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Similar to", + "type": "text" + }, + { + "bbox": [ + 394, + 262, + 405, + 272 + ], + "score": 0.52, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 260, + 505, + 274 + ], + "score": 1.0, + "content": ", in each training session", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 111, + 282 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 272, + 334, + 285 + ], + "score": 1.0, + "content": "of IFL, the model has only access to the training data", + "type": "text" + }, + { + "bbox": [ + 334, + 272, + 347, + 282 + ], + "score": 0.88, + "content": "\\mathcal { D } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "and possibly a small amount of saved", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 356, + 295 + ], + "score": 1.0, + "content": "exemplars from previous sessions. When the training of session", + "type": "text" + }, + { + "bbox": [ + 356, + 284, + 361, + 293 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "is completed, the model is evaluated", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 294, + 308 + ], + "score": 1.0, + "content": "on test samples from all encountered classes", + "type": "text" + }, + { + "bbox": [ + 295, + 294, + 350, + 308 + ], + "score": 0.94, + "content": "{ \\mathcal { C } } = \\textstyle \\bigcup _ { i = 1 } ^ { t } { \\mathcal { C } } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 294, + 506, + 308 + ], + "score": 1.0, + "content": ", where it is assumed that there is no", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 439, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 321, + 321 + ], + "score": 1.0, + "content": "overlap between the classes of different sessions, i.e.,", + "type": "text" + }, + { + "bbox": [ + 321, + 309, + 340, + 319 + ], + "score": 0.86, + "content": "\\forall i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 307, + 358, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 307, + 381, + 320 + ], + "score": 0.82, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 307, + 385, + 321 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 385, + 307, + 435, + 320 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { i } \\cap { \\mathcal { C } } ^ { j } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 307, + 439, + 321 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 185, + 506, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 430, + 338 + ], + "score": 1.0, + "content": "Catastrophic forgetting. IFL is undoubtedly a more challenging problem than", + "type": "text" + }, + { + "bbox": [ + 430, + 325, + 442, + 335 + ], + "score": 0.51, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "due to the data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "scarcity setting. IL suffers from catastrophic forgetting, a well-known phenomenon and long-standing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "issue, which refers to the drastic drop in test performance on previous (old) classes, caused by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "inaccessibility of old data in the current training session. Unfortunately, catastrophic forgetting is an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "even bigger issue for IFL, because data scarcity makes it difficult to adapt well to new tasks and learn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "new concepts, while the adaptation process could easily lead to the forgetting of base classes. In the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 430, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 430, + 403 + ], + "score": 1.0, + "content": "following, we illustrate this point by evaluating a simple baseline model for IFL.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 324, + 506, + 403 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 418, + 270, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 417, + 273, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 273, + 431 + ], + "score": 1.0, + "content": "3.2 A Simple Baseline Model for IFL", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "We consider an intransigent model that simply does not adapt to new tasks. Particularly, the model", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 489, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 280, + 463 + ], + "score": 1.0, + "content": "only needs to be trained in the base session", + "type": "text" + }, + { + "bbox": [ + 281, + 451, + 294, + 461 + ], + "score": 0.88, + "content": "\\bar { \\mathcal { D } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 451, + 489, + 463 + ], + "score": 1.0, + "content": "and is directly used for inference in all sessions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 439, + 505, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 148, + 480 + ], + "score": 1.0, + "content": "Training", + "type": "text" + }, + { + "bbox": [ + 148, + 468, + 174, + 478 + ], + "score": 0.85, + "content": "( t = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 465, + 297, + 480 + ], + "score": 1.0, + "content": "). We train a feature extractor", + "type": "text" + }, + { + "bbox": [ + 297, + 468, + 304, + 479 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 465, + 379, + 480 + ], + "score": 1.0, + "content": "parameterized by", + "type": "text" + }, + { + "bbox": [ + 379, + 468, + 386, + 478 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 465, + 505, + 480 + ], + "score": 1.0, + "content": "with a fully-connected layer", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 469, + 492 + ], + "score": 1.0, + "content": "as classifier by minimizing the standard cross-entropy loss using the training examples of", + "type": "text" + }, + { + "bbox": [ + 469, + 478, + 482, + 488 + ], + "score": 0.88, + "content": "\\mathcal { D } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 477, + 506, + 492 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 174, + 502 + ], + "score": 1.0, + "content": "feature extractor", + "type": "text" + }, + { + "bbox": [ + 174, + 489, + 181, + 501 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 488, + 319, + 502 + ], + "score": 1.0, + "content": "is fixed for the following sessions", + "type": "text" + }, + { + "bbox": [ + 319, + 489, + 344, + 500 + ], + "score": 0.85, + "content": "\\left( t \\geq 2 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 488, + 505, + 502 + ], + "score": 1.0, + "content": ") without any fine-tuning on new classes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 465, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "Inference (test). 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Specifically, all the training and test samples are mapped to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 276, + 540 + ], + "score": 1.0, + "content": "embedding space of the feature extractor", + "type": "text" + }, + { + "bbox": [ + 277, + 528, + 284, + 539 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 527, + 385, + 540 + ], + "score": 1.0, + "content": ", and Euclidean distance", + "type": "text" + }, + { + "bbox": [ + 386, + 527, + 410, + 539 + ], + "score": 0.92, + "content": "d ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "is used to measure the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 537, + 310, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 310, + 551 + ], + "score": 1.0, + "content": "similarity between them. 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Note that we save the prototypes of all classes in", + "type": "text" + }, + { + "bbox": [ + 376, + 618, + 387, + 628 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 617, + 468, + 630 + ], + "score": 1.0, + "content": "for later evaluation.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 596, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "The baseline model outperforms state-of-the-art IFL and IL methods. We compare the above", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "baseline model against state-of-the-art IFL methods including FSLL [34], IDLVQC [8] and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "TOPIC [47], IL methods including Rebalance [20] and iCarl [40], and a joint-training method", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "that uses all previously seen data including the base and the following few-shot tasks for training,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "for IFL. The performance is evaluated on miniImageNet, CIFAR-100, and CUB-200. We tune the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "methods re-implemented by us to the best performance. For the other methods, we use the results", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "reported in the original papers. The experimental details are provided in Sec. 5. As shown in Fig. 1,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "the baseline model consistently outperforms all the compared methods including the joint-training", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "score": 1.0, + "content": "method (which suffers from severe data imbalance) on every dataset3. The fact that an intransigent", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 251, + 245, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 245, + 264 + ], + "score": 1.0, + "content": "model performs best suggests that", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 634, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 70, + 500, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 70, + 500, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 500, + 186 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 500, + 186 + ], + "score": 0.97, + "type": "image", + "image_path": "a3479acb9b05770da7d6c9d886cf4da1b35bd210974b3e33e48344c785552875.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 70, + 500, + 108.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 108.66666666666666, + 500, + 147.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 147.33333333333331, + 500, + 185.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 197, + 504, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 439, + 209 + ], + "score": 1.0, + "content": "Figure 1: Comparison of the proposed baseline model with state-of-the-art IFL and", + "type": "text" + }, + { + "bbox": [ + 440, + 197, + 451, + 207 + ], + "score": 0.32, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "methods and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 207, + 429, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 429, + 220 + ], + "score": 1.0, + "content": "the joint-training method.The baseline model outperforms all the other methods.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 504, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "score": 1.0, + "content": "method (which suffers from severe data imbalance) on every dataset3. The fact that an intransigent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 251, + 245, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 245, + 264 + ], + "score": 1.0, + "content": "model performs best suggests that", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 133, + 271, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 133, + 271, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 133, + 271, + 507, + 285 + ], + "score": 1.0, + "content": "• For IFL, preserving the old (base classes) may be more critical than adapting to the new.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 141, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "Due to data scarcity, the performance gain on new classes is limited and cannot make up for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 294, + 340, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 340, + 307 + ], + "score": 1.0, + "content": "the significant performance drop on base classes.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 133, + 308, + 507, + 323 + ], + "spans": [ + { + "bbox": [ + 133, + 308, + 507, + 323 + ], + "score": 1.0, + "content": "• Prior works [47, 8, 34, 20, 40] that enforce strong constraints on model parameters or exem-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "plars during fine-tuning on new classes cannot effectively prevent catastrophic forgetting in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 330, + 444, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 444, + 345 + ], + "score": 1.0, + "content": "IFL, indicating that actions may need to be taken in the base training stage.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 106, + 362, + 476, + 377 + ], + "lines": [ + { + "bbox": [ + 103, + 361, + 478, + 379 + ], + "spans": [ + { + "bbox": [ + 103, + 361, + 478, + 379 + ], + "score": 1.0, + "content": "4 Overcoming Catastrophic Forgetting in IFL by Finding Flat Minima", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "The goal of IFL is to preserve the old while adapting to the new efficiently. 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(c) Finding flat local minima of base classes and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "clamping the parameters after trained on new classes to make them fall within the flat region can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 277, + 281, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 281, + 290 + ], + "score": 1.0, + "content": "effectively mitigate catastrophic forgetting.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "In practice, it is hard to find the flat local minima that strictly satisfies the above definition, which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 317, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 506, + 332 + ], + "score": 1.0, + "content": "may not even exist. Hence, our goal is to find an approximately flat local minima of the base training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "objective function. To this end, we propose to add some small random noise to the model parameters.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "The noise can be added for multiple times to obtain similar but different loss functions, which will be", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "optimized together to locate the flat minima region. The intuition is clear – the parameter vectors", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 350, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 350, + 374 + ], + "score": 1.0, + "content": "around the flat local minima also have small function values.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 414, + 391 + ], + "score": 1.0, + "content": "To formally state the idea, we assume that the model is parameterized by", + "type": "text" + }, + { + "bbox": [ + 415, + 378, + 464, + 390 + ], + "score": 0.93, + "content": "\\theta = \\{ \\phi , \\psi \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 377, + 496, + 391 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 497, + 379, + 504, + 390 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 332, + 402 + ], + "score": 1.0, + "content": "denotes the parameters of the embedding network and", + "type": "text" + }, + { + "bbox": [ + 333, + 390, + 340, + 401 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 389, + 507, + 402 + ], + "score": 1.0, + "content": "denotes the parameters of the classifier.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 113, + 410 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 400, + 381, + 413 + ], + "score": 1.0, + "content": "denotes a labelled training sample. 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(c) Finding flat local minima of base classes and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "clamping the parameters after trained on new classes to make them fall within the flat region can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 277, + 281, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 281, + 290 + ], + "score": 1.0, + "content": "effectively mitigate catastrophic forgetting.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "In practice, it is hard to find the flat local minima that strictly satisfies the above definition, which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 317, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 506, + 332 + ], + "score": 1.0, + "content": "may not even exist. 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The intuition is clear – the parameter vectors", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 350, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 350, + 374 + ], + "score": 1.0, + "content": "around the flat local minima also have small function values.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 307, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 414, + 391 + ], + "score": 1.0, + "content": "To formally state the idea, we assume that the model is parameterized by", + "type": "text" + }, + { + "bbox": [ + 415, + 378, + 464, + 390 + ], + "score": 0.93, + "content": "\\theta = \\{ \\phi , \\psi \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 377, + 496, + 391 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 497, + 379, + 504, + 390 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 332, + 402 + ], + "score": 1.0, + "content": "denotes the parameters of the embedding network and", + "type": "text" + }, + { + "bbox": [ + 333, + 390, + 340, + 401 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 389, + 507, + 402 + ], + "score": 1.0, + "content": "denotes the parameters of the classifier.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 113, + 410 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 400, + 381, + 413 + ], + "score": 1.0, + "content": "denotes a labelled training sample. 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After fine-tuning, we evaluate the model using the nearest class", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "mean classifier as in Eq. 1, with previously saved prototypes and newly computed ones. The whole", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "training process is described in Algorithm 1. Note that to calibrate the estimates of the classifier, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 519, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 507, + 531 + ], + "score": 1.0, + "content": "normalize all prototypes to make those of base classes and those of new classes have the same norm.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 542, + 224, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 225, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 225, + 556 + ], + "score": 1.0, + "content": "4.3 Convergence Analysis", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "Our aim is to find a flat region within which all parameter vectors work well. We then minimize the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 308, + 586 + ], + "score": 1.0, + "content": "expected loss w.r.t. the joint distribution of noise", + "type": "text" + }, + { + "bbox": [ + 308, + 575, + 314, + 583 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 572, + 351, + 586 + ], + "score": 1.0, + "content": "and data", + "type": "text" + }, + { + "bbox": [ + 352, + 575, + 358, + 583 + ], + "score": 0.68, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 572, + 507, + 586 + ], + "score": 1.0, + "content": ". 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Given", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "score": 1.0, + "content": "the non-convex loss function in Eq. 3, we prove the convergence of our proposed method. The proof", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 348, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 348, + 630 + ], + "score": 1.0, + "content": "idea is inspired by the convergence analysis of SGD [3, 27].", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 504, + 656 + ], + "lines": [ + { + "bbox": [ + 103, + 629, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 103, + 629, + 200, + 650 + ], + "score": 1.0, + "content": "Formally, in each batch", + "type": "text" + }, + { + "bbox": [ + 201, + 634, + 207, + 643 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 629, + 223, + 650 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 223, + 635, + 234, + 644 + ], + "score": 0.84, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 629, + 323, + 650 + ], + "score": 1.0, + "content": "denote the batch data,", + "type": "text" + }, + { + "bbox": [ + 324, + 632, + 356, + 647 + ], + "score": 0.93, + "content": "\\{ \\epsilon _ { j } \\} _ { j = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 629, + 465, + 650 + ], + "score": 1.0, + "content": "be the sampled noises, and", + "type": "text" + }, + { + "bbox": [ + 465, + 635, + 478, + 644 + ], + "score": 0.85, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 629, + 507, + 650 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 436, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 436, + 657 + ], + "score": 1.0, + "content": "step size. 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After fine-tuning, we evaluate the model using the nearest class", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "mean classifier as in Eq. 1, with previously saved prototypes and newly computed ones. The whole", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "training process is described in Algorithm 1. Note that to calibrate the estimates of the classifier, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 519, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 507, + 531 + ], + "score": 1.0, + "content": "normalize all prototypes to make those of base classes and those of new classes have the same norm.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 419, + 507, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 542, + 224, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 225, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 225, + 556 + ], + "score": 1.0, + "content": "4.3 Convergence Analysis", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "Our aim is to find a flat region within which all parameter vectors work well. 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Given", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "score": 1.0, + "content": "the non-convex loss function in Eq. 3, we prove the convergence of our proposed method. The proof", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 617, + 348, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 348, + 630 + ], + "score": 1.0, + "content": "idea is inspired by the convergence analysis of SGD [3, 27].", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 563, + 507, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 504, + 656 + ], + "lines": [ + { + "bbox": [ + 103, + 629, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 103, + 629, + 200, + 650 + ], + "score": 1.0, + "content": "Formally, in each batch", + "type": "text" + }, + { + "bbox": [ + 201, + 634, + 207, + 643 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 629, + 223, + 650 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 223, + 635, + 234, + 644 + ], + "score": 0.84, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 629, + 323, + 650 + ], + "score": 1.0, + "content": "denote the batch data,", + "type": "text" + }, + { + "bbox": [ + 324, + 632, + 356, + 647 + ], + "score": 0.93, + "content": "\\{ \\epsilon _ { j } \\} _ { j = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 629, + 465, + 650 + ], + "score": 1.0, + "content": "be the sampled noises, and", + "type": "text" + }, + { + "bbox": [ + 465, + 635, + 478, + 644 + ], + "score": 0.85, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 629, + 507, + 650 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 436, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 436, + 657 + ], + "score": 1.0, + "content": "step size. 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The expected loss function", + "type": "text" + }, + { + "bbox": [ + 402, + 72, + 461, + 83 + ], + "score": 0.9, + "content": "R : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 70, + 506, + 86 + ], + "score": 1.0, + "content": "(Eq. 2) is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 396, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 232, + 95 + ], + "score": 1.0, + "content": "continuously differentiable and", + "type": "text" + }, + { + "bbox": [ + 233, + 84, + 240, + 93 + ], + "score": 0.64, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 83, + 329, + 95 + ], + "score": 1.0, + "content": "-smooth with constant", + "type": "text" + }, + { + "bbox": [ + 329, + 84, + 355, + 94 + ], + "score": 0.89, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 83, + 396, + 95 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 100, + 378, + 114 + ], + "lines": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "spans": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "score": 0.91, + "content": "\\| \\nabla R ( \\theta ) - \\nabla R ( \\theta ^ { \\prime } ) \\| _ { 2 } \\leq L \\| \\theta - \\theta ^ { \\prime } \\| .", + "type": "interline_equation", + "image_path": "c79ae89ed0149ed19421c38fca4ad7578dc6b6f98eb0b23f11ea9772ff205480.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 506, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 124, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 137 + ], + "score": 1.0, + "content": "This assumption is significant for the convergence analysis of gradient-based optimization algorithms,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 484, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 484, + 148 + ], + "score": 1.0, + "content": "since it limits how fast the gradient of the loss function can change w.r.t. the parameter vector.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 108, + 150, + 424, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 424, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 424, + 164 + ], + "score": 1.0, + "content": "Assumption 4.2. The expected loss function satisfies the following conditions:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 128, + 170, + 500, + 184 + ], + "lines": [ + { + "bbox": [ + 132, + 168, + 498, + 186 + ], + "spans": [ + { + "bbox": [ + 132, + 168, + 196, + 186 + ], + "score": 1.0, + "content": "• Condition 1:", + "type": "text" + }, + { + "bbox": [ + 196, + 172, + 205, + 181 + ], + "score": 0.48, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 168, + 325, + 186 + ], + "score": 1.0, + "content": "is bounded below by a scalar", + "type": "text" + }, + { + "bbox": [ + 325, + 172, + 338, + 181 + ], + "score": 0.87, + "content": "R ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 168, + 478, + 186 + ], + "score": 1.0, + "content": ", given the sequence of parameters", + "type": "text" + }, + { + "bbox": [ + 479, + 171, + 498, + 183 + ], + "score": 0.91, + "content": "\\{ \\theta _ { k } \\}", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 134, + 189, + 317, + 202 + ], + "lines": [ + { + "bbox": [ + 133, + 187, + 317, + 203 + ], + "spans": [ + { + "bbox": [ + 133, + 187, + 225, + 203 + ], + "score": 1.0, + "content": "• Condition 2: For all", + "type": "text" + }, + { + "bbox": [ + 226, + 190, + 252, + 200 + ], + "score": 0.9, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 187, + 270, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 190, + 314, + 202 + ], + "score": 0.91, + "content": "j \\in [ 1 , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 187, + 317, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 206, + 401, + 221 + ], + "lines": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "spans": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5bc16b0b8564721592171d4aa24ec24ad8a4402750b2216466dfbe55ca25b01d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 133, + 230, + 480, + 244 + ], + "lines": [ + { + "bbox": [ + 133, + 230, + 480, + 245 + ], + "spans": [ + { + "bbox": [ + 133, + 230, + 185, + 245 + ], + "score": 1.0, + "content": "• Condition", + "type": "text" + }, + { + "bbox": [ + 185, + 233, + 191, + 241 + ], + "score": 0.32, + "content": "^ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 230, + 273, + 245 + ], + "score": 1.0, + "content": ": There exist scalars", + "type": "text" + }, + { + "bbox": [ + 273, + 232, + 306, + 243 + ], + "score": 0.91, + "content": "m _ { 1 } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 230, + 325, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 232, + 358, + 243 + ], + "score": 0.9, + "content": "m _ { 2 } \\geq 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 230, + 388, + 245 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 388, + 232, + 415, + 242 + ], + "score": 0.89, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 230, + 433, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 231, + 477, + 244 + ], + "score": 0.92, + "content": "j \\in [ 1 , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 230, + 480, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 248, + 428, + 264 + ], + "lines": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "spans": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5096183562af805022f33c150d89bccb14263ee7a806f9ee4921f24c132bfe54.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 141, + 287 + ], + "score": 0.89, + "content": "\\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 271, + 442, + 291 + ], + "score": 1.0, + "content": "denotes the expectation w.r.t. the joint distribution of random variables", + "type": "text" + }, + { + "bbox": [ + 442, + 277, + 453, + 286 + ], + "score": 0.85, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 271, + 473, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 473, + 277, + 483, + 287 + ], + "score": 0.85, + "content": "\\epsilon _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 271, + 507, + 291 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 283, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 141, + 299 + ], + "score": 0.88, + "content": "\\mathbb { V } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 283, + 423, + 303 + ], + "score": 1.0, + "content": "denotes the variance. 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Condition 2", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 214, + 321 + ], + "score": 1.0, + "content": "assumes that the gradient", + "type": "text" + }, + { + "bbox": [ + 214, + 309, + 290, + 321 + ], + "score": 0.92, + "content": "g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 308, + 402, + 321 + ], + "score": 1.0, + "content": "is an unbiased estimate of", + "type": "text" + }, + { + "bbox": [ + 403, + 309, + 437, + 321 + ], + "score": 0.92, + "content": "\\nabla R ( \\theta _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 308, + 506, + 321 + ], + "score": 1.0, + "content": ". This is a strict", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "assumption made to simplify the proof, but it can be easily relaxed to a general and easily-met", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 329, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 211, + 345 + ], + "score": 1.0, + "content": "condition that there exist", + "type": "text" + }, + { + "bbox": [ + 211, + 331, + 267, + 342 + ], + "score": 0.91, + "content": "\\mu _ { 1 } \\geq \\mu _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 329, + 311, + 345 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 311, + 330, + 504, + 343 + ], + "score": 0.91, + "content": "\\| \\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 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 123, + 357 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 342, + 336, + 357 + ], + "score": 0.9, + "content": "\\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 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 341, + 506, + 357 + ], + "score": 1.0, + "content": ". Therefore, the convergence can be proved", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "in a similar way using the techniques presented in the Appendix. Condition 3 assumes the variance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 166, + 378 + ], + "score": 1.0, + "content": "of the gradient", + "type": "text" + }, + { + "bbox": [ + 167, + 365, + 241, + 378 + ], + "score": 0.91, + "content": "g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "cannot be arbitrarily large, which is also reasonable in practice. To", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 408, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 408, + 388 + ], + "score": 1.0, + "content": "facilitate later analysis, similar to [43], we restrict the step sizes as follows.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 286, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 287, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 287, + 406 + ], + "score": 1.0, + "content": "Assumption 4.3. The learning rates satisfy:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 407, + 363, + 440 + ], + "lines": [ + { + "bbox": [ + 246, + 407, + 363, + 440 + ], + "spans": [ + { + "bbox": [ + 246, + 407, + 363, + 440 + ], + "score": 0.94, + "content": "\\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } = \\infty , \\ \\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } ^ { 2 } < \\infty .", + "type": "interline_equation", + "image_path": "11b47e921b3b0cb564364a1725626d441628fc2d3166e3268b98c43dcd1fe07f.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 407, + 363, + 423.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 246, + 423.5, + 363, + 440.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 450, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 378, + 462 + ], + "score": 1.0, + "content": "This assumption can be easily met, since in practice the learning rate", + "type": "text" + }, + { + "bbox": [ + 378, + 452, + 390, + 461 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "is usually far less than 1 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 461, + 465, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 169, + 473 + ], + "score": 1.0, + "content": "decreases w.r.t.", + "type": "text" + }, + { + "bbox": [ + 170, + 462, + 176, + 471 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 461, + 465, + 473 + ], + "score": 1.0, + "content": ". Based on the above assumptions, we can derive the following theorem.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 485, + 487 + ], + "score": 1.0, + "content": "Theorem 4.1. Under assumptions 4.1, 4.2 and 4.3, we further assume that the risk function", + "type": "text" + }, + { + "bbox": [ + 486, + 476, + 495, + 486 + ], + "score": 0.72, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 486, + 469, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 224, + 500 + ], + "score": 1.0, + "content": "twice differentiable, and that", + "type": "text" + }, + { + "bbox": [ + 224, + 487, + 268, + 499 + ], + "score": 0.94, + "content": "\\| \\nabla R ( \\theta ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 486, + 278, + 500 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 278, + 487, + 290, + 498 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 486, + 380, + 500 + ], + "score": 1.0, + "content": "-smooth with constant", + "type": "text" + }, + { + "bbox": [ + 380, + 487, + 411, + 498 + ], + "score": 0.91, + "content": "L _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 486, + 469, + 500 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 503, + 358, + 523 + ], + "lines": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "spans": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "score": 0.92, + "content": "\\operatorname* { l i m } _ { k \\to \\infty } \\mathbb { E } [ \\| \\nabla R ( \\theta _ { k } ) \\| _ { 2 } ^ { 2 } ] = 0 .", + "type": "interline_equation", + "image_path": "19bd76fc1ebc8d0c84ad22f6f47d9db08d8dec6c80681989c406d1873676a886.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 532, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "This theorem establishes the convergence of our algorithm. The proof is provided in Appendix A.1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 106, + 559, + 192, + 573 + ], + "lines": [ + { + "bbox": [ + 104, + 558, + 192, + 577 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 192, + 577 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "In this section, we empirically evaluate our proposed method for incremental few-shot learning and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 595, + 405, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 405, + 608 + ], + "score": 1.0, + "content": "demonstrate its effectiveness by comparison with state-of-the-art methods.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 619, + 216, + 631 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 217, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 217, + 636 + ], + "score": 1.0, + "content": "5.1 Experimental Setup", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "Datasets. For CIFAR-100 and miniImageNet, we randomly select 60 classes as the base classes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "and the remaining 40 classes as the new classes. In each incremental learning session, we construct", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "5-way 5-shot tasks by randomly picking 5 classes and sampling 5 examples for each class. For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "CUB-200-2011 with 200 classes, we select 100 classes as the base classes and 100 classes as the new", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 683, + 307, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 307, + 696 + ], + "score": 1.0, + "content": "ones. We test 10-way 5-shot tasks on this dataset.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "Baselines. We compare our method F2M with 8 methods: the Baseline proposed in Sec. 3, a joint-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "training method that uses all previously seen data including the base and the following few-shot", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 401, + 86 + ], + "score": 1.0, + "content": "Assumption 4.1 (L-smooth risk function). The expected loss function", + "type": "text" + }, + { + "bbox": [ + 402, + 72, + 461, + 83 + ], + "score": 0.9, + "content": "R : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 70, + 506, + 86 + ], + "score": 1.0, + "content": "(Eq. 2) is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 396, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 232, + 95 + ], + "score": 1.0, + "content": "continuously differentiable and", + "type": "text" + }, + { + "bbox": [ + 233, + 84, + 240, + 93 + ], + "score": 0.64, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 83, + 329, + 95 + ], + "score": 1.0, + "content": "-smooth with constant", + "type": "text" + }, + { + "bbox": [ + 329, + 84, + 355, + 94 + ], + "score": 0.89, + "content": "L > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 83, + 396, + 95 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 70, + 506, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 100, + 378, + 114 + ], + "lines": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "spans": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "score": 0.91, + "content": "\\| \\nabla R ( \\theta ) - \\nabla R ( \\theta ^ { \\prime } ) \\| _ { 2 } \\leq L \\| \\theta - \\theta ^ { \\prime } \\| .", + "type": "interline_equation", + "image_path": "c79ae89ed0149ed19421c38fca4ad7578dc6b6f98eb0b23f11ea9772ff205480.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 233, + 100, + 378, + 114 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 506, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 124, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 137 + ], + "score": 1.0, + "content": "This assumption is significant for the convergence analysis of gradient-based optimization algorithms,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 484, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 484, + 148 + ], + "score": 1.0, + "content": "since it limits how fast the gradient of the loss function can change w.r.t. the parameter vector.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 124, + 506, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 150, + 424, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 424, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 424, + 164 + ], + "score": 1.0, + "content": "Assumption 4.2. The expected loss function satisfies the following conditions:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 148, + 424, + 164 + ] + }, + { + "type": "text", + "bbox": [ + 128, + 170, + 500, + 184 + ], + "lines": [ + { + "bbox": [ + 132, + 168, + 498, + 186 + ], + "spans": [ + { + "bbox": [ + 132, + 168, + 196, + 186 + ], + "score": 1.0, + "content": "• Condition 1:", + "type": "text" + }, + { + "bbox": [ + 196, + 172, + 205, + 181 + ], + "score": 0.48, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 168, + 325, + 186 + ], + "score": 1.0, + "content": "is bounded below by a scalar", + "type": "text" + }, + { + "bbox": [ + 325, + 172, + 338, + 181 + ], + "score": 0.87, + "content": "R ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 168, + 478, + 186 + ], + "score": 1.0, + "content": ", given the sequence of parameters", + "type": "text" + }, + { + "bbox": [ + 479, + 171, + 498, + 183 + ], + "score": 0.91, + "content": "\\{ \\theta _ { k } \\}", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 132, + 168, + 498, + 186 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 189, + 317, + 202 + ], + "lines": [ + { + "bbox": [ + 133, + 187, + 317, + 203 + ], + "spans": [ + { + "bbox": [ + 133, + 187, + 225, + 203 + ], + "score": 1.0, + "content": "• Condition 2: For all", + "type": "text" + }, + { + "bbox": [ + 226, + 190, + 252, + 200 + ], + "score": 0.9, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 187, + 270, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 190, + 314, + 202 + ], + "score": 0.91, + "content": "j \\in [ 1 , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 187, + 317, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 133, + 187, + 317, + 203 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 206, + 401, + 221 + ], + "lines": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "spans": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5bc16b0b8564721592171d4aa24ec24ad8a4402750b2216466dfbe55ca25b01d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 245, + 206, + 401, + 221 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 133, + 230, + 480, + 244 + ], + "lines": [ + { + "bbox": [ + 133, + 230, + 480, + 245 + ], + "spans": [ + { + "bbox": [ + 133, + 230, + 185, + 245 + ], + "score": 1.0, + "content": "• Condition", + "type": "text" + }, + { + "bbox": [ + 185, + 233, + 191, + 241 + ], + "score": 0.32, + "content": "^ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 230, + 273, + 245 + ], + "score": 1.0, + "content": ": There exist scalars", + "type": "text" + }, + { + "bbox": [ + 273, + 232, + 306, + 243 + ], + "score": 0.91, + "content": "m _ { 1 } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 230, + 325, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 232, + 358, + 243 + ], + "score": 0.9, + "content": "m _ { 2 } \\geq 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 230, + 388, + 245 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 388, + 232, + 415, + 242 + ], + "score": 0.89, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 230, + 433, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 433, + 231, + 477, + 244 + ], + "score": 0.92, + "content": "j \\in [ 1 , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 230, + 480, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 133, + 230, + 480, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 248, + 428, + 264 + ], + "lines": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "spans": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "score": 0.9, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5096183562af805022f33c150d89bccb14263ee7a806f9ee4921f24c132bfe54.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 218, + 248, + 428, + 264 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 141, + 287 + ], + "score": 0.89, + "content": "\\mathbb { E } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 271, + 442, + 291 + ], + "score": 1.0, + "content": "denotes the expectation w.r.t. the joint distribution of random variables", + "type": "text" + }, + { + "bbox": [ + 442, + 277, + 453, + 286 + ], + "score": 0.85, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 271, + 473, + 291 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 473, + 277, + 483, + 287 + ], + "score": 0.85, + "content": "\\epsilon _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 271, + 507, + 291 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 283, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 141, + 299 + ], + "score": 0.88, + "content": "\\mathbb { V } _ { z _ { k } , \\epsilon _ { j } } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 283, + 423, + 303 + ], + "score": 1.0, + "content": "denotes the variance. 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Condition 2", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 214, + 321 + ], + "score": 1.0, + "content": "assumes that the gradient", + "type": "text" + }, + { + "bbox": [ + 214, + 309, + 290, + 321 + ], + "score": 0.92, + "content": "g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 308, + 402, + 321 + ], + "score": 1.0, + "content": "is an unbiased estimate of", + "type": "text" + }, + { + "bbox": [ + 403, + 309, + 437, + 321 + ], + "score": 0.92, + "content": "\\nabla R ( \\theta _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 308, + 506, + 321 + ], + "score": 1.0, + "content": ". This is a strict", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "assumption made to simplify the proof, but it can be easily relaxed to a general and easily-met", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 329, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 211, + 345 + ], + "score": 1.0, + "content": "condition that there exist", + "type": "text" + }, + { + "bbox": [ + 211, + 331, + 267, + 342 + ], + "score": 0.91, + "content": "\\mu _ { 1 } \\geq \\mu _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 329, + 311, + 345 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 311, + 330, + 504, + 343 + ], + "score": 0.91, + "content": "\\| \\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 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 123, + 357 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 342, + 336, + 357 + ], + "score": 0.9, + "content": "\\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 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 341, + 506, + 357 + ], + "score": 1.0, + "content": ". Therefore, the convergence can be proved", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "in a similar way using the techniques presented in the Appendix. Condition 3 assumes the variance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 166, + 378 + ], + "score": 1.0, + "content": "of the gradient", + "type": "text" + }, + { + "bbox": [ + 167, + 365, + 241, + 378 + ], + "score": 0.91, + "content": "g ( z _ { k } ; \\phi _ { k } + \\epsilon _ { j } , \\psi _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "cannot be arbitrarily large, which is also reasonable in practice. To", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 408, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 408, + 388 + ], + "score": 1.0, + "content": "facilitate later analysis, similar to [43], we restrict the step sizes as follows.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 271, + 507, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 286, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 287, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 287, + 406 + ], + "score": 1.0, + "content": "Assumption 4.3. The learning rates satisfy:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 388, + 287, + 406 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 407, + 363, + 440 + ], + "lines": [ + { + "bbox": [ + 246, + 407, + 363, + 440 + ], + "spans": [ + { + "bbox": [ + 246, + 407, + 363, + 440 + ], + "score": 0.94, + "content": "\\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } = \\infty , \\ \\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } ^ { 2 } < \\infty .", + "type": "interline_equation", + "image_path": "11b47e921b3b0cb564364a1725626d441628fc2d3166e3268b98c43dcd1fe07f.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 407, + 363, + 423.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 246, + 423.5, + 363, + 440.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 450, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 378, + 462 + ], + "score": 1.0, + "content": "This assumption can be easily met, since in practice the learning rate", + "type": "text" + }, + { + "bbox": [ + 378, + 452, + 390, + 461 + ], + "score": 0.87, + "content": "\\alpha _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "is usually far less than 1 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 461, + 465, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 169, + 473 + ], + "score": 1.0, + "content": "decreases w.r.t.", + "type": "text" + }, + { + "bbox": [ + 170, + 462, + 176, + 471 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 461, + 465, + 473 + ], + "score": 1.0, + "content": ". Based on the above assumptions, we can derive the following theorem.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 450, + 505, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 506, + 499 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 485, + 487 + ], + "score": 1.0, + "content": "Theorem 4.1. Under assumptions 4.1, 4.2 and 4.3, we further assume that the risk function", + "type": "text" + }, + { + "bbox": [ + 486, + 476, + 495, + 486 + ], + "score": 0.72, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 486, + 469, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 224, + 500 + ], + "score": 1.0, + "content": "twice differentiable, and that", + "type": "text" + }, + { + "bbox": [ + 224, + 487, + 268, + 499 + ], + "score": 0.94, + "content": "\\| \\nabla R ( \\theta ) \\| _ { 2 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 486, + 278, + 500 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 278, + 487, + 290, + 498 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 486, + 380, + 500 + ], + "score": 1.0, + "content": "-smooth with constant", + "type": "text" + }, + { + "bbox": [ + 380, + 487, + 411, + 498 + ], + "score": 0.91, + "content": "L _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 486, + 469, + 500 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 475, + 505, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 503, + 358, + 523 + ], + "lines": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "spans": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "score": 0.92, + "content": "\\operatorname* { l i m } _ { k \\to \\infty } \\mathbb { E } [ \\| \\nabla R ( \\theta _ { k } ) \\| _ { 2 } ^ { 2 } ] = 0 .", + "type": "interline_equation", + "image_path": "19bd76fc1ebc8d0c84ad22f6f47d9db08d8dec6c80681989c406d1873676a886.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 252, + 503, + 358, + 523 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 532, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "This theorem establishes the convergence of our algorithm. The proof is provided in Appendix A.1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 532, + 505, + 546 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 559, + 192, + 573 + ], + "lines": [ + { + "bbox": [ + 104, + 558, + 192, + 577 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 192, + 577 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 584, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "In this section, we empirically evaluate our proposed method for incremental few-shot learning and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 595, + 405, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 405, + 608 + ], + "score": 1.0, + "content": "demonstrate its effectiveness by comparison with state-of-the-art methods.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 584, + 505, + 608 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 619, + 216, + 631 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 217, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 217, + 636 + ], + "score": 1.0, + "content": "5.1 Experimental Setup", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "Datasets. For CIFAR-100 and miniImageNet, we randomly select 60 classes as the base classes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "and the remaining 40 classes as the new classes. In each incremental learning session, we construct", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "5-way 5-shot tasks by randomly picking 5 classes and sampling 5 examples for each class. For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "CUB-200-2011 with 200 classes, we select 100 classes as the base classes and 100 classes as the new", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 683, + 307, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 307, + 696 + ], + "score": 1.0, + "content": "ones. We test 10-way 5-shot tasks on this dataset.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 639, + 506, + 696 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "Baselines. We compare our method F2M with 8 methods: the Baseline proposed in Sec. 3, a joint-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "training method that uses all previously seen data including the base and the following few-shot", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "tasks for training, the classifier re-training method (cRT) [25] for long-tailed classification trained", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "with all encountered data, iCaRL [40], Rebalance [20], TOPIC [47], FSLL [34], and IDLVQ-C [8].", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "For a fair comparison, we re-implement cRT [25], iCaRL [40], Rebalance [20], FSLL [34], and the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "joint-training method and tune them to their best performance. We also provide the results reported in", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "the original papers for comparison. The results of TOPIC [47] and IDLVQ-C [8] are copied from the", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "original papers. Note that for IL, joint-training is naturally the upper bound of incremental learning", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "algorithms, however, for IFL, joint-training is not a good approximation of the upper bound because", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "data imbalance makes the model perform significantly poorer on new classes (long-tailed classes). To", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "score": 1.0, + "content": "address the data imbalance issue, we re-implement the cRT method as the approximate upper bound.", + "type": "text", + "cross_page": true + } + ], + "index": 18 + } + ], + "index": 39.5, + "bbox_fs": [ + 106, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 99, + 484, + 245 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 76, + 503, + 98 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 74, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 505, + 90 + ], + "score": 1.0, + "content": "Table 1: Classification accuracy on CIFAR-100 for 5-way 5-shot incremental learning. ∗ indicates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "score": 1.0, + "content": "our re-implementation.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 99, + 484, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 99, + 484, + 245 + ], + "spans": [ + { + "bbox": [ + 128, + 99, + 484, + 245 + ], + "score": 0.985, + "html": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*65.1863.8960.2057.2353.7150.3948.7747.2945.28=
Joint-training*65.1861.4557.3653.6850.8447.3344.7942.6240.08-5.20
Baseline65.1861.6758.6155.1151.8649.4347.6045.6443.83-1.45
iCaRL[40]*66.5257.2654.2750.6247.3344.9943.1441.1639.49-5.79
Rebalance [20]*66.6661.4257.2953.0248.8545.6843.0640.5638.35-6.93
FSLL [34]*65.1856.2454.5551.6149.1147.2745.3543.9542.22-3.08
iCaRL[40]64.1053.2841.6934.1327.9325.0620.4115.4813.73-31.55
Rebalance [20]64.1053.0543.9636.9731.6126.7321.2316.7813.54-31.74
TOPIC [47]64.1055.8847.0745.1640.1136.3833.9631.5529.37-15.91
FSLL [34]64.1055.8551.7148.5945.3443.2541.5239.8138.16-7.12
FSLL+SS [34]66.7655.5252.2049.1746.2344.6443.0741.2039.57-5.71
F2M64.7162.0559.0155.5852.5549.9648.0846.2844.67-0.61
", + "type": "table", + "image_path": "3835b07fcacb259d9fc8f7adfa333925ec8fddfe46906fed2c496f5cf8b28580.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 128, + 99, + 484, + 147.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 128, + 147.66666666666666, + 484, + 196.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 128, + 196.33333333333331, + 484, + 244.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "table", + "bbox": [ + 128, + 303, + 484, + 458 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 281, + 504, + 302 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "Table 2: Classification accuracy on miniImageNet for 5-way 5-shot incremental learning. ∗ indicates", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 292, + 200, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 200, + 303 + ], + "score": 1.0, + "content": "our re-implementation.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 303, + 484, + 458 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 128, + 303, + 484, + 458 + ], + "spans": [ + { + "bbox": [ + 128, + 303, + 484, + 458 + ], + "score": 0.985, + "html": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*67.3064.1560.5957.3254.2251.4348.9246.7844.85=
Joint-training*67.3062.3457.7954.0850.9347.6544.6442.6140.29-4.56
Baseline67.3063.1859.6256.3353.2850.5047.9645.8543.88-0.97
iCaRL [40]*67.3559.9155.6452.6049.4346.7344.1342.1740.29-4.56
Rebalance [20]*67.9163.1158.7554.8350.6847.1143.8841.1938.72-6.13
FSLL [34]*67.3059.8157.2654.5752.0549.4246.9544.9442.87-1.11
iCaRL[40]61.3146.3242.9437.6330.4924.0020.8918.8017.21-27.64
Rebalance [20]61.3147.8039.3131.9125.6821.3518.6717.2414.17-30.68
TOPIC [47]61.3150.0945.1741.1637.4835.5232.1929.4624.42-20.43
FSLL [34]66.4861.7558.1654.1651.1048.5346.5444.2042.28-2.57
FSLL+SS[34]68.8563.1459.2455.2352.2449.6547.7445.2343.92-0.93
IDLVQ-C[8]64.7759.8755.9352.6249.8847.5544.8343.1441.84-3.01
F2M67.2863.8060.3857.0654.0851.3948.8246.5844.65-0.20
", + "type": "table", + "image_path": "a4fbc03180e63e236407b6f9ed5fcce1cee014bd347b0f0a3ce14d1440d31067.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 128, + 303, + 484, + 354.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 128, + 354.6666666666667, + 484, + 406.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 128, + 406.33333333333337, + 484, + 458.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "tasks for training, the classifier re-training method (cRT) [25] for long-tailed classification trained", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "with all encountered data, iCaRL [40], Rebalance [20], TOPIC [47], FSLL [34], and IDLVQ-C [8].", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "For a fair comparison, we re-implement cRT [25], iCaRL [40], Rebalance [20], FSLL [34], and the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "joint-training method and tune them to their best performance. We also provide the results reported in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "the original papers for comparison. The results of TOPIC [47] and IDLVQ-C [8] are copied from the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "original papers. Note that for IL, joint-training is naturally the upper bound of incremental learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "algorithms, however, for IFL, joint-training is not a good approximation of the upper bound because", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "data imbalance makes the model perform significantly poorer on new classes (long-tailed classes). To", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "score": 1.0, + "content": "address the data imbalance issue, we re-implement the cRT method as the approximate upper bound.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 613 + ], + "score": 1.0, + "content": "Experimental details. The experiments are conducted with NVIDIA GPU RTX3090 on CUDA 11.0.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "We randomly split each dataset into multiple tasks (sessions). For each dataset (with a fixed split),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "we run each algorithm for 10 times and report the mean accuracy. We adopt ResNet18 [16] as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "backbone network. For data augmentation, we use standard random crop and horizontal flip. In the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "base training stage, we select the last 4 or 8 convolution layers to inject noise, because these layers", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 366, + 669 + ], + "score": 1.0, + "content": "output higher-level feature representations. The flat region bound", + "type": "text" + }, + { + "bbox": [ + 367, + 657, + 372, + 666 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "is set as 0.01. We set the number", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 228, + 680 + ], + "score": 1.0, + "content": "of times for noise sampling as", + "type": "text" + }, + { + "bbox": [ + 228, + 667, + 277, + 677 + ], + "score": 0.91, + "content": "M = 2 \\sim 4", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 667, + 336, + 680 + ], + "score": 1.0, + "content": ", since a larger", + "type": "text" + }, + { + "bbox": [ + 336, + 667, + 348, + 677 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "will increase the training time. In each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "incremental few-shot learning session, the total number of training epochs is 6, and the learning rate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "is 0.02. To verify the correctness of our implementation, we conduct experiments on incremental", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "learning and compare our results to those reported on CIFAR-100 in Appendix A.3. More experiment", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 710, + 259, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 259, + 723 + ], + "score": 1.0, + "content": "details are provided in Appendix A.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 99, + 484, + 245 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 76, + 503, + 98 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 74, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 505, + 90 + ], + "score": 1.0, + "content": "Table 1: Classification accuracy on CIFAR-100 for 5-way 5-shot incremental learning. ∗ indicates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "score": 1.0, + "content": "our re-implementation.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 99, + 484, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 99, + 484, + 245 + ], + "spans": [ + { + "bbox": [ + 128, + 99, + 484, + 245 + ], + "score": 0.985, + "html": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*65.1863.8960.2057.2353.7150.3948.7747.2945.28=
Joint-training*65.1861.4557.3653.6850.8447.3344.7942.6240.08-5.20
Baseline65.1861.6758.6155.1151.8649.4347.6045.6443.83-1.45
iCaRL[40]*66.5257.2654.2750.6247.3344.9943.1441.1639.49-5.79
Rebalance [20]*66.6661.4257.2953.0248.8545.6843.0640.5638.35-6.93
FSLL [34]*65.1856.2454.5551.6149.1147.2745.3543.9542.22-3.08
iCaRL[40]64.1053.2841.6934.1327.9325.0620.4115.4813.73-31.55
Rebalance [20]64.1053.0543.9636.9731.6126.7321.2316.7813.54-31.74
TOPIC [47]64.1055.8847.0745.1640.1136.3833.9631.5529.37-15.91
FSLL [34]64.1055.8551.7148.5945.3443.2541.5239.8138.16-7.12
FSLL+SS [34]66.7655.5252.2049.1746.2344.6443.0741.2039.57-5.71
F2M64.7162.0559.0155.5852.5549.9648.0846.2844.67-0.61
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MethodsessionsThe gap with cRT
123456789
cRT[25]*67.3064.1560.5957.3254.2251.4348.9246.7844.85=
Joint-training*67.3062.3457.7954.0850.9347.6544.6442.6140.29-4.56
Baseline67.3063.1859.6256.3353.2850.5047.9645.8543.88-0.97
iCaRL [40]*67.3559.9155.6452.6049.4346.7344.1342.1740.29-4.56
Rebalance [20]*67.9163.1158.7554.8350.6847.1143.8841.1938.72-6.13
FSLL [34]*67.3059.8157.2654.5752.0549.4246.9544.9442.87-1.11
iCaRL[40]61.3146.3242.9437.6330.4924.0020.8918.8017.21-27.64
Rebalance [20]61.3147.8039.3131.9125.6821.3518.6717.2414.17-30.68
TOPIC [47]61.3150.0945.1741.1637.4835.5232.1929.4624.42-20.43
FSLL [34]66.4861.7558.1654.1651.1048.5346.5444.2042.28-2.57
FSLL+SS[34]68.8563.1459.2455.2352.2449.6547.7445.2343.92-0.93
IDLVQ-C[8]64.7759.8755.9352.6249.8847.5544.8343.1441.84-3.01
F2M67.2863.8060.3857.0654.0851.3948.8246.5844.65-0.20
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The experiments are conducted with NVIDIA GPU RTX3090 on CUDA 11.0.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 625 + ], + "score": 1.0, + "content": "We randomly split each dataset into multiple tasks (sessions). For each dataset (with a fixed split),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "we run each algorithm for 10 times and report the mean accuracy. We adopt ResNet18 [16] as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "backbone network. For data augmentation, we use standard random crop and horizontal flip. In the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "base training stage, we select the last 4 or 8 convolution layers to inject noise, because these layers", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 366, + 669 + ], + "score": 1.0, + "content": "output higher-level feature representations. The flat region bound", + "type": "text" + }, + { + "bbox": [ + 367, + 657, + 372, + 666 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "is set as 0.01. We set the number", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 228, + 680 + ], + "score": 1.0, + "content": "of times for noise sampling as", + "type": "text" + }, + { + "bbox": [ + 228, + 667, + 277, + 677 + ], + "score": 0.91, + "content": "M = 2 \\sim 4", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 667, + 336, + 680 + ], + "score": 1.0, + "content": ", since a larger", + "type": "text" + }, + { + "bbox": [ + 336, + 667, + 348, + 677 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "will increase the training time. In each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "incremental few-shot learning session, the total number of training epochs is 6, and the learning rate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "is 0.02. To verify the correctness of our implementation, we conduct experiments on incremental", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "learning and compare our results to those reported on CIFAR-100 in Appendix A.3. More experiment", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 710, + 259, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 259, + 723 + ], + "score": 1.0, + "content": "details are provided in Appendix A.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 602, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 99, + 504, + 248 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 76, + 503, + 98 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 75, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 75, + 505, + 90 + ], + "score": 1.0, + "content": "Table 3: Classification accuracy on CUB-200-2011 for 10-way 5-shot incremental learning.∗ indicates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 200, + 99 + ], + "score": 1.0, + "content": "our re-implementation.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 99, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 99, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 108, + 99, + 504, + 248 + ], + "score": 0.983, + "html": "
MethodsessionsThe gap with cRT
1234567891011
cRT[25]*80.8378.5176.1273.9371.4668.9667.7366.7564.2262.5361.08
Joint-training*80.8377.5774.1170.7568.5265.9764.5862.2260.1858.4956.78-4.30
Baseline80.8777.1574.4672.2669.4767.1865.6263.6861.3059.7258.12-2.96
iCaRL [40]*79.5867.6364.1761.8058.1055.5153.3450.8948.6247.3445.60-15.48
Rebalance [20]*80.9470.3262.9657.1951.0646.7044.0340.1536.7534.8832.09-28.99
FSLL [34]*80.8377.3872.3771.8467.5165.3063.7561.1659.0558.0355.82-5.26
iCaRL [40]68.6852.6548.6144.1636.6229.5227.8326.2624.0123.8921.16-39.92
Rebalance [20]68.6857.1244.2128.7826.7125.6624.6221.5220.1220.0619.87-41.21
TOPIC[47]68.6862.4954.8149.9945.2541.4038.3535.3632.2228.3126.28-34.80
FSLL [34]72.7769.3365.5162.6661.1058.6557.7857.2655.5955.3954.21-6.87
FSLL+SS [34]75.6371.8168.1664.3262.6160.1058.8258.7056.4556.4155.82-5.26
IDLVQ-C [8]77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81-3.27
F2M81.0778.1675.5772.8970.8668.1767.0165.2663.3661.7660.26-0.82
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MethodIndicator IVariance g2
Training SetTesting SetTraining SetTesting Set
Baseline0.29930.45820.14510.2395
F2M0.05060.08000.02960.0334
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The main results on CIFAR-100, miniImageNet", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "and CUB-200-2011 are presented in Table 1, Table 2 and Table 3 respectively. Based on the experi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "ment results, we have the following observations: 1) The Baseline introduced in Sec. 3 outperforms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 408, + 507, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 507, + 423 + ], + "score": 1.0, + "content": "the state-of-the-art approaches on all incremental sessions. 2) As expected, cRT consistently outper-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 206, + 433 + ], + "score": 1.0, + "content": "forms the Baseline up to", + "type": "text" + }, + { + "bbox": [ + 206, + 420, + 221, + 431 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 421, + 232, + 433 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 232, + 420, + 247, + 431 + ], + "score": 0.87, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "by considering the data imbalance problem and applying proper", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "techniques to tackle the long-tailed classification problem to improve performance. Hence, it is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "reasonable to use cRT as the approximate upper bound of IFL. 3) Our F2M outperforms the state-of-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "the-art methods and the Baseline. Moreover, the performance of F2M is very close to the approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 282, + 477 + ], + "score": 1.0, + "content": "upper bound, i.e., the gap with cRT is only", + "type": "text" + }, + { + "bbox": [ + 283, + 464, + 306, + 474 + ], + "score": 0.87, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "in the last session on miniImageNet. The results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "show that even with strong constraints [20, 40, 34] and saved examplars of base classes [20, 40, 8],", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "current methods cannot effectively address the catastrophic forgetting problem. In contrast, finding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 496, + 401, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 401, + 510 + ], + "score": 1.0, + "content": "flat minima seems a promising approach to overcome this harsh problem.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 252, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 253, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 253, + 536 + ], + "score": 1.0, + "content": "5.3 Ablation Study and Analysis", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 543, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "Analysis on the flatness of local minima. Here, we verify that our method can find a more flat local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 554, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 310, + 564 + ], + "score": 1.0, + "content": "minima than the Baseline. For a found local minima", + "type": "text" + }, + { + "bbox": [ + 310, + 554, + 321, + 564 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 555, + 504, + 564 + ], + "score": 1.0, + "content": ", we measure its flatness as follows. We sample", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 392, + 577 + ], + "score": 1.0, + "content": "the noise for 1000 times. For each time, we inject the sampled noise to", + "type": "text" + }, + { + "bbox": [ + 392, + 565, + 402, + 575 + ], + "score": 0.88, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 564, + 491, + 577 + ], + "score": 1.0, + "content": "and calculate the loss", + "type": "text" + }, + { + "bbox": [ + 492, + 565, + 503, + 576 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 564, + 506, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 102, + 570, + 509, + 608 + ], + "spans": [ + { + "bbox": [ + 102, + 581, + 203, + 608 + ], + "score": 1.0, + "content": "to measure the flatness.", + "type": "text" + }, + { + "bbox": [ + 104, + 575, + 291, + 592 + ], + "score": 1.0, + "content": "Then, we adopt the indicator I = 11000 P1000i=1", + "type": "text" + }, + { + "bbox": [ + 203, + 591, + 216, + 601 + ], + "score": 0.87, + "content": "\\mathcal { L } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 581, + 294, + 608 + ], + "score": 1.0, + "content": "denotes the loss of", + "type": "text" + }, + { + "bbox": [ + 224, + 576, + 336, + 591 + ], + "score": 0.93, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 591, + 304, + 601 + ], + "score": 0.85, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 581, + 325, + 608 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 325, + 590, + 334, + 601 + ], + "score": 0.8, + "content": "\\overline { { \\mathcal { L } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 581, + 445, + 608 + ], + "score": 1.0, + "content": "1000denotes the average loss of", + "type": "text" + }, + { + "bbox": [ + 336, + 570, + 390, + 595 + ], + "score": 1.0, + "content": "and variance", + "type": "text" + }, + { + "bbox": [ + 391, + 576, + 504, + 591 + ], + "score": 0.91, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 590, + 482, + 603 + ], + "score": 0.9, + "content": "\\{ \\mathcal { L } _ { i } \\} _ { i = 1 } ^ { 1 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 581, + 509, + 608 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "values of the indicator and variance of F2M and the Baseline are presented in Table 4, which clearly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 612, + 360, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 360, + 624 + ], + "score": 1.0, + "content": "demonstrate that our method can find a more flat local minima.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "Ablation study on the designs of our method. Here, we study the effectiveness of each design of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 402, + 653 + ], + "score": 1.0, + "content": "our method, including adding noise to the model parameters for finding", + "type": "text" + }, + { + "bbox": [ + 402, + 640, + 407, + 650 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "-flat local minima (FM)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 651, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 506, + 663 + ], + "score": 1.0, + "content": "during the base training session, the prototype fixing term (PF) used in the base training objective", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "(Eq. 4), parameter clamping (PC) during incremental learning, and prototype normalization (PN). We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "conduct an ablation study by removing each component in turn and report the experimental results in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 683, + 141, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 141, + 695 + ], + "score": 1.0, + "content": "Table 5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "score": 1.0, + "content": "Finding b-flat local minima. Standard supervised training with SGD as the optimizer tends to converge", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 711, + 504, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 504, + 724 + ], + "score": 1.0, + "content": "to a sharp local minima. 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MethodsessionsThe gap with cRT
1234567891011
cRT[25]*80.8378.5176.1273.9371.4668.9667.7366.7564.2262.5361.08
Joint-training*80.8377.5774.1170.7568.5265.9764.5862.2260.1858.4956.78-4.30
Baseline80.8777.1574.4672.2669.4767.1865.6263.6861.3059.7258.12-2.96
iCaRL [40]*79.5867.6364.1761.8058.1055.5153.3450.8948.6247.3445.60-15.48
Rebalance [20]*80.9470.3262.9657.1951.0646.7044.0340.1536.7534.8832.09-28.99
FSLL [34]*80.8377.3872.3771.8467.5165.3063.7561.1659.0558.0355.82-5.26
iCaRL [40]68.6852.6548.6144.1636.6229.5227.8326.2624.0123.8921.16-39.92
Rebalance [20]68.6857.1244.2128.7826.7125.6624.6221.5220.1220.0619.87-41.21
TOPIC[47]68.6862.4954.8149.9945.2541.4038.3535.3632.2228.3126.28-34.80
FSLL [34]72.7769.3365.5162.6661.1058.6557.7857.2655.5955.3954.21-6.87
FSLL+SS [34]75.6371.8168.1664.3262.6160.1058.8258.7056.4556.4155.82-5.26
IDLVQ-C [8]77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81-3.27
F2M81.0778.1675.5772.8970.8668.1767.0165.2663.3661.7660.26-0.82
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MethodIndicator IVariance g2
Training SetTesting SetTraining SetTesting Set
Baseline0.29930.45820.14510.2395
F2M0.05060.08000.02960.0334
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The main results on CIFAR-100, miniImageNet", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "and CUB-200-2011 are presented in Table 1, Table 2 and Table 3 respectively. Based on the experi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "ment results, we have the following observations: 1) The Baseline introduced in Sec. 3 outperforms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 408, + 507, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 507, + 423 + ], + "score": 1.0, + "content": "the state-of-the-art approaches on all incremental sessions. 2) As expected, cRT consistently outper-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 206, + 433 + ], + "score": 1.0, + "content": "forms the Baseline up to", + "type": "text" + }, + { + "bbox": [ + 206, + 420, + 221, + 431 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 421, + 232, + 433 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 232, + 420, + 247, + 431 + ], + "score": 0.87, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "by considering the data imbalance problem and applying proper", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "techniques to tackle the long-tailed classification problem to improve performance. Hence, it is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "reasonable to use cRT as the approximate upper bound of IFL. 3) Our F2M outperforms the state-of-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "the-art methods and the Baseline. Moreover, the performance of F2M is very close to the approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 282, + 477 + ], + "score": 1.0, + "content": "upper bound, i.e., the gap with cRT is only", + "type": "text" + }, + { + "bbox": [ + 283, + 464, + 306, + 474 + ], + "score": 0.87, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "in the last session on miniImageNet. The results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "show that even with strong constraints [20, 40, 34] and saved examplars of base classes [20, 40, 8],", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "current methods cannot effectively address the catastrophic forgetting problem. In contrast, finding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 496, + 401, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 401, + 510 + ], + "score": 1.0, + "content": "flat minima seems a promising approach to overcome this harsh problem.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 376, + 507, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 252, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 253, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 253, + 536 + ], + "score": 1.0, + "content": "5.3 Ablation Study and Analysis", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 543, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "Analysis on the flatness of local minima. Here, we verify that our method can find a more flat local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 554, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 310, + 564 + ], + "score": 1.0, + "content": "minima than the Baseline. For a found local minima", + "type": "text" + }, + { + "bbox": [ + 310, + 554, + 321, + 564 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 555, + 504, + 564 + ], + "score": 1.0, + "content": ", we measure its flatness as follows. We sample", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 392, + 577 + ], + "score": 1.0, + "content": "the noise for 1000 times. For each time, we inject the sampled noise to", + "type": "text" + }, + { + "bbox": [ + 392, + 565, + 402, + 575 + ], + "score": 0.88, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 564, + 491, + 577 + ], + "score": 1.0, + "content": "and calculate the loss", + "type": "text" + }, + { + "bbox": [ + 492, + 565, + 503, + 576 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 564, + 506, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 102, + 570, + 509, + 608 + ], + "spans": [ + { + "bbox": [ + 102, + 581, + 203, + 608 + ], + "score": 1.0, + "content": "to measure the flatness.", + "type": "text" + }, + { + "bbox": [ + 104, + 575, + 291, + 592 + ], + "score": 1.0, + "content": "Then, we adopt the indicator I = 11000 P1000i=1", + "type": "text" + }, + { + "bbox": [ + 203, + 591, + 216, + 601 + ], + "score": 0.87, + "content": "\\mathcal { L } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 581, + 294, + 608 + ], + "score": 1.0, + "content": "denotes the loss of", + "type": "text" + }, + { + "bbox": [ + 224, + 576, + 336, + 591 + ], + "score": 0.93, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 591, + 304, + 601 + ], + "score": 0.85, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 581, + 325, + 608 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 325, + 590, + 334, + 601 + ], + "score": 0.8, + "content": "\\overline { { \\mathcal { L } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 581, + 445, + 608 + ], + "score": 1.0, + "content": "1000denotes the average loss of", + "type": "text" + }, + { + "bbox": [ + 336, + 570, + 390, + 595 + ], + "score": 1.0, + "content": "and variance", + "type": "text" + }, + { + "bbox": [ + 391, + 576, + 504, + 591 + ], + "score": 0.91, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 590, + 482, + 603 + ], + "score": 0.9, + "content": "\\{ \\mathcal { L } _ { i } \\} _ { i = 1 } ^ { 1 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 581, + 509, + 608 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "values of the indicator and variance of F2M and the Baseline are presented in Table 4, which clearly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 612, + 360, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 360, + 624 + ], + "score": 1.0, + "content": "demonstrate that our method can find a more flat local minima.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 102, + 542, + 509, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "Ablation study on the designs of our method. Here, we study the effectiveness of each design of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 402, + 653 + ], + "score": 1.0, + "content": "our method, including adding noise to the model parameters for finding", + "type": "text" + }, + { + "bbox": [ + 402, + 640, + 407, + 650 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "-flat local minima (FM)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 651, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 506, + 663 + ], + "score": 1.0, + "content": "during the base training session, the prototype fixing term (PF) used in the base training objective", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "(Eq. 4), parameter clamping (PC) during incremental learning, and prototype normalization (PN). We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "conduct an ablation study by removing each component in turn and report the experimental results in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 683, + 141, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 141, + 695 + ], + "score": 1.0, + "content": "Table 5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 628, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 714 + ], + "score": 1.0, + "content": "Finding b-flat local minima. Standard supervised training with SGD as the optimizer tends to converge", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 711, + 504, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 504, + 724 + ], + "score": 1.0, + "content": "to a sharp local minima. It leads to a significant drop in performance because the loss changes quickly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 312, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 505, + 329 + ], + "score": 1.0, + "content": "in the neighborhood of the sharp local minima. As shown in Table 5, even with parameter clamping", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "during incremental learning, the performance still drops significantly. 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It can be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 434, + 491, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 491, + 447 + ], + "score": 1.0, + "content": "seen from Table 5 that removing parameter clamping leads to a significant drop in performance.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "Prototype normalization. As mentioned in Sec. 4.2, we normalize the class prototypes to calibrate the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 474 + ], + "score": 1.0, + "content": "estimates of the class mean classifier. 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We report the test accuracy in session 1 (base session) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 261, + 523 + ], + "score": 1.0, + "content": "session 9 (last session) w.r.t. different", + "type": "text" + }, + { + "bbox": [ + 261, + 511, + 267, + 521 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "in Table 6. 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Input: the flat region bound b,randomly initialized 0 = {Φ,ψ},the step sizes α and β. Training over base classes t = 1
for epoch k = 1,2.... do
for j = 1,2.., M do
Sample a noise vector ∈j ~ P(ε), s.t.-b ≤ ∈j ≤b;
Add the noise to the parameters of the embedding network,i.e.,0 = {Φ + ∈j,γ}; Compute the base loss Lbase with Eq. 4;
Reset the parameters,i.e.,0= {,ψ};
end
Update θ=0-αVL(θ) with the loss L defined in Eq.3.
Normalize and save the prototype of each base class;
// Incremental learning t ≥ 2
Combine the training data Dt and the exemplars saved in previous few-shot sessions 2 ≤ te <t;
for epoch k = 1,2.... do
Compute the metric-based classification loss Lm by Eq. 5;
Update Φ= Φ- βVLm(z;Φ); Clamp the parameters to ensure they fall in the flat minima region;
end
Randomly select and save a few exemplars from the training data Dt;
Normalize and save the prototype of each new class;
Output: Model parameters θ = {Φ,}.
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MethodsessionsThe gap with cRT
123456789
cRT[25]*67.3064.1560.5957.3254.2251.4348.9246.7844.85=
Joint-training*67.3062.3457.7954.0850.9347.6544.6442.6140.29-4.56
Baseline67.3063.1859.6256.3353.2850.5047.9645.8543.88-0.97
iCaRL [40]*67.3559.9155.6452.6049.4346.7344.1342.1740.29-4.56
Rebalance [20]*67.9163.1158.7554.8350.6847.1143.8841.1938.72-6.13
FSLL [34]*67.3059.8157.2654.5752.0549.4246.9544.9442.87-1.11
iCaRL[40]61.3146.3242.9437.6330.4924.0020.8918.8017.21-27.64
Rebalance [20]61.3147.8039.3131.9125.6821.3518.6717.2414.17-30.68
TOPIC [47]61.3150.0945.1741.1637.4835.5232.1929.4624.42-20.43
FSLL [34]66.4861.7558.1654.1651.1048.5346.5444.2042.28-2.57
FSLL+SS[34]68.8563.1459.2455.2352.2449.6547.7445.2343.92-0.93
IDLVQ-C[8]64.7759.8755.9352.6249.8847.5544.8343.1441.84-3.01
F2M67.2863.8060.3857.0654.0851.3948.8246.5844.65-0.20
" + }, + { + "category_id": 5, + "poly": [ + 356, + 276, + 1345, + 276, + 1345, + 682, + 356, + 682 + ], + "score": 0.985, + "html": "
MethodsessionsThe gap with cRT
123456789
cRT[25]*65.1863.8960.2057.2353.7150.3948.7747.2945.28=
Joint-training*65.1861.4557.3653.6850.8447.3344.7942.6240.08-5.20
Baseline65.1861.6758.6155.1151.8649.4347.6045.6443.83-1.45
iCaRL[40]*66.5257.2654.2750.6247.3344.9943.1441.1639.49-5.79
Rebalance [20]*66.6661.4257.2953.0248.8545.6843.0640.5638.35-6.93
FSLL [34]*65.1856.2454.5551.6149.1147.2745.3543.9542.22-3.08
iCaRL[40]64.1053.2841.6934.1327.9325.0620.4115.4813.73-31.55
Rebalance [20]64.1053.0543.9636.9731.6126.7321.2316.7813.54-31.74
TOPIC [47]64.1055.8847.0745.1640.1136.3833.9631.5529.37-15.91
FSLL [34]64.1055.8551.7148.5945.3443.2541.5239.8138.16-7.12
FSLL+SS [34]66.7655.5252.2049.1746.2344.6443.0741.2039.57-5.71
F2M64.7162.0559.0155.5852.5549.9648.0846.2844.67-0.61
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MethodsessionsThe gap with cRT
1234567891011
cRT[25]*80.8378.5176.1273.9371.4668.9667.7366.7564.2262.5361.08
Joint-training*80.8377.5774.1170.7568.5265.9764.5862.2260.1858.4956.78-4.30
Baseline80.8777.1574.4672.2669.4767.1865.6263.6861.3059.7258.12-2.96
iCaRL [40]*79.5867.6364.1761.8058.1055.5153.3450.8948.6247.3445.60-15.48
Rebalance [20]*80.9470.3262.9657.1951.0646.7044.0340.1536.7534.8832.09-28.99
FSLL [34]*80.8377.3872.3771.8467.5165.3063.7561.1659.0558.0355.82-5.26
iCaRL [40]68.6852.6548.6144.1636.6229.5227.8326.2624.0123.8921.16-39.92
Rebalance [20]68.6857.1244.2128.7826.7125.6624.6221.5220.1220.0619.87-41.21
TOPIC[47]68.6862.4954.8149.9945.2541.4038.3535.3632.2228.3126.28-34.80
FSLL [34]72.7769.3365.5162.6661.1058.6557.7857.2655.5955.3954.21-6.87
FSLL+SS [34]75.6371.8168.1664.3262.6160.1058.8258.7056.4556.4155.82-5.26
IDLVQ-C [8]77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81-3.27
F2M81.0778.1675.5772.8970.8668.1767.0165.2663.3661.7660.26-0.82
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MethodIndicator IVariance g2
Training SetTesting SetTraining SetTesting Set
Baseline0.29930.45820.14510.2395
F2M0.05060.08000.02960.0334
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SessionThe hyperparameter b
0.00250.0050.010.020.040.08
Session 1(60 bases classes)64.8564.6764.8164.7163.3062.25
Session 9 (All 100 classes)44.1644.5444.5844.6743.7543.04
Session 9 (60 base classes)59.5859.6959.7359.4458.3857.21
Session 9 (40 new classes)21.0321.8121.8622.5221.8021.77
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FMPFPCPNsessionsPD↓
123456789
广65.1860.8353.1343.5723.7510.7608.2607.2406.4558.73
√65.1859.4856.7752.9950.0947.8045.9244.2042.5522.63
√√64.7159.5453.0345.0941.6839.0438.6437.1936.0128.70
√√64.5561.2758.3354.8251.6049.2247.4845.7844.0820.47
√√√64.7161.7558.8055.3352.2749.7547.7246.0144.4320.28
√√√64.7161.9958.9955.5852.5549.9648.0846.2844.6720.04
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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. 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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. \ No newline at end of file diff --git a/parse/train/BJij4yg0Z/BJij4yg0Z_content_list.json b/parse/train/BJij4yg0Z/BJij4yg0Z_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e08c1691165cf64fd1c031f7e400cee2f7e92252 --- /dev/null +++ b/parse/train/BJij4yg0Z/BJij4yg0Z_content_list.json @@ -0,0 +1,1650 @@ +[ + { + "type": "text", + "text": "A BAYESIAN PERSPECTIVE ON GENERALIZATION AND STOCHASTIC GRADIENT DESCENT ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 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, + 764, + 491 + ], + "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": [ + 174, + 554, + 825, + 720 + ], + "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": [ + 174, + 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": [ + 217, + 103, + 823, + 160 + ], + "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": [ + 174, + 169, + 825, + 253 + ], + "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, + 825, + 359 + ], + "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": [ + 351, + 359, + 647, + 393 + ], + "page_idx": 1 + }, + { + "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, ", + "bbox": [ + 173, + 395, + 825, + 502 + ], + "page_idx": 1 + }, + { + "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", + "bbox": [ + 299, + 502, + 699, + 571 + ], + "page_idx": 1 + }, + { + "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": [ + 179, + 573, + 821, + 614 + ], + "page_idx": 1 + }, + { + "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", + "bbox": [ + 375, + 614, + 622, + 650 + ], + "page_idx": 1 + }, + { + "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": [ + 176, + 650, + 823, + 718 + ], + "page_idx": 1 + }, + { + "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": [ + 341, + 720, + 656, + 790 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 789, + 823, + 832 + ], + "page_idx": 1 + }, + { + "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": [ + 302, + 832, + 696, + 905 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1Equivalently, there is an optimal learning rate proportional to the batch size and the training set size. ", + "bbox": [ + 192, + 910, + 789, + 924 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 102, + 825, + 191 + ], + "page_idx": 2 + }, + { + "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": [ + 318, + 194, + 678, + 238 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 242, + 825, + 371 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 376, + 825, + 434 + ], + "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": [ + 401, + 438, + 593, + 472 + ], + "page_idx": 2 + }, + { + "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": [ + 174, + 474, + 826, + 503 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 510, + 825, + 650 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BAYES THEOREM AND GENERALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 670, + 544, + 685 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 700, + 825, + 825 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 832, + 823, + 888 + ], + "page_idx": 2 + }, + { + "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": [], + "bbox": [ + 191, + 108, + 807, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 366, + 825, + 421 + ], + "page_idx": 3 + }, + { + "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": [ + 173, + 429, + 825, + 582 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 588, + 825, + 672 + ], + "page_idx": 3 + }, + { + "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": [ + 189, + 710, + 807, + 881 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 4 + }, + { + "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": [ + 174, + 166, + 825, + 265 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 BAYES THEOREM AND STOCHASTIC GRADIENT DESCENT ", + "text_level": 1, + "bbox": [ + 176, + 287, + 676, + 301 + ], + "page_idx": 4 + }, + { + "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": [ + 174, + 318, + 825, + 472 + ], + "page_idx": 4 + }, + { + "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": [ + 174, + 478, + 825, + 589 + ], + "page_idx": 4 + }, + { + "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": [ + 174, + 597, + 825, + 666 + ], + "page_idx": 4 + }, + { + "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": [ + 191, + 700, + 805, + 881 + ], + "page_idx": 4 + }, + { + "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, + 111, + 810, + 291 + ], + "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": [ + 189, + 343, + 807, + 522 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 599, + 825, + 656 + ], + "page_idx": 5 + }, + { + "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": [ + 173, + 662, + 825, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 STOCHASTIC DIFFERENTIAL EQUATIONS AND THE SCALING RULES ", + "text_level": 1, + "bbox": [ + 173, + 767, + 758, + 782 + ], + "page_idx": 5 + }, + { + "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, ", + "bbox": [ + 173, + 797, + 825, + 882 + ], + "page_idx": 5 + }, + { + "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": [ + 357, + 886, + 638, + 929 + ], + "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, + 807, + 291 + ], + "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, + 556, + 823, + 617 + ], + "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, + 621, + 782, + 689 + ], + "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, + 693, + 825, + 792 + ], + "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": [ + 173, + 797, + 825, + 924 + ], + "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, + 112, + 807, + 292 + ], + "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, + 825, + 501 + ], + "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, + 508, + 825, + 606 + ], + "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, + 612, + 825, + 741 + ], + "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": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 328, + 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, + 127, + 823, + 156 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 178, + 287, + 193 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Alessandro Achille and Stefano Soatto. On the emergence of invariance and disentangling in deep representations. arXiv preprint arXiv:1706.01350, 2017. ", + "bbox": [ + 171, + 202, + 823, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lukas Balles, Javier Romero, and Philipp Hennig. 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", + "bbox": [ + 174, + 486, + 825, + 515 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A BAYESIAN POSTERIOR SAMPLING AND LANGEVIN DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 541, + 720, + 558 + ], + "page_idx": 9 + }, + { + "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, + 825, + 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, + 825, + 820 + ], + "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, + 825, + 825, + 854 + ], + "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, + 825, + 188 + ], + "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, + 825, + 236 + ], + "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": [ + 173, + 342, + 825, + 397 + ], + "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, + 867, + 823, + 924 + ], + "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, + 821, + 924 + ], + "page_idx": 12 + } +] \ No newline at end of file diff --git a/parse/train/BJij4yg0Z/BJij4yg0Z_middle.json b/parse/train/BJij4yg0Z/BJij4yg0Z_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..e2aca115239bb03a58672f4369350a00110028c8 --- /dev/null +++ b/parse/train/BJij4yg0Z/BJij4yg0Z_middle.json @@ -0,0 +1,34340 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "A BAYESIAN PERSPECTIVE ON GENERALIZATION AND", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 364, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 364, + 118 + ], + "score": 1.0, + "content": "STOCHASTIC GRADIENT DESCENT", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 262, + 169 + ], + "lines": [ + { + "bbox": [ + 112, + 134, + 249, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 134, + 249, + 147 + ], + "score": 1.0, + "content": "Samuel L. Smith∗ & Quoc V. Le", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 170, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 170, + 158 + ], + "score": 1.0, + "content": "Google Brain", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 157, + 263, + 170 + ], + "spans": [ + { + "bbox": [ + 113, + 157, + 263, + 170 + ], + "score": 1.0, + "content": "{slsmith, qvl}@google.com", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 209 + ], + "lines": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "spans": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 224, + 468, + 389 + ], + "lines": [ + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "score": 1.0, + "content": "We consider two questions at the heart of machine learning; how can we predict", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "score": 1.0, + "content": "if a minimum will generalize to the test set, and why does stochastic gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 259 + ], + "score": 1.0, + "content": "descent find minima that generalize well? Our work responds to Zhang et al.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "(2016), who showed deep neural networks can easily memorize randomly labeled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "score": 1.0, + "content": "training data, despite generalizing well on real labels of the same inputs. We show", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "score": 1.0, + "content": "that the same phenomenon occurs in small linear models. These observations are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 290, + 470, + 302 + ], + "spans": [ + { + "bbox": [ + 142, + 290, + 470, + 302 + ], + "score": 1.0, + "content": "explained by the Bayesian evidence, which penalizes sharp minima but is invariant", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 313 + ], + "score": 1.0, + "content": "to model parameterization. We also demonstrate that, when one holds the learning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "score": 1.0, + "content": "rate fixed, there is an optimum batch size which maximizes the test set accuracy.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 323, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 323, + 469, + 334 + ], + "score": 1.0, + "content": "We propose that the noise introduced by small mini-batches drives the parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "towards minima whose evidence is large. Interpreting stochastic gradient descent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 402, + 358 + ], + "score": 1.0, + "content": "as a stochastic differential equation, we identify the “noise scale”", + "type": "text" + }, + { + "bbox": [ + 402, + 343, + 469, + 357 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\overline { { g } } = \\epsilon ( \\frac { N } { B } - 1 ) \\approx } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 355, + 169, + 367 + ], + "score": 0.91, + "content": "\\epsilon N / B", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 356, + 200, + 367 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 200, + 357, + 206, + 365 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 356, + 288, + 367 + ], + "score": 1.0, + "content": "is the learning rate,", + "type": "text" + }, + { + "bbox": [ + 288, + 356, + 298, + 366 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 356, + 398, + 367 + ], + "score": 1.0, + "content": "the training set size and", + "type": "text" + }, + { + "bbox": [ + 398, + 356, + 408, + 365 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 356, + 469, + 367 + ], + "score": 1.0, + "content": "the batch size.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 366, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 470, + 379 + ], + "score": 1.0, + "content": "Consequently the optimum batch size is proportional to both the learning rate and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 377, + 461, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 250, + 390 + ], + "score": 1.0, + "content": "the size of the training set,", + "type": "text" + }, + { + "bbox": [ + 250, + 378, + 297, + 390 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 377, + 461, + 390 + ], + "score": 1.0, + "content": ". We verify these predictions empirically.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 205, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 208, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 208, + 429 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "score": 1.0, + "content": "This paper shows Bayesian principles can explain many recent observations in the deep learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "literature, while also discovering practical new insights. Zhang et al. (2016) trained deep convolu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "tional networks on ImageNet and CIFAR10, achieving excellent accuracy on both training and test", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "sets. They then took the same input images, but randomized the labels, and found that while their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "networks were now unable to generalize to the test set, they still memorized the training labels. They", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "claimed these results contradict learning theory, although this claim is disputed (Kawaguchi et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "2017; Dziugaite & Roy, 2017). Nonetheless, their results beg the question; if our models can assign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "arbitrary labels to the training set, why do they work so well in practice? Meanwhile Keskar et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "score": 1.0, + "content": "(2016) observed that if we hold the learning rate fixed and increase the batch size, the test accuracy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "usually falls. This striking result shows improving our estimate of the full-batch gradient can harm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "performance. Goyal et al. (2017) observed a linear scaling rule between batch size and learning rate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 560, + 488, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 488, + 573 + ], + "score": 1.0, + "content": "in a deep ResNet, while Hoffer et al. (2017) proposed a square root rule on theoretical grounds.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Many authors have suggested “broad minima” whose curvature is small may generalize better than", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "“sharp minima” whose curvature is large (Chaudhari et al., 2016; Hochreiter & Schmidhuber, 1997).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "score": 1.0, + "content": "Indeed, Dziugaite & Roy (2017) argued the results of Zhang et al. (2016) can be understood using", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "“nonvacuous” PAC-Bayes generalization bounds which penalize sharp minima, while Keskar et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "(2016) showed stochastic gradient descent (SGD) finds wider minima as the batch size is reduced.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "However Dinh et al. (2017) challenged this interpretation, by arguing that the curvature of a mini-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 497, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 497, + 655 + ], + "score": 1.0, + "content": "mum can be arbitrarily increased by changing the model parameterization. In this work we show:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 133, + 666, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 132, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 132, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "• The results of Zhang et al. (2016) are not unique to deep learning; we observe the same", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 141, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "phenomenon in a small “over-parameterized” linear model. We demonstrate that this phe-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "nomenon is straightforwardly understood by evaluating the Bayesian evidence in favor of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 699, + 502, + 712 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 502, + 712 + ], + "score": 1.0, + "content": "each model, which penalizes sharp minima but is invariant to the model parameterization.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 721, + 468, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 469, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 469, + 734 + ], + "score": 1.0, + "content": "∗Work done as a member of the Google Brain Residency Program (g.co/brainresidency)", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "A BAYESIAN PERSPECTIVE ON GENERALIZATION AND", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 364, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 364, + 118 + ], + "score": 1.0, + "content": "STOCHASTIC GRADIENT DESCENT", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 262, + 169 + ], + "lines": [ + { + "bbox": [ + 112, + 134, + 249, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 134, + 249, + 147 + ], + "score": 1.0, + "content": "Samuel L. Smith∗ & Quoc V. Le", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 170, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 170, + 158 + ], + "score": 1.0, + "content": "Google Brain", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 157, + 263, + 170 + ], + "spans": [ + { + "bbox": [ + 113, + 157, + 263, + 170 + ], + "score": 1.0, + "content": "{slsmith, qvl}@google.com", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 112, + 134, + 263, + 170 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 209 + ], + "lines": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "spans": [ + { + "bbox": [ + 276, + 196, + 336, + 212 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 224, + 468, + 389 + ], + "lines": [ + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 142, + 224, + 469, + 236 + ], + "score": 1.0, + "content": "We consider two questions at the heart of machine learning; how can we predict", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 247 + ], + "score": 1.0, + "content": "if a minimum will generalize to the test set, and why does stochastic gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 259 + ], + "score": 1.0, + "content": "descent find minima that generalize well? Our work responds to Zhang et al.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "(2016), who showed deep neural networks can easily memorize randomly labeled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 469, + 280 + ], + "score": 1.0, + "content": "training data, despite generalizing well on real labels of the same inputs. We show", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 291 + ], + "score": 1.0, + "content": "that the same phenomenon occurs in small linear models. These observations are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 290, + 470, + 302 + ], + "spans": [ + { + "bbox": [ + 142, + 290, + 470, + 302 + ], + "score": 1.0, + "content": "explained by the Bayesian evidence, which penalizes sharp minima but is invariant", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 313 + ], + "score": 1.0, + "content": "to model parameterization. We also demonstrate that, when one holds the learning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 325 + ], + "score": 1.0, + "content": "rate fixed, there is an optimum batch size which maximizes the test set accuracy.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 323, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 323, + 469, + 334 + ], + "score": 1.0, + "content": "We propose that the noise introduced by small mini-batches drives the parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "towards minima whose evidence is large. Interpreting stochastic gradient descent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 402, + 358 + ], + "score": 1.0, + "content": "as a stochastic differential equation, we identify the “noise scale”", + "type": "text" + }, + { + "bbox": [ + 402, + 343, + 469, + 357 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\overline { { g } } = \\epsilon ( \\frac { N } { B } - 1 ) \\approx } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 355, + 169, + 367 + ], + "score": 0.91, + "content": "\\epsilon N / B", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 356, + 200, + 367 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 200, + 357, + 206, + 365 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 356, + 288, + 367 + ], + "score": 1.0, + "content": "is the learning rate,", + "type": "text" + }, + { + "bbox": [ + 288, + 356, + 298, + 366 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 356, + 398, + 367 + ], + "score": 1.0, + "content": "the training set size and", + "type": "text" + }, + { + "bbox": [ + 398, + 356, + 408, + 365 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 356, + 469, + 367 + ], + "score": 1.0, + "content": "the batch size.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 366, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 470, + 379 + ], + "score": 1.0, + "content": "Consequently the optimum batch size is proportional to both the learning rate and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 377, + 461, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 250, + 390 + ], + "score": 1.0, + "content": "the size of the training set,", + "type": "text" + }, + { + "bbox": [ + 250, + 378, + 297, + 390 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 377, + 461, + 390 + ], + "score": 1.0, + "content": ". We verify these predictions empirically.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 224, + 470, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 413, + 205, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 208, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 208, + 429 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "score": 1.0, + "content": "This paper shows Bayesian principles can explain many recent observations in the deep learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "literature, while also discovering practical new insights. Zhang et al. (2016) trained deep convolu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "tional networks on ImageNet and CIFAR10, achieving excellent accuracy on both training and test", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "sets. They then took the same input images, but randomized the labels, and found that while their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "networks were now unable to generalize to the test set, they still memorized the training labels. They", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "claimed these results contradict learning theory, although this claim is disputed (Kawaguchi et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "2017; Dziugaite & Roy, 2017). Nonetheless, their results beg the question; if our models can assign", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "arbitrary labels to the training set, why do they work so well in practice? Meanwhile Keskar et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "score": 1.0, + "content": "(2016) observed that if we hold the learning rate fixed and increase the batch size, the test accuracy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "usually falls. This striking result shows improving our estimate of the full-batch gradient can harm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "performance. Goyal et al. (2017) observed a linear scaling rule between batch size and learning rate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 560, + 488, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 488, + 573 + ], + "score": 1.0, + "content": "in a deep ResNet, while Hoffer et al. (2017) proposed a square root rule on theoretical grounds.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 438, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Many authors have suggested “broad minima” whose curvature is small may generalize better than", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "“sharp minima” whose curvature is large (Chaudhari et al., 2016; Hochreiter & Schmidhuber, 1997).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 613 + ], + "score": 1.0, + "content": "Indeed, Dziugaite & Roy (2017) argued the results of Zhang et al. (2016) can be understood using", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "“nonvacuous” PAC-Bayes generalization bounds which penalize sharp minima, while Keskar et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "(2016) showed stochastic gradient descent (SGD) finds wider minima as the batch size is reduced.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "However Dinh et al. (2017) challenged this interpretation, by arguing that the curvature of a mini-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 497, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 497, + 655 + ], + "score": 1.0, + "content": "mum can be arbitrarily increased by changing the model parameterization. In this work we show:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 577, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 666, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 132, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 132, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "• The results of Zhang et al. (2016) are not unique to deep learning; we observe the same", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 141, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "phenomenon in a small “over-parameterized” linear model. We demonstrate that this phe-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "nomenon is straightforwardly understood by evaluating the Bayesian evidence in favor of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 699, + 502, + 712 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 502, + 712 + ], + "score": 1.0, + "content": "each model, which penalizes sharp minima but is invariant to the model parameterization.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 132, + 666, + 505, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 133, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 133, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 133, + 83, + 426, + 95 + ], + "score": 1.0, + "content": "• SGD integrates a stochastic differential equation whose “noise scale”", + "type": "text" + }, + { + "bbox": [ + 426, + 82, + 474, + 95 + ], + "score": 0.92, + "content": "g \\approx \\epsilon N / B", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 83, + 505, + 95 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 93, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 142, + 96, + 148, + 104 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 93, + 230, + 105 + ], + "score": 1.0, + "content": "is the learning rate,", + "type": "text" + }, + { + "bbox": [ + 230, + 94, + 241, + 104 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 93, + 325, + 105 + ], + "score": 1.0, + "content": "training set size and", + "type": "text" + }, + { + "bbox": [ + 326, + 94, + 335, + 104 + ], + "score": 0.76, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 93, + 504, + 105 + ], + "score": 1.0, + "content": "batch size. Noise drives SGD away from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 142, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "sharp minima, and therefore there is an optimal batch size which maximizes the test set", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "accuracy. This optimal batch size is proportional to the learning rate and training set size1.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 134, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "score": 1.0, + "content": "We describe Bayesian model comparison in section 2. In section 3 we replicate the observations of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 159 + ], + "score": 1.0, + "content": "Zhang et al. (2016) in a linear model, and show they are explained by the Bayesian evidence. In", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "section 4 we show there is an optimum batch size which maximizes the test set accuracy, and in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "section 5 we derive scaling rules between the optimum batch size, learning rate, training set size and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "momentum coefficient. Throughout this work, “generalization gap” refers to the gap in test accuracy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "between small and large batch SGD training, not the gap in accuracy between training and test sets.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 217, + 290, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 292, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 292, + 231 + ], + "score": 1.0, + "content": "2 BAYESIAN MODEL COMPARISON", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "Bayesian model comparison was first applied to neural networks in MacKay (1992). We provide a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 263 + ], + "score": 1.0, + "content": "brief tutorial here, since the theory is central to the remainder of the paper. For simplicity we first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 231, + 276 + ], + "score": 1.0, + "content": "consider a classification model", + "type": "text" + }, + { + "bbox": [ + 231, + 263, + 243, + 272 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 263, + 339, + 276 + ], + "score": 1.0, + "content": "with a single parameter", + "type": "text" + }, + { + "bbox": [ + 340, + 264, + 347, + 272 + ], + "score": 0.73, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 263, + 411, + 276 + ], + "score": 1.0, + "content": ", training inputs", + "type": "text" + }, + { + "bbox": [ + 411, + 264, + 418, + 272 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 263, + 495, + 276 + ], + "score": 1.0, + "content": "and training labels", + "type": "text" + }, + { + "bbox": [ + 495, + 264, + 501, + 274 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 263, + 505, + 276 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 489, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 489, + 286 + ], + "score": 1.0, + "content": "We can infer a posterior probability distribution over the parameter by applying Bayes theorem,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 285, + 396, + 312 + ], + "lines": [ + { + "bbox": [ + 215, + 285, + 396, + 312 + ], + "spans": [ + { + "bbox": [ + 215, + 285, + 396, + 312 + ], + "score": 0.92, + "content": "\\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}", + "type": "interline_equation", + "image_path": "638dd4ddf5315d75a1652360e2d31405a240f2094aae9efc929ab7fea5338e6a.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 285, + 396, + 298.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 215, + 298.5, + 396, + 312.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 311, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 179, + 327 + ], + "score": 1.0, + "content": "The likelihood,", + "type": "text" + }, + { + "bbox": [ + 179, + 313, + 407, + 327 + ], + "score": 0.78, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 311, + 446, + 327 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 446, + 314, + 504, + 326 + ], + "score": 0.89, + "content": "\\begin{array} { r l } { H ( \\omega ; M ) } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 211, + 338 + ], + "score": 0.91, + "content": "\\begin{array} { r } { - \\sum _ { i } \\ln \\left( P ( y _ { i } | \\omega , x _ { i } ; \\dot { M } ) \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "denotes the cross-entropy of unique categorical labels. We typically", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 198, + 353 + ], + "score": 1.0, + "content": "use a Gaussian prior,", + "type": "text" + }, + { + "bbox": [ + 198, + 336, + 319, + 352 + ], + "score": 0.91, + "content": "P ( \\omega ; M ) = \\sqrt { \\lambda / 2 \\pi } e ^ { - \\lambda \\omega ^ { 2 } / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 335, + 506, + 353 + ], + "score": 1.0, + "content": ", and therefore the posterior probability den-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 347, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 284, + 367 + ], + "score": 1.0, + "content": "sity of the parameter given the training data,", + "type": "text" + }, + { + "bbox": [ + 284, + 351, + 424, + 365 + ], + "score": 0.93, + "content": "P ( \\omega | y , x ; M ) \\propto \\sqrt { \\lambda / 2 \\pi } e ^ { - C ( \\omega ; M ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 347, + 455, + 367 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 455, + 351, + 506, + 364 + ], + "score": 0.91, + "content": "C ( \\omega ; M ) =", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 186, + 376 + ], + "score": 0.91, + "content": "H ( \\omega ; M ) + \\lambda \\omega ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 363, + 448, + 378 + ], + "score": 1.0, + "content": "denotes the L2 regularized cross entropy, or “cost function”, and", + "type": "text" + }, + { + "bbox": [ + 449, + 365, + 456, + 374 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 363, + 505, + 378 + ], + "score": 1.0, + "content": "is the regu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 237, + 388 + ], + "score": 1.0, + "content": "larization coefficient. The value", + "type": "text" + }, + { + "bbox": [ + 237, + 376, + 249, + 387 + ], + "score": 0.86, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "which minimizes the cost function lies at the maximum of this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 385, + 478, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 263, + 399 + ], + "score": 1.0, + "content": "posterior. To predict an unknown label", + "type": "text" + }, + { + "bbox": [ + 263, + 388, + 272, + 398 + ], + "score": 0.84, + "content": "y _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 385, + 333, + 399 + ], + "score": 1.0, + "content": "of a new input", + "type": "text" + }, + { + "bbox": [ + 334, + 388, + 343, + 397 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 385, + 478, + 399 + ], + "score": 1.0, + "content": ", we should compute the integral,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 398, + 428, + 453 + ], + "lines": [ + { + "bbox": [ + 183, + 398, + 428, + 453 + ], + "spans": [ + { + "bbox": [ + 183, + 398, + 428, + 453 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5a66ef55f712d0bbc7f7a762975b13c55f437b17b7e6894f815d285a9f163ce5.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 183, + 398, + 428, + 416.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 416.3333333333333, + 428, + 434.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 183, + 434.66666666666663, + 428, + 452.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 110, + 454, + 503, + 487 + ], + "lines": [ + { + "bbox": [ + 108, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 108, + 453, + 342, + 466 + ], + "score": 1.0, + "content": "However these integrals are dominated by the region near", + "type": "text" + }, + { + "bbox": [ + 342, + 456, + 353, + 465 + ], + "score": 0.76, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 453, + 399, + 466 + ], + "score": 1.0, + "content": ", and since", + "type": "text" + }, + { + "bbox": [ + 399, + 454, + 461, + 466 + ], + "score": 0.9, + "content": "P ( y _ { t } | \\omega , x _ { t } ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "is smooth", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 108, + 464, + 205, + 478 + ], + "score": 1.0, + "content": "we usually approximate", + "type": "text" + }, + { + "bbox": [ + 205, + 465, + 356, + 477 + ], + "score": 0.91, + "content": "P ( y _ { t } | x _ { t } , x , y ; M ) \\approx P ( y _ { t } | \\omega _ { 0 } , x _ { t } ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 464, + 437, + 478 + ], + "score": 1.0, + "content": ". Having minimized", + "type": "text" + }, + { + "bbox": [ + 437, + 466, + 475, + 477 + ], + "score": 0.92, + "content": "C ( \\omega ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "to find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 475, + 492, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 479, + 118, + 487 + ], + "score": 0.74, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 475, + 492, + 489 + ], + "score": 1.0, + "content": ", we now wish to compare two different models and select the best one. The probability ratio,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 487, + 381, + 515 + ], + "lines": [ + { + "bbox": [ + 230, + 487, + 381, + 515 + ], + "spans": [ + { + "bbox": [ + 230, + 487, + 381, + 515 + ], + "score": 0.93, + "content": "\\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 } ) } .", + "type": "interline_equation", + "image_path": "dcefd7747160699b3d2c79f64482749cf659571784bd1b9a54cfb67fdc34e033.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 487, + 381, + 501.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 230, + 501.0, + 381, + 515.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 515, + 504, + 569 + ], + "lines": [ + { + "bbox": [ + 107, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "The second factor on the right is the prior ratio, which describes which model is most plausible. To", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "avoid unnecessary subjectivity, we usually set this to 1. Meanwhile the first factor on the right is the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "evidence ratio, which controls how much the training data changes our prior beliefs. 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In", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "section 4 we show there is an optimum batch size which maximizes the test set accuracy, and in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "section 5 we derive scaling rules between the optimum batch size, learning rate, training set size and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "momentum coefficient. 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The value", + "type": "text" + }, + { + "bbox": [ + 237, + 376, + 249, + 387 + ], + "score": 0.86, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "which minimizes the cost function lies at the maximum of this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 385, + 478, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 263, + 399 + ], + "score": 1.0, + "content": "posterior. To predict an unknown label", + "type": "text" + }, + { + "bbox": [ + 263, + 388, + 272, + 398 + ], + "score": 0.84, + "content": "y _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 385, + 333, + 399 + ], + "score": 1.0, + "content": "of a new input", + "type": "text" + }, + { + "bbox": [ + 334, + 388, + 343, + 397 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 385, + 478, + 399 + ], + "score": 1.0, + "content": ", we should compute the integral,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 311, + 506, + 399 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 398, + 428, + 453 + ], + "lines": [ + { + "bbox": [ + 183, + 398, + 428, + 453 + ], + "spans": [ + { + "bbox": [ + 183, + 398, + 428, + 453 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "5a66ef55f712d0bbc7f7a762975b13c55f437b17b7e6894f815d285a9f163ce5.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 183, + 398, + 428, + 416.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 416.3333333333333, + 428, + 434.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 183, + 434.66666666666663, + 428, + 452.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 110, + 454, + 503, + 487 + ], + "lines": [ + { + "bbox": [ + 108, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 108, + 453, + 342, + 466 + ], + "score": 1.0, + "content": "However these integrals are dominated by the region near", + "type": "text" + }, + { + "bbox": [ + 342, + 456, + 353, + 465 + ], + "score": 0.76, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 453, + 399, + 466 + ], + "score": 1.0, + "content": ", and since", + "type": "text" + }, + { + "bbox": [ + 399, + 454, + 461, + 466 + ], + "score": 0.9, + "content": "P ( y _ { t } | \\omega , x _ { t } ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "is smooth", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 108, + 464, + 205, + 478 + ], + "score": 1.0, + "content": "we usually approximate", + "type": "text" + }, + { + "bbox": [ + 205, + 465, + 356, + 477 + ], + "score": 0.91, + "content": "P ( y _ { t } | x _ { t } , x , y ; M ) \\approx P ( y _ { t } | \\omega _ { 0 } , x _ { t } ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 464, + 437, + 478 + ], + "score": 1.0, + "content": ". Having minimized", + "type": "text" + }, + { + "bbox": [ + 437, + 466, + 475, + 477 + ], + "score": 0.92, + "content": "C ( \\omega ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "to find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 475, + 492, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 479, + 118, + 487 + ], + "score": 0.74, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 475, + 492, + 489 + ], + "score": 1.0, + "content": ", we now wish to compare two different models and select the best one. The probability ratio,", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 107, + 453, + 505, + 489 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 487, + 381, + 515 + ], + "lines": [ + { + "bbox": [ + 230, + 487, + 381, + 515 + ], + "spans": [ + { + "bbox": [ + 230, + 487, + 381, + 515 + ], + "score": 0.93, + "content": "\\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 } ) } .", + "type": "interline_equation", + "image_path": "dcefd7747160699b3d2c79f64482749cf659571784bd1b9a54cfb67fdc34e033.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 487, + 381, + 501.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 230, + 501.0, + 381, + 515.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 515, + 504, + 569 + ], + "lines": [ + { + "bbox": [ + 107, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "The second factor on the right is the prior ratio, which describes which model is most plausible. To", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "avoid unnecessary subjectivity, we usually set this to 1. Meanwhile the first factor on the right is the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "evidence ratio, which controls how much the training data changes our prior beliefs. Germain et al.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 561 + ], + "score": 1.0, + "content": "(2016) showed that maximizing the evidence (or “marginal likelihood”) minimizes a PAC-Bayes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 559, + 464, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 464, + 572 + ], + "score": 1.0, + "content": "generalization bound. To compute it, we evaluate the normalizing constant of equation 1,", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 515, + 505, + 572 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 571, + 402, + 626 + ], + "lines": [ + { + "bbox": [ + 209, + 571, + 402, + 626 + ], + "spans": [ + { + "bbox": [ + 209, + 571, + 402, + 626 + ], + "score": 0.94, + "content": "\\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}", + "type": "interline_equation", + "image_path": "7273c681ce2981db1167b6c0ac81b39d37907cba1378976e53e9f1d2f50ba37e.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 571, + 402, + 584.75 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 209, + 584.75, + 402, + 598.5 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 209, + 598.5, + 402, + 612.25 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 209, + 612.25, + 402, + 626.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 625, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "Notice that the evidence is computed by integrating out the parameters; and consequently it is invari-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 636, + 504, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 504, + 648 + ], + "score": 1.0, + "content": "ant to the model parameterization. Since this integral is dominated by the region near the minimum", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 646, + 502, + 660 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 118, + 658 + ], + "score": 0.8, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 646, + 325, + 660 + ], + "score": 1.0, + "content": ", we can estimate the evidence by Taylor expanding", + "type": "text" + }, + { + "bbox": [ + 325, + 646, + 499, + 659 + ], + "score": 0.89, + "content": "C ( \\omega ; M ) \\approx \\dot { C } ( \\omega _ { 0 } ) + \\dot { C } ^ { \\prime \\prime } ( \\omega _ { 0 } ) ( \\omega - \\omega _ { 0 } ) ^ { 2 } / 2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 646, + 502, + 660 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 624, + 505, + 660 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 659, + 426, + 717 + ], + "lines": [ + { + "bbox": [ + 185, + 659, + 426, + 717 + ], + "spans": [ + { + "bbox": [ + 185, + 659, + 426, + 717 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "4d35e3757beb121e0bd4a872e8801447132e1c4aea77012af4697d8b49190471.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 659, + 426, + 673.5 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 185, + 673.5, + 426, + 688.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 185, + 688.0, + 426, + 702.5 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 185, + 702.5, + 426, + 717.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 118, + 721, + 483, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 484, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 484, + 734 + ], + "score": 1.0, + "content": "1Equivalently, there is an optimal learning rate proportional to the batch size and the training set size.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48, + "bbox_fs": [ + 119, + 720, + 484, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Within this “Laplace” approximation, the evidence is controlled by the value of the cost function at", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the minimum, and by the logarithm of the ratio of the curvature about this minimum compared to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "the regularization constant. Thus far we have considered models of a single parameter; in realistic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 103, + 112, + 508, + 135 + ], + "spans": [ + { + "bbox": [ + 103, + 112, + 228, + 135 + ], + "score": 1.0, + "content": "models with many parameters", + "type": "text" + }, + { + "bbox": [ + 228, + 115, + 394, + 130 + ], + "score": 0.92, + "content": "P ( y | x ; M ) \\approx \\lambda ^ { \\frac { p } { 2 } } e ^ { - C ( \\omega _ { 0 } ) } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 112, + 425, + 135 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 426, + 117, + 479, + 130 + ], + "score": 0.92, + "content": "| \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 112, + 508, + 135 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 235, + 142 + ], + "score": 1.0, + "content": "determinant of the Hessian, and", + "type": "text" + }, + { + "bbox": [ + 235, + 131, + 241, + 140 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 128, + 505, + 142 + ], + "score": 1.0, + "content": "denotes the number of model parameters (Kass & Raftery, 1995).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 480, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 395, + 154 + ], + "score": 1.0, + "content": "The determinant of the Hessian is simply the product of its eigenvalues,", + "type": "text" + }, + { + "bbox": [ + 396, + 140, + 437, + 153 + ], + "score": 0.92, + "content": "( \\prod _ { i = 1 } ^ { p } \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 139, + 480, + 154 + ], + "score": 1.0, + "content": ", and thus,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 154, + 415, + 189 + ], + "lines": [ + { + "bbox": [ + 195, + 154, + 415, + 189 + ], + "spans": [ + { + "bbox": [ + 195, + 154, + 415, + 189 + ], + "score": 0.94, + "content": "P ( y | x ; M ) \\approx \\exp \\Bigg \\{ - \\left( C ( \\omega _ { 0 } ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { p } \\ln ( \\lambda _ { i } / \\lambda ) \\right) \\Bigg \\} .", + "type": "interline_equation", + "image_path": "4078e8618d773291ae1f3438d48fd4149d358d3f863d89216c7f967746d94848.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 154, + 415, + 171.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 195, + 171.5, + 415, + 189.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 103, + 190, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 103, + 190, + 177, + 210 + ], + "score": 1.0, + "content": "The contribution", + "type": "text" + }, + { + "bbox": [ + 178, + 191, + 260, + 207 + ], + "score": 0.93, + "content": "( \\lambda ^ { \\frac { p } { 2 } } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 190, + 507, + 210 + ], + "score": 1.0, + "content": "is often called the “Occam factor”, because it enforces Oc-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "cam’s razor; when two models describe the data equally well, the simpler model is usually better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "(Gull, 1988). Minima with low curvature are simple, because the parameters do not have to be fine-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "tuned to fit the data. Intuitively, the Occam factor describes the fraction of the prior parameter space", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "consistent with the data. Since this fraction is always less than one, we propose to approximate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 469, + 262 + ], + "score": 1.0, + "content": "equation 9 away from local minima by only performing the summation over eigenvalues", + "type": "text" + }, + { + "bbox": [ + 469, + 249, + 501, + 261 + ], + "score": 0.91, + "content": "\\lambda _ { i } \\geq \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 249, + 506, + 262 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "The evidence can be reframed in the language of information theory, whereby Occam’s factor penal-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "izes the amount of information the model must learn about the parameters to accurately model the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 495, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 495, + 295 + ], + "score": 1.0, + "content": "training data (Hinton & Van Camp, 1993; Achille & Soatto, 2017; Shwartz-Ziv & Tishby, 2017).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "In this work, we will compare the evidence against a null model which assumes the labels are entirely", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "random, assigning equal probability to each class. This unusual model has no parameters, and so the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 320, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 298, + 335 + ], + "score": 1.0, + "content": "evidence is controlled by the likelihood alone,", + "type": "text" + }, + { + "bbox": [ + 298, + 321, + 464, + 334 + ], + "score": 0.92, + "content": "P ( y | x ; N U L L ) = ( 1 / n ) ^ { N } \\stackrel { - } { = } e ^ { - N \\ln { ( n ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 320, + 496, + 335 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 497, + 324, + 504, + 332 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 503, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 272, + 345 + ], + "score": 1.0, + "content": "denotes the number of model classes and", + "type": "text" + }, + { + "bbox": [ + 272, + 334, + 282, + 343 + ], + "score": 0.85, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 333, + 503, + 345 + ], + "score": 1.0, + "content": "the number of training labels. Thus the evidence ratio,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 347, + 363, + 374 + ], + "lines": [ + { + "bbox": [ + 246, + 347, + 363, + 374 + ], + "spans": [ + { + "bbox": [ + 246, + 347, + 363, + 374 + ], + "score": 0.94, + "content": "\\frac { P ( y | x ; M ) } { P ( y | x ; N U L L ) } = e ^ { - E ( \\omega _ { 0 } ) } ,", + "type": "interline_equation", + "image_path": "308e11f4919465f708f251adcb3731dc2ceb7ea6fcb80afc3aa7e9027ac7d11e.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 347, + 363, + 360.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 246, + 360.5, + 363, + 374.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 135, + 390 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 135, + 376, + 333, + 390 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E ( \\omega _ { 0 } ) = C ( \\omega _ { 0 } ) + ( 1 / 2 ) \\sum _ { i } \\ln ( \\lambda _ { i } / \\lambda ) - N \\ln ( n ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "is the log evidence ratio in favor of the null", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 456, + 400 + ], + "score": 1.0, + "content": "model. Clearly, we should only assign any confidence to the predictions of our model if", + "type": "text" + }, + { + "bbox": [ + 456, + 388, + 501, + 400 + ], + "score": 0.92, + "content": "E ( \\omega _ { 0 } ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 387, + 505, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "The evidence supports the intuition that broad minima generalize better than sharp minima, but", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "unlike the curvature it does not depend on the model parameterization. Dinh et al. (2017) showed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "one can increase the Hessian eigenvalues by rescaling the parameters, but they must simultaneously", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "rescale the regularization coefficients, otherwise the model changes. Since Occam’s factor arises", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 181, + 461 + ], + "score": 1.0, + "content": "from the log ratio,", + "type": "text" + }, + { + "bbox": [ + 181, + 448, + 219, + 461 + ], + "score": 0.92, + "content": "\\ln \\left( \\lambda _ { i } / \\lambda \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 449, + 506, + 461 + ], + "score": 1.0, + "content": ", these two effects cancel out2. It is difficult to evaluate the evidence for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "deep networks, as we cannot compute the Hessian of millions of parameters. Additionally, neural", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "networks exhibit many equivalent minima, since we can permute the hidden units without changing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "the model. To compute the evidence we must carefully account for this “degeneracy”. We argue", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "these issues are not a major limitation, since the intuition we build studying the evidence in simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 504, + 495, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 495, + 515 + ], + "score": 1.0, + "content": "cases will be sufficient to explain the results of both Zhang et al. (2016) and Keskar et al. (2016).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 531, + 333, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 335, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 335, + 546 + ], + "score": 1.0, + "content": "3 BAYES THEOREM AND GENERALIZATION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "Zhang et al. (2016) showed that deep neural networks generalize well on training inputs with in-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "score": 1.0, + "content": "formative labels, and yet the same model can drastically overfit on the same input images when the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "labels are randomized; perfectly memorizing the training set. To demonstrate that these observations", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "are not unique to deep networks, let’s consider a far simpler model; logistic regression. We form a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 489, + 611 + ], + "score": 1.0, + "content": "small balanced training set comprising 800 images from MNIST, of which half have true label", + "type": "text" + }, + { + "bbox": [ + 490, + 599, + 504, + 609 + ], + "score": 0.27, + "content": "\\mathbf { \\vec { \\Delta } } ^ { 6 } 0 ^ { 9 }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "and half true label “1”. Our test set is also balanced, comprising 5000 MNIST images of zeros and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "5000 MNIST images of ones. There are two tasks. In the first task, the labels of both the training and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "score": 1.0, + "content": "test sets are randomized. In the second task, the labels are informative, matching the true MNIST", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 643, + 455, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 455, + 655 + ], + "score": 1.0, + "content": "labels. Since the images contain 784 pixels, our model has just 784 weights and 1 bias.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "We show the accuracy of the model predictions on both the training and test sets in figure 1. When", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "trained on the informative labels, the model generalizes well to the test set, so long as it is weakly", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "regularized. However the model also perfectly memorizes the random labels, replicating the obser-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "score": 1.0, + "content": "vations of Zhang et al. (2016) in deep networks. No significant improvement in model performance", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 710, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 710, + 504, + 723 + ], + "score": 1.0, + "content": "2Note however that while the evidence itself is invariant to model parameterization, one can find reparame-", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 411, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 411, + 732 + ], + "score": 1.0, + "content": "terizations which change the approximate evidence after the Laplace approximation.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Within this “Laplace” approximation, the evidence is controlled by the value of the cost function at", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the minimum, and by the logarithm of the ratio of the curvature about this minimum compared to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "the regularization constant. Thus far we have considered models of a single parameter; in realistic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 103, + 112, + 508, + 135 + ], + "spans": [ + { + "bbox": [ + 103, + 112, + 228, + 135 + ], + "score": 1.0, + "content": "models with many parameters", + "type": "text" + }, + { + "bbox": [ + 228, + 115, + 394, + 130 + ], + "score": 0.92, + "content": "P ( y | x ; M ) \\approx \\lambda ^ { \\frac { p } { 2 } } e ^ { - C ( \\omega _ { 0 } ) } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 112, + 425, + 135 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 426, + 117, + 479, + 130 + ], + "score": 0.92, + "content": "| \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 112, + 508, + 135 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 235, + 142 + ], + "score": 1.0, + "content": "determinant of the Hessian, and", + "type": "text" + }, + { + "bbox": [ + 235, + 131, + 241, + 140 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 128, + 505, + 142 + ], + "score": 1.0, + "content": "denotes the number of model parameters (Kass & Raftery, 1995).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 480, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 395, + 154 + ], + "score": 1.0, + "content": "The determinant of the Hessian is simply the product of its eigenvalues,", + "type": "text" + }, + { + "bbox": [ + 396, + 140, + 437, + 153 + ], + "score": 0.92, + "content": "( \\prod _ { i = 1 } ^ { p } \\lambda _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 139, + 480, + 154 + ], + "score": 1.0, + "content": ", and thus,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 103, + 82, + 508, + 154 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 154, + 415, + 189 + ], + "lines": [ + { + "bbox": [ + 195, + 154, + 415, + 189 + ], + "spans": [ + { + "bbox": [ + 195, + 154, + 415, + 189 + ], + "score": 0.94, + "content": "P ( y | x ; M ) \\approx \\exp \\Bigg \\{ - \\left( C ( \\omega _ { 0 } ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { p } \\ln ( \\lambda _ { i } / \\lambda ) \\right) \\Bigg \\} .", + "type": "interline_equation", + "image_path": "4078e8618d773291ae1f3438d48fd4149d358d3f863d89216c7f967746d94848.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 154, + 415, + 171.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 195, + 171.5, + 415, + 189.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 505, + 294 + ], + "lines": [ + { + "bbox": [ + 103, + 190, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 103, + 190, + 177, + 210 + ], + "score": 1.0, + "content": "The contribution", + "type": "text" + }, + { + "bbox": [ + 178, + 191, + 260, + 207 + ], + "score": 0.93, + "content": "( \\lambda ^ { \\frac { p } { 2 } } / | \\nabla \\nabla C ( \\omega ) | _ { \\omega _ { 0 } } ^ { 1 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 190, + 507, + 210 + ], + "score": 1.0, + "content": "is often called the “Occam factor”, because it enforces Oc-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "cam’s razor; when two models describe the data equally well, the simpler model is usually better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 229 + ], + "score": 1.0, + "content": "(Gull, 1988). Minima with low curvature are simple, because the parameters do not have to be fine-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "tuned to fit the data. Intuitively, the Occam factor describes the fraction of the prior parameter space", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "consistent with the data. Since this fraction is always less than one, we propose to approximate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 469, + 262 + ], + "score": 1.0, + "content": "equation 9 away from local minima by only performing the summation over eigenvalues", + "type": "text" + }, + { + "bbox": [ + 469, + 249, + 501, + 261 + ], + "score": 0.91, + "content": "\\lambda _ { i } \\geq \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 249, + 506, + 262 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "The evidence can be reframed in the language of information theory, whereby Occam’s factor penal-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "izes the amount of information the model must learn about the parameters to accurately model the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 495, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 495, + 295 + ], + "score": 1.0, + "content": "training data (Hinton & Van Camp, 1993; Achille & Soatto, 2017; Shwartz-Ziv & Tishby, 2017).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 103, + 190, + 507, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "In this work, we will compare the evidence against a null model which assumes the labels are entirely", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "random, assigning equal probability to each class. This unusual model has no parameters, and so the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 320, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 298, + 335 + ], + "score": 1.0, + "content": "evidence is controlled by the likelihood alone,", + "type": "text" + }, + { + "bbox": [ + 298, + 321, + 464, + 334 + ], + "score": 0.92, + "content": "P ( y | x ; N U L L ) = ( 1 / n ) ^ { N } \\stackrel { - } { = } e ^ { - N \\ln { ( n ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 320, + 496, + 335 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 497, + 324, + 504, + 332 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 503, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 272, + 345 + ], + "score": 1.0, + "content": "denotes the number of model classes and", + "type": "text" + }, + { + "bbox": [ + 272, + 334, + 282, + 343 + ], + "score": 0.85, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 333, + 503, + 345 + ], + "score": 1.0, + "content": "the number of training labels. Thus the evidence ratio,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 299, + 505, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 347, + 363, + 374 + ], + "lines": [ + { + "bbox": [ + 246, + 347, + 363, + 374 + ], + "spans": [ + { + "bbox": [ + 246, + 347, + 363, + 374 + ], + "score": 0.94, + "content": "\\frac { P ( y | x ; M ) } { P ( y | x ; N U L L ) } = e ^ { - E ( \\omega _ { 0 } ) } ,", + "type": "interline_equation", + "image_path": "308e11f4919465f708f251adcb3731dc2ceb7ea6fcb80afc3aa7e9027ac7d11e.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 347, + 363, + 360.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 246, + 360.5, + 363, + 374.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 135, + 390 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 135, + 376, + 333, + 390 + ], + "score": 0.92, + "content": "\\begin{array} { r } { E ( \\omega _ { 0 } ) = C ( \\omega _ { 0 } ) + ( 1 / 2 ) \\sum _ { i } \\ln ( \\lambda _ { i } / \\lambda ) - N \\ln ( n ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "is the log evidence ratio in favor of the null", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 456, + 400 + ], + "score": 1.0, + "content": "model. Clearly, we should only assign any confidence to the predictions of our model if", + "type": "text" + }, + { + "bbox": [ + 456, + 388, + 501, + 400 + ], + "score": 0.92, + "content": "E ( \\omega _ { 0 } ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 387, + 505, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 376, + 506, + 400 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "The evidence supports the intuition that broad minima generalize better than sharp minima, but", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "unlike the curvature it does not depend on the model parameterization. Dinh et al. (2017) showed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "one can increase the Hessian eigenvalues by rescaling the parameters, but they must simultaneously", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "rescale the regularization coefficients, otherwise the model changes. Since Occam’s factor arises", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 181, + 461 + ], + "score": 1.0, + "content": "from the log ratio,", + "type": "text" + }, + { + "bbox": [ + 181, + 448, + 219, + 461 + ], + "score": 0.92, + "content": "\\ln \\left( \\lambda _ { i } / \\lambda \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 449, + 506, + 461 + ], + "score": 1.0, + "content": ", these two effects cancel out2. It is difficult to evaluate the evidence for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "deep networks, as we cannot compute the Hessian of millions of parameters. Additionally, neural", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "networks exhibit many equivalent minima, since we can permute the hidden units without changing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "the model. To compute the evidence we must carefully account for this “degeneracy”. We argue", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "these issues are not a major limitation, since the intuition we build studying the evidence in simple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 504, + 495, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 495, + 515 + ], + "score": 1.0, + "content": "cases will be sufficient to explain the results of both Zhang et al. (2016) and Keskar et al. (2016).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 405, + 506, + 515 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 531, + 333, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 335, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 335, + 546 + ], + "score": 1.0, + "content": "3 BAYES THEOREM AND GENERALIZATION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "Zhang et al. (2016) showed that deep neural networks generalize well on training inputs with in-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 504, + 578 + ], + "score": 1.0, + "content": "formative labels, and yet the same model can drastically overfit on the same input images when the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "labels are randomized; perfectly memorizing the training set. To demonstrate that these observations", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "are not unique to deep networks, let’s consider a far simpler model; logistic regression. We form a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 489, + 611 + ], + "score": 1.0, + "content": "small balanced training set comprising 800 images from MNIST, of which half have true label", + "type": "text" + }, + { + "bbox": [ + 490, + 599, + 504, + 609 + ], + "score": 0.27, + "content": "\\mathbf { \\vec { \\Delta } } ^ { 6 } 0 ^ { 9 }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "and half true label “1”. Our test set is also balanced, comprising 5000 MNIST images of zeros and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "5000 MNIST images of ones. There are two tasks. In the first task, the labels of both the training and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "score": 1.0, + "content": "test sets are randomized. In the second task, the labels are informative, matching the true MNIST", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 643, + 455, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 455, + 655 + ], + "score": 1.0, + "content": "labels. Since the images contain 784 pixels, our model has just 784 weights and 1 bias.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 555, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "We show the accuracy of the model predictions on both the training and test sets in figure 1. When", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "trained on the informative labels, the model generalizes well to the test set, so long as it is weakly", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "regularized. However the model also perfectly memorizes the random labels, replicating the obser-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "score": 1.0, + "content": "vations of Zhang et al. (2016) in deep networks. No significant improvement in model performance", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 291, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 302 + ], + "score": 1.0, + "content": "is observed as the regularization coefficient increases. For completeness, we also evaluate the mean", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "margin between training examples and the decision boundary. For both random and informative", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "labels, the margin drops significantly as we reduce the regularization coefficient. When weakly", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 499, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 268, + 335 + ], + "score": 1.0, + "content": "regularized, the mean margin is roughly", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 268, + 323, + 288, + 334 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 288, + 324, + 499, + 335 + ], + "score": 1.0, + "content": "larger for informative labels than for random labels.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 659, + 506, + 706 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 86, + 494, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 86, + 494, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 86, + 494, + 223 + ], + "spans": [ + { + "bbox": [ + 117, + 86, + 494, + 223 + ], + "score": 0.953, + "type": "image", + "image_path": "b54a96e24409024f1dba39245529ce32aa588511ceab8602abf0ec1f279994eb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 86, + 494, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 131.66666666666666, + 494, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 177.33333333333331, + 494, + 222.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 233, + 505, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "Figure 1: Prediction accuracy and mean training set margin as a function of regularization coeffi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "cient, for a logistic regression trained on random (a) and informative (b) labels of the same inputs.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "The weakly regularized model generalizes well on informative labels but memorizes random labels.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 302 + ], + "score": 1.0, + "content": "is observed as the regularization coefficient increases. For completeness, we also evaluate the mean", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "margin between training examples and the decision boundary. For both random and informative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "labels, the margin drops significantly as we reduce the regularization coefficient. When weakly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 323, + 499, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 268, + 335 + ], + "score": 1.0, + "content": "regularized, the mean margin is roughly", + "type": "text" + }, + { + "bbox": [ + 268, + 323, + 288, + 334 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 324, + 499, + 335 + ], + "score": 1.0, + "content": "larger for informative labels than for random labels.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Now consider figure 2, where we plot the mean cross-entropy of the model predictions, evaluated on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "both training and test sets, as well as the Bayesian log evidence ratio defined in the previous section.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "Looking first at the random label experiment in figure 2a, while the cross-entropy on the training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "set vanishes when the model is weakly regularized, the cross-entropy on the test set explodes. Not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "only does the model make random predictions, but it is extremely confident in those predictions. As", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 442, + 407 + ], + "score": 1.0, + "content": "the regularization coefficient is increased the test set cross-entropy falls, settling at", + "type": "text" + }, + { + "bbox": [ + 443, + 395, + 459, + 406 + ], + "score": 0.45, + "content": "\\ln 2", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 396, + 505, + 407 + ], + "score": 1.0, + "content": ", the cross-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "entropy of assigning equal probability to both classes. Now consider the Bayesian evidence, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "we evaluate on the training set. The log evidence ratio is large and positive when the model is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "weakly regularized, indicating that the model is exponentially less plausible than assigning equal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "probabilities to each class. As the regularization parameter is increased, the log evidence ratio falls,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 450, + 472, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 472, + 462 + ], + "score": 1.0, + "content": "but it is always positive, indicating that the model can never be expected to generalize well.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "Now consider figure 2b (informative labels). Once again, the training cross-entropy falls to zero", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "when the model is weakly regularized, while the test cross-entropy is high. Even though the model", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "makes accurate predictions, those predictions are overconfident. As the regularization coefficient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 287, + 513 + ], + "score": 1.0, + "content": "increases, the test cross-entropy falls below", + "type": "text" + }, + { + "bbox": [ + 288, + 500, + 304, + 510 + ], + "score": 0.33, + "content": "\\ln 2", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 499, + 506, + 513 + ], + "score": 1.0, + "content": ", indicating that the model is successfully gener-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "alizing to the test set. Now consider the Bayesian evidence. The log evidence ratio is large and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "positive when the model is weakly regularized, but as the regularization coefficient increases, the", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "image", + "bbox": [ + 116, + 563, + 494, + 698 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 563, + 494, + 698 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 563, + 494, + 698 + ], + "spans": [ + { + "bbox": [ + 116, + 563, + 494, + 698 + ], + "score": 0.971, + "type": "image", + "image_path": "0f494f2c4f939d075fe13acb31a80bfe22b0322c78ac900bdee3b4d30d954879.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 116, + 563, + 494, + 608.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 116, + 608.0, + 494, + 653.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 116, + 653.0, + 494, + 698.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 707, + 505, + 730 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "Figure 2: The cross-entropy and log evidence ratio, evaluated on random (a) or informative (b)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 717, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 505, + 732 + ], + "score": 1.0, + "content": "labels. The evidence, evaluated on the training set, is strongly correlated with the test cross-entropy.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 86, + 494, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 86, + 494, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 86, + 494, + 223 + ], + "spans": [ + { + "bbox": [ + 117, + 86, + 494, + 223 + ], + "score": 0.953, + "type": "image", + "image_path": "b54a96e24409024f1dba39245529ce32aa588511ceab8602abf0ec1f279994eb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 86, + 494, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 131.66666666666666, + 494, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 177.33333333333331, + 494, + 222.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 233, + 505, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "Figure 1: Prediction accuracy and mean training set margin as a function of regularization coeffi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "cient, for a logistic regression trained on random (a) and informative (b) labels of the same inputs.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "The weakly regularized model generalizes well on informative labels but memorizes random labels.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 505, + 334 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 291, + 505, + 335 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Now consider figure 2, where we plot the mean cross-entropy of the model predictions, evaluated on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "both training and test sets, as well as the Bayesian log evidence ratio defined in the previous section.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "score": 1.0, + "content": "Looking first at the random label experiment in figure 2a, while the cross-entropy on the training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "set vanishes when the model is weakly regularized, the cross-entropy on the test set explodes. Not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "only does the model make random predictions, but it is extremely confident in those predictions. As", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 442, + 407 + ], + "score": 1.0, + "content": "the regularization coefficient is increased the test set cross-entropy falls, settling at", + "type": "text" + }, + { + "bbox": [ + 443, + 395, + 459, + 406 + ], + "score": 0.45, + "content": "\\ln 2", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 396, + 505, + 407 + ], + "score": 1.0, + "content": ", the cross-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "entropy of assigning equal probability to both classes. Now consider the Bayesian evidence, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "we evaluate on the training set. The log evidence ratio is large and positive when the model is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "weakly regularized, indicating that the model is exponentially less plausible than assigning equal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "probabilities to each class. As the regularization parameter is increased, the log evidence ratio falls,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 450, + 472, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 472, + 462 + ], + "score": 1.0, + "content": "but it is always positive, indicating that the model can never be expected to generalize well.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 340, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "Now consider figure 2b (informative labels). Once again, the training cross-entropy falls to zero", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "when the model is weakly regularized, while the test cross-entropy is high. Even though the model", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "makes accurate predictions, those predictions are overconfident. As the regularization coefficient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 287, + 513 + ], + "score": 1.0, + "content": "increases, the test cross-entropy falls below", + "type": "text" + }, + { + "bbox": [ + 288, + 500, + 304, + 510 + ], + "score": 0.33, + "content": "\\ln 2", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 499, + 506, + 513 + ], + "score": 1.0, + "content": ", indicating that the model is successfully gener-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "alizing to the test set. Now consider the Bayesian evidence. The log evidence ratio is large and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "positive when the model is weakly regularized, but as the regularization coefficient increases, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "log evidence ratio drops below zero, indicating that the model is exponentially more plausible than", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "assigning equal probabilities to each class. As we further increase the regularization, the log evi-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 350, + 118 + ], + "score": 1.0, + "content": "dence ratio rises to zero while the test cross-entropy rises to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 351, + 105, + 367, + 115 + ], + "score": 0.52, + "content": "\\ln 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 367, + 105, + 506, + 118 + ], + "score": 1.0, + "content": ". 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The evidence, evaluated on the training set, is strongly correlated with the test cross-entropy.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "log evidence ratio drops below zero, indicating that the model is exponentially more plausible than", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "assigning equal probabilities to each class. As we further increase the regularization, the log evi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 350, + 118 + ], + "score": 1.0, + "content": "dence ratio rises to zero while the test cross-entropy rises to", + "type": "text" + }, + { + "bbox": [ + 351, + 105, + 367, + 115 + ], + "score": 0.52, + "content": "\\ln 2", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 105, + 506, + 118 + ], + "score": 1.0, + "content": ". Test cross-entropy and Bayesian", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 433, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 433, + 128 + ], + "score": 1.0, + "content": "evidence are strongly correlated, with minima at the same regularization strength.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "Bayesian model comparison has explained our results in a logistic regression. Meanwhile, Krueger", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "et al. (2017) showed the largest Hessian eigenvalue also increased when training on random labels", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "in deep networks, implying the evidence is falling. We conclude that Bayesian model comparison", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "is quantitatively consistent with the results of Zhang et al. (2016) in linear models where we can", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "compute the evidence, and qualitatively consistent with their results in deep networks where we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "score": 1.0, + "content": "cannot. Dziugaite & Roy (2017) recently demonstrated the results of Zhang et al. (2016) can also", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 493, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 493, + 211 + ], + "score": 1.0, + "content": "be understood by minimising a PAC-Bayes generalization bound which penalizes sharp minima.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 228, + 414, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 416, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 416, + 242 + ], + "score": 1.0, + "content": "4 BAYES THEOREM AND STOCHASTIC GRADIENT DESCENT", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "We showed above that generalization is strongly correlated with the Bayesian evidence, a weighted", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "combination of the depth of a minimum (the cost function) and its breadth (the Occam factor).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 504, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 504, + 287 + ], + "score": 1.0, + "content": "Consequently Bayesians often add isotropic Gaussian noise to the gradient (Welling & Teh, 2011).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "In appendix A, we show this drives the parameters towards broad minima whose evidence is large.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "The noise introduced by small batch training is not isotropic, and its covariance matrix is a function", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "of the parameter values, but empirically Keskar et al. 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Looking first", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "score": 1.0, + "content": "at figure 3a, small batch training takes longer to converge, but after a thousand gradient updates", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "a clear generalization gap in model accuracy emerges between small and large training batches.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "Now consider figure 3b. While the test cross-entropy for small batch training is lower at the end", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "of training; the cross-entropy of both small and large training batches is increasing, indicative of", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "score": 1.0, + "content": "over-fitting. Both models exhibit a minimum test cross-entropy, although after different numbers of", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "gradient updates. Intriguingly, we show in appendix B that the generalization gap between small", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 509, + 438, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 438, + 520 + ], + "score": 1.0, + "content": "and large batch training shrinks significantly when we introduce L2 regularization.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 471, + 506, + 531 + ] + }, + { + "type": "image", + "bbox": [ + 117, + 555, + 493, + 698 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 555, + 493, + 698 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 555, + 493, + 698 + ], + "spans": [ + { + "bbox": [ + 117, + 555, + 493, + 698 + ], + "score": 0.969, + "type": "image", + "image_path": "574a250cd29b8f09e8b5e4a578f9b9c70694cd5769423b16b172e086607be672.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 117, + 555, + 493, + 602.6666666666666 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 117, + 602.6666666666666, + 493, + 650.3333333333333 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 117, + 650.3333333333333, + 493, + 697.9999999999999 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 707, + 504, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 706, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 505, + 721 + ], + "score": 1.0, + "content": "Figure 3: The evolution during training of the test accuracy (a), and the test set cross-entropy (b).", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 716, + 447, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 447, + 732 + ], + "score": 1.0, + "content": "Full batches are composed of 1000 images, while small batches comprise 30 images.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "index": 38.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 88, + 496, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 88, + 496, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 88, + 496, + 231 + ], + "spans": [ + { + "bbox": [ + 117, + 88, + 496, + 231 + ], + "score": 0.95, + "type": "image", + "image_path": "cb26d4d2e14c04d96237c32396fee654dc10720f0b6d94dc657d790de21c8f2c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 88, + 496, + 135.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 135.66666666666666, + 496, + 183.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 183.33333333333331, + 496, + 230.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 240, + 502, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 238, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 504, + 254 + ], + "score": 1.0, + "content": "Figure 4: The test accuracy for a range of batch sizes, during training (a) and after 10000 steps (b).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "image", + "bbox": [ + 116, + 272, + 494, + 414 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 272, + 494, + 414 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 272, + 494, + 414 + ], + "spans": [ + { + "bbox": [ + 116, + 272, + 494, + 414 + ], + "score": 0.967, + "type": "image", + "image_path": "f3aa0fc16a2e26860bdf9e12234b09eee37b95e9b1f81d398ddde5651cd5ec96.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 116, + 272, + 494, + 319.3333333333333 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 116, + 319.3333333333333, + 494, + 366.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 116, + 366.66666666666663, + 494, + 413.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 423, + 505, + 456 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 475, + 436 + ], + "score": 1.0, + "content": "Figure 5: a) The test set accuracy as a function of batch size, for a range of learning rates", + "type": "text" + }, + { + "bbox": [ + 475, + 425, + 481, + 433 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 423, + 506, + 436 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 312, + 447 + ], + "score": 1.0, + "content": "performance peak shifts to the right as we increase", + "type": "text" + }, + { + "bbox": [ + 313, + 436, + 318, + 444 + ], + "score": 0.47, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 433, + 477, + 447 + ], + "score": 1.0, + "content": ", but the overall performance falls once", + "type": "text" + }, + { + "bbox": [ + 477, + 434, + 501, + 446 + ], + "score": 0.9, + "content": "\\epsilon \\gtrsim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 433, + 505, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 445, + 503, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 503, + 457 + ], + "score": 1.0, + "content": "b) The best observed batch size is proportional to the learning rate across two orders of magnitude.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 506, + 488 + ], + "score": 1.0, + "content": "of training; the cross-entropy of both small and large training batches is increasing, indicative of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 498 + ], + "score": 1.0, + "content": "over-fitting. Both models exhibit a minimum test cross-entropy, although after different numbers of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "gradient updates. Intriguingly, we show in appendix B that the generalization gap between small", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 509, + 438, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 438, + 520 + ], + "score": 1.0, + "content": "and large batch training shrinks significantly when we introduce L2 regularization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "From now on we focus on the test set accuracy (since this converges as the number of gradient", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "updates increases). In figure 4a, we exhibit training curves for a range of batch sizes between 1 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 351, + 560 + ], + "score": 1.0, + "content": "1000. We find that the model cannot train when the batch size", + "type": "text" + }, + { + "bbox": [ + 351, + 547, + 384, + 559 + ], + "score": 0.91, + "content": "B \\lessapprox \\bar { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 546, + 506, + 560 + ], + "score": 1.0, + "content": ". In figure 4b we plot the mean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "test set accuracy after 10000 training steps. A clear peak emerges, indicating that there is indeed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "an optimum batch size which maximizes the test accuracy, consistent with Bayesian intuition. The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 580, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 504, + 593 + ], + "score": 1.0, + "content": "results of Keskar et al. (2016) focused on the decay in test accuracy above this optimum batch size.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 106, + 608, + 464, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 466, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 466, + 623 + ], + "score": 1.0, + "content": "5 STOCHASTIC DIFFERENTIAL EQUATIONS AND THE SCALING RULES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "We showed above that the test accuracy peaks at an optimal batch size, if one holds the other SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "hyper-parameters constant. We argued that this peak arises from the tradeoff between depth and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "breadth in the Bayesian evidence. However it is not the batch size itself which controls this tradeoff,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "but the underlying scale of random fluctuations in the SGD dynamics. We now identify this SGD", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "“noise scale”, and use it to derive three scaling rules which predict how the optimal batch size", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 687, + 477, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 477, + 700 + ], + "score": 1.0, + "content": "depends on the learning rate, training set size and momentum coefficient. A gradient update,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 702, + 391, + 736 + ], + "lines": [ + { + "bbox": [ + 219, + 702, + 391, + 736 + ], + "spans": [ + { + "bbox": [ + 219, + 702, + 391, + 736 + ], + "score": 0.95, + "content": "\\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}", + "type": "interline_equation", + "image_path": "ca2c1619d6fe760545e15462bc4d39b3b0f599e15e0f990417a56ab1927ff7e2.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 702, + 391, + 719.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 219, + 719.0, + 391, + 736.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 88, + 496, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 88, + 496, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 88, + 496, + 231 + ], + "spans": [ + { + "bbox": [ + 117, + 88, + 496, + 231 + ], + "score": 0.95, + "type": "image", + "image_path": "cb26d4d2e14c04d96237c32396fee654dc10720f0b6d94dc657d790de21c8f2c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 88, + 496, + 135.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 135.66666666666666, + 496, + 183.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 183.33333333333331, + 496, + 230.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 240, + 502, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 238, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 504, + 254 + ], + "score": 1.0, + "content": "Figure 4: The test accuracy for a range of batch sizes, during training (a) and after 10000 steps (b).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "image", + "bbox": [ + 116, + 272, + 494, + 414 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 272, + 494, + 414 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 272, + 494, + 414 + ], + "spans": [ + { + "bbox": [ + 116, + 272, + 494, + 414 + ], + "score": 0.967, + "type": "image", + "image_path": "f3aa0fc16a2e26860bdf9e12234b09eee37b95e9b1f81d398ddde5651cd5ec96.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 116, + 272, + 494, + 319.3333333333333 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 116, + 319.3333333333333, + 494, + 366.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 116, + 366.66666666666663, + 494, + 413.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 423, + 505, + 456 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 475, + 436 + ], + "score": 1.0, + "content": "Figure 5: a) The test set accuracy as a function of batch size, for a range of learning rates", + "type": "text" + }, + { + "bbox": [ + 475, + 425, + 481, + 433 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 423, + 506, + 436 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 312, + 447 + ], + "score": 1.0, + "content": "performance peak shifts to the right as we increase", + "type": "text" + }, + { + "bbox": [ + 313, + 436, + 318, + 444 + ], + "score": 0.47, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 433, + 477, + 447 + ], + "score": 1.0, + "content": ", but the overall performance falls once", + "type": "text" + }, + { + "bbox": [ + 477, + 434, + 501, + 446 + ], + "score": 0.9, + "content": "\\epsilon \\gtrsim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 433, + 505, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 445, + 503, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 503, + 457 + ], + "score": 1.0, + "content": "b) The best observed batch size is proportional to the learning rate across two orders of magnitude.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 520 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 476, + 506, + 520 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "From now on we focus on the test set accuracy (since this converges as the number of gradient", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "updates increases). In figure 4a, we exhibit training curves for a range of batch sizes between 1 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 351, + 560 + ], + "score": 1.0, + "content": "1000. We find that the model cannot train when the batch size", + "type": "text" + }, + { + "bbox": [ + 351, + 547, + 384, + 559 + ], + "score": 0.91, + "content": "B \\lessapprox \\bar { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 546, + 506, + 560 + ], + "score": 1.0, + "content": ". In figure 4b we plot the mean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "test set accuracy after 10000 training steps. A clear peak emerges, indicating that there is indeed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "an optimum batch size which maximizes the test accuracy, consistent with Bayesian intuition. The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 580, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 504, + 593 + ], + "score": 1.0, + "content": "results of Keskar et al. (2016) focused on the decay in test accuracy above this optimum batch size.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 525, + 506, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 608, + 464, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 466, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 466, + 623 + ], + "score": 1.0, + "content": "5 STOCHASTIC DIFFERENTIAL EQUATIONS AND THE SCALING RULES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "We showed above that the test accuracy peaks at an optimal batch size, if one holds the other SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "hyper-parameters constant. We argued that this peak arises from the tradeoff between depth and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "breadth in the Bayesian evidence. However it is not the batch size itself which controls this tradeoff,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "but the underlying scale of random fluctuations in the SGD dynamics. We now identify this SGD", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "“noise scale”, and use it to derive three scaling rules which predict how the optimal batch size", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 687, + 477, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 477, + 700 + ], + "score": 1.0, + "content": "depends on the learning rate, training set size and momentum coefficient. A gradient update,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 632, + 506, + 700 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 702, + 391, + 736 + ], + "lines": [ + { + "bbox": [ + 219, + 702, + 391, + 736 + ], + "spans": [ + { + "bbox": [ + 219, + 702, + 391, + 736 + ], + "score": 0.95, + "content": "\\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}", + "type": "interline_equation", + "image_path": "ca2c1619d6fe760545e15462bc4d39b3b0f599e15e0f990417a56ab1927ff7e2.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 702, + 391, + 719.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 219, + 719.0, + 391, + 736.0 + ], + "spans": [], + "index": 28 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 89, + 494, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 89, + 494, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 89, + 494, + 231 + ], + "spans": [ + { + "bbox": [ + 116, + 89, + 494, + 231 + ], + "score": 0.964, + "type": "image", + "image_path": "98b0bc094f9ff0d411a45850540a3de33d0a1a5554b328c700b05f4ca887c2a0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 89, + 494, + 136.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 136.33333333333334, + 494, + 183.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 183.66666666666669, + 494, + 231.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 240, + 505, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "Figure 6: a) The test accuracy as a function of batch size, for a range of training set sizes. To reduce", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "noise, we average each curve over five experiments. The performance peak shift to the right as we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "increase the size of the training set. Unsurprisingly, the overall model performance also improves.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 273, + 484, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 432, + 286 + ], + "score": 1.0, + "content": "b) The best observed batch size is proportional to the size of the training set once", + "type": "text" + }, + { + "bbox": [ + 432, + 273, + 480, + 285 + ], + "score": 0.91, + "content": "N \\gtrsim 2 0 0 0 \\bar { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 273, + 484, + 286 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 335, + 321 + ], + "score": 1.0, + "content": "where \u000f is the learning rate, N the training set size, dCdω", + "type": "text" + }, + { + "bbox": [ + 318, + 302, + 387, + 318 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d C } { d \\omega } = \\sum _ { i = 1 } ^ { N } \\frac { d C _ { i } } { d \\omega } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 299, + 479, + 320 + ], + "score": 1.0, + "content": "the true gradient, and", + "type": "text" + }, + { + "bbox": [ + 479, + 301, + 505, + 318 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { d \\hat { C } } { d \\omega } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 314, + 507, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 314, + 507, + 334 + ], + "score": 1.0, + "content": "NB PBi=1 dCidω the estimated gradient evaluated on a mini-batch. The expected gradient of a single", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 328, + 509, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 145, + 354 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 145, + 333, + 206, + 348 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left. \\frac { d C _ { i } } { d \\omega } \\right. = \\frac { 1 } { N } \\frac { d C } { d \\omega } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 328, + 235, + 354 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 235, + 331, + 396, + 351 + ], + "score": 0.9, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 328, + 509, + 354 + ], + "score": 1.0, + "content": "is a matrix describing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 100, + 348, + 406, + 387 + ], + "spans": [ + { + "bbox": [ + 100, + 348, + 257, + 387 + ], + "score": 1.0, + "content": "gradient covariances, which are a functheorem and model the gradient error", + "type": "text" + }, + { + "bbox": [ + 257, + 360, + 322, + 376 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 348, + 344, + 387 + ], + "score": 1.0, + "content": "nt par with", + "type": "text" + }, + { + "bbox": [ + 378, + 348, + 406, + 387 + ], + "score": 1.0, + "content": "alues. n rando", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 374, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 344, + 390 + ], + "score": 1.0, + "content": "approximation briefly in appendix C). It is easy to show that", + "type": "text" + }, + { + "bbox": [ + 345, + 375, + 378, + 388 + ], + "score": 0.93, + "content": "\\langle \\alpha \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 374, + 501, + 389 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\langle \\alpha ^ { 2 } \\rangle = N ( \\frac { N } { B } - 1 ) F ( \\omega ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 374, + 506, + 390 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 147, + 401 + ], + "score": 1.0, + "content": "Typically", + "type": "text" + }, + { + "bbox": [ + 147, + 388, + 185, + 399 + ], + "score": 0.9, + "content": "N \\gg B", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 230, + 401 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 230, + 388, + 314, + 401 + ], + "score": 0.94, + "content": "\\langle \\alpha ^ { 2 } \\rangle \\approx N ^ { 2 } F ( \\omega ) / B", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 387, + 506, + 401 + ], + "score": 1.0, + "content": ". To continue, we interpret equation 11 as the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 399, + 445, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 445, + 412 + ], + "score": 1.0, + "content": "discrete update of a stochastic differential equation (Li et al., 2017; Gardiner, 1985),", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 414, + 347, + 438 + ], + "lines": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "spans": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "score": 0.95, + "content": "\\frac { d \\omega } { d t } = - \\frac { d C } { d \\omega } + \\eta ( t ) ,", + "type": "interline_equation", + "image_path": "8158d684a1a27720bd6009bc60d70996048f1cedc99bee78a4e04e9ca506dc5c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 504, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 135, + 454 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 135, + 443, + 140, + 451 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 441, + 237, + 454 + ], + "score": 1.0, + "content": "is a continuous variable,", + "type": "text" + }, + { + "bbox": [ + 238, + 442, + 255, + 453 + ], + "score": 0.91, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 441, + 324, + 454 + ], + "score": 1.0, + "content": "represents noise,", + "type": "text" + }, + { + "bbox": [ + 325, + 441, + 368, + 453 + ], + "score": 0.93, + "content": "\\langle \\eta ( t ) \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 441, + 385, + 454 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 441, + 501, + 453 + ], + "score": 0.92, + "content": "\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = g F ( \\omega ) \\delta ( t { - } t ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 505, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 160, + 464 + ], + "score": 1.0, + "content": "The constant", + "type": "text" + }, + { + "bbox": [ + 160, + 455, + 167, + 464 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "controls the scale of random fluctuations in the dynamics. To relate this differential", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 463, + 501, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 320, + 479 + ], + "score": 1.0, + "content": "equation to the SGD, we compute a gradient update", + "type": "text" + }, + { + "bbox": [ + 320, + 463, + 501, + 480 + ], + "score": 0.92, + "content": "\\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}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 477, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 185, + 491 + ], + "score": 1.0, + "content": "Finally, to measure", + "type": "text" + }, + { + "bbox": [ + 185, + 479, + 191, + 489 + ], + "score": 0.3, + "content": "\\mathbf { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 477, + 504, + 491 + ], + "score": 1.0, + "content": ", we equate the variance in this gradient update to the variance in equation 11,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 492, + 479, + 546 + ], + "lines": [ + { + "bbox": [ + 132, + 492, + 479, + 546 + ], + "spans": [ + { + "bbox": [ + 132, + 492, + 479, + 546 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "fb3ddc24790224f84790dc706c3ab25786d957d852bd27c1a3eb6ed8c54924b1.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 132, + 492, + 479, + 510.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 132, + 510.0, + 479, + 528.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 132, + 528.0, + 479, + 546.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 102, + 545, + 509, + 568 + ], + "spans": [ + { + "bbox": [ + 102, + 545, + 247, + 568 + ], + "score": 1.0, + "content": "Rearranging, the SGD noise scale", + "type": "text" + }, + { + "bbox": [ + 247, + 549, + 349, + 563 + ], + "score": 0.92, + "content": "\\begin{array} { r } { g = \\epsilon ( \\frac { N } { B } - 1 ) \\approx \\epsilon N / B } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 545, + 509, + 568 + ], + "score": 1.0, + "content": ". The noise scale falls when the batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 425, + 575 + ], + "score": 1.0, + "content": "size increases, consistent with our earlier observation of an optimal batch size", + "type": "text" + }, + { + "bbox": [ + 425, + 561, + 445, + 573 + ], + "score": 0.93, + "content": "B _ { o p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 559, + 506, + 575 + ], + "score": 1.0, + "content": "while holding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "the other hyper-parameters fixed. Notice that one would equivalently observe an optimal learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "rate if one held the batch size constant. A similar analysis of the SGD was recently performed by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Mandt et al. (2017), although their treatment only holds near local minima where the covariances", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 129, + 617 + ], + "score": 0.9, + "content": "F ( \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "are stationary. Our analysis holds throughout training, which is necessary since Keskar et al.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 615, + 488, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 488, + 629 + ], + "score": 1.0, + "content": "(2016) found that the beneficial influence of noise was most pronounced at the start of training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "When we vary the learning rate or the training set size, we should keep the noise scale fixed, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 157, + 656 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 158, + 644, + 208, + 656 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 643, + 506, + 656 + ], + "score": 1.0, + "content": ". In figure 5a, we plot the test accuracy as a function of batch size after", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 108, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 108, + 655, + 149, + 667 + ], + "score": 0.85, + "content": "( 1 0 0 0 0 / \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "training steps, for a range of learning rates. Exactly as predicted, the peak moves to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 138, + 679 + ], + "score": 1.0, + "content": "right as", + "type": "text" + }, + { + "bbox": [ + 139, + 668, + 145, + 676 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "increases. Additionally, the peak test accuracy achieved at a given learning rate does not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 178, + 689 + ], + "score": 1.0, + "content": "begin to fall until", + "type": "text" + }, + { + "bbox": [ + 178, + 677, + 203, + 687 + ], + "score": 0.88, + "content": "\\epsilon \\sim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ", indicating that there is no significant discretization error in integrating the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "stochastic differential equation below this point. Above this point, the discretization error begins to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "dominate and the peak test accuracy falls rapidly. In figure 5b, we plot the best observed batch size", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 351, + 723 + ], + "score": 1.0, + "content": "as a function of learning rate, observing a clear linear trend,", + "type": "text" + }, + { + "bbox": [ + 352, + 710, + 390, + 722 + ], + "score": 0.93, + "content": "B _ { o p t } \\propto \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ". The error bars indicate the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 478, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 478, + 734 + ], + "score": 1.0, + "content": "distance from the best observed batch size to the next batch size sampled in our experiments.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 89, + 494, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 89, + 494, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 89, + 494, + 231 + ], + "spans": [ + { + "bbox": [ + 116, + 89, + 494, + 231 + ], + "score": 0.964, + "type": "image", + "image_path": "98b0bc094f9ff0d411a45850540a3de33d0a1a5554b328c700b05f4ca887c2a0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 89, + 494, + 136.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 136.33333333333334, + 494, + 183.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 183.66666666666669, + 494, + 231.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 240, + 505, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "Figure 6: a) The test accuracy as a function of batch size, for a range of training set sizes. To reduce", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "noise, we average each curve over five experiments. The performance peak shift to the right as we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "increase the size of the training set. Unsurprisingly, the overall model performance also improves.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 273, + 484, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 432, + 286 + ], + "score": 1.0, + "content": "b) The best observed batch size is proportional to the size of the training set once", + "type": "text" + }, + { + "bbox": [ + 432, + 273, + 480, + 285 + ], + "score": 0.91, + "content": "N \\gtrsim 2 0 0 0 \\bar { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 273, + 484, + 286 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 335, + 321 + ], + "score": 1.0, + "content": "where \u000f is the learning rate, N the training set size, dCdω", + "type": "text" + }, + { + "bbox": [ + 318, + 302, + 387, + 318 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { d C } { d \\omega } = \\sum _ { i = 1 } ^ { N } \\frac { d C _ { i } } { d \\omega } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 299, + 479, + 320 + ], + "score": 1.0, + "content": "the true gradient, and", + "type": "text" + }, + { + "bbox": [ + 479, + 301, + 505, + 318 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { d \\hat { C } } { d \\omega } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 314, + 507, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 314, + 507, + 334 + ], + "score": 1.0, + "content": "NB PBi=1 dCidω the estimated gradient evaluated on a mini-batch. The expected gradient of a single", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 328, + 509, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 145, + 354 + ], + "score": 1.0, + "content": "example,", + "type": "text" + }, + { + "bbox": [ + 145, + 333, + 206, + 348 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left. \\frac { d C _ { i } } { d \\omega } \\right. = \\frac { 1 } { N } \\frac { d C } { d \\omega } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 328, + 235, + 354 + ], + "score": 1.0, + "content": ", while", + "type": "text" + }, + { + "bbox": [ + 235, + 331, + 396, + 351 + ], + "score": 0.9, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 328, + 509, + 354 + ], + "score": 1.0, + "content": "is a matrix describing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 100, + 348, + 406, + 387 + ], + "spans": [ + { + "bbox": [ + 100, + 348, + 257, + 387 + ], + "score": 1.0, + "content": "gradient covariances, which are a functheorem and model the gradient error", + "type": "text" + }, + { + "bbox": [ + 257, + 360, + 322, + 376 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 348, + 344, + 387 + ], + "score": 1.0, + "content": "nt par with", + "type": "text" + }, + { + "bbox": [ + 378, + 348, + 406, + 387 + ], + "score": 1.0, + "content": "alues. n rando", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 374, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 344, + 390 + ], + "score": 1.0, + "content": "approximation briefly in appendix C). It is easy to show that", + "type": "text" + }, + { + "bbox": [ + 345, + 375, + 378, + 388 + ], + "score": 0.93, + "content": "\\langle \\alpha \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 374, + 501, + 389 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\langle \\alpha ^ { 2 } \\rangle = N ( \\frac { N } { B } - 1 ) F ( \\omega ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 374, + 506, + 390 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 147, + 401 + ], + "score": 1.0, + "content": "Typically", + "type": "text" + }, + { + "bbox": [ + 147, + 388, + 185, + 399 + ], + "score": 0.9, + "content": "N \\gg B", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 230, + 401 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 230, + 388, + 314, + 401 + ], + "score": 0.94, + "content": "\\langle \\alpha ^ { 2 } \\rangle \\approx N ^ { 2 } F ( \\omega ) / B", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 387, + 506, + 401 + ], + "score": 1.0, + "content": ". To continue, we interpret equation 11 as the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 399, + 445, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 445, + 412 + ], + "score": 1.0, + "content": "discrete update of a stochastic differential equation (Li et al., 2017; Gardiner, 1985),", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 100, + 299, + 509, + 412 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 414, + 347, + 438 + ], + "lines": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "spans": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "score": 0.95, + "content": "\\frac { d \\omega } { d t } = - \\frac { d C } { d \\omega } + \\eta ( t ) ,", + "type": "interline_equation", + "image_path": "8158d684a1a27720bd6009bc60d70996048f1cedc99bee78a4e04e9ca506dc5c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 263, + 414, + 347, + 438 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 504, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 135, + 454 + ], + "score": 1.0, + "content": "Where", + "type": "text" + }, + { + "bbox": [ + 135, + 443, + 140, + 451 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 441, + 237, + 454 + ], + "score": 1.0, + "content": "is a continuous variable,", + "type": "text" + }, + { + "bbox": [ + 238, + 442, + 255, + 453 + ], + "score": 0.91, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 441, + 324, + 454 + ], + "score": 1.0, + "content": "represents noise,", + "type": "text" + }, + { + "bbox": [ + 325, + 441, + 368, + 453 + ], + "score": 0.93, + "content": "\\langle \\eta ( t ) \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 441, + 385, + 454 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 441, + 501, + 453 + ], + "score": 0.92, + "content": "\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = g F ( \\omega ) \\delta ( t { - } t ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 441, + 505, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 160, + 464 + ], + "score": 1.0, + "content": "The constant", + "type": "text" + }, + { + "bbox": [ + 160, + 455, + 167, + 464 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "controls the scale of random fluctuations in the dynamics. To relate this differential", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 463, + 501, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 320, + 479 + ], + "score": 1.0, + "content": "equation to the SGD, we compute a gradient update", + "type": "text" + }, + { + "bbox": [ + 320, + 463, + 501, + 480 + ], + "score": 0.92, + "content": "\\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}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 477, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 185, + 491 + ], + "score": 1.0, + "content": "Finally, to measure", + "type": "text" + }, + { + "bbox": [ + 185, + 479, + 191, + 489 + ], + "score": 0.3, + "content": "\\mathbf { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 477, + 504, + 491 + ], + "score": 1.0, + "content": ", we equate the variance in this gradient update to the variance in equation 11,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 441, + 505, + 491 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 492, + 479, + 546 + ], + "lines": [ + { + "bbox": [ + 132, + 492, + 479, + 546 + ], + "spans": [ + { + "bbox": [ + 132, + 492, + 479, + 546 + ], + "score": 0.93, + "content": "\\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}", + "type": "interline_equation", + "image_path": "fb3ddc24790224f84790dc706c3ab25786d957d852bd27c1a3eb6ed8c54924b1.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 132, + 492, + 479, + 510.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 132, + 510.0, + 479, + 528.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 132, + 528.0, + 479, + 546.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 102, + 545, + 509, + 568 + ], + "spans": [ + { + "bbox": [ + 102, + 545, + 247, + 568 + ], + "score": 1.0, + "content": "Rearranging, the SGD noise scale", + "type": "text" + }, + { + "bbox": [ + 247, + 549, + 349, + 563 + ], + "score": 0.92, + "content": "\\begin{array} { r } { g = \\epsilon ( \\frac { N } { B } - 1 ) \\approx \\epsilon N / B } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 545, + 509, + 568 + ], + "score": 1.0, + "content": ". The noise scale falls when the batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 425, + 575 + ], + "score": 1.0, + "content": "size increases, consistent with our earlier observation of an optimal batch size", + "type": "text" + }, + { + "bbox": [ + 425, + 561, + 445, + 573 + ], + "score": 0.93, + "content": "B _ { o p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 559, + 506, + 575 + ], + "score": 1.0, + "content": "while holding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "the other hyper-parameters fixed. Notice that one would equivalently observe an optimal learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "rate if one held the batch size constant. A similar analysis of the SGD was recently performed by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Mandt et al. (2017), although their treatment only holds near local minima where the covariances", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 129, + 617 + ], + "score": 0.9, + "content": "F ( \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "are stationary. Our analysis holds throughout training, which is necessary since Keskar et al.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 615, + 488, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 488, + 629 + ], + "score": 1.0, + "content": "(2016) found that the beneficial influence of noise was most pronounced at the start of training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 102, + 545, + 509, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "When we vary the learning rate or the training set size, we should keep the noise scale fixed, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 157, + 656 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 158, + 644, + 208, + 656 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 643, + 506, + 656 + ], + "score": 1.0, + "content": ". In figure 5a, we plot the test accuracy as a function of batch size after", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 108, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 108, + 655, + 149, + 667 + ], + "score": 0.85, + "content": "( 1 0 0 0 0 / \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "training steps, for a range of learning rates. Exactly as predicted, the peak moves to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 138, + 679 + ], + "score": 1.0, + "content": "right as", + "type": "text" + }, + { + "bbox": [ + 139, + 668, + 145, + 676 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "increases. Additionally, the peak test accuracy achieved at a given learning rate does not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 178, + 689 + ], + "score": 1.0, + "content": "begin to fall until", + "type": "text" + }, + { + "bbox": [ + 178, + 677, + 203, + 687 + ], + "score": 0.88, + "content": "\\epsilon \\sim 3", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ", indicating that there is no significant discretization error in integrating the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "stochastic differential equation below this point. Above this point, the discretization error begins to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "dominate and the peak test accuracy falls rapidly. In figure 5b, we plot the best observed batch size", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 351, + 723 + ], + "score": 1.0, + "content": "as a function of learning rate, observing a clear linear trend,", + "type": "text" + }, + { + "bbox": [ + 352, + 710, + 390, + 722 + ], + "score": 0.93, + "content": "B _ { o p t } \\propto \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ". The error bars indicate the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 478, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 478, + 734 + ], + "score": 1.0, + "content": "distance from the best observed batch size to the next batch size sampled in our experiments.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 632, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 89, + 494, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 89, + 494, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 89, + 494, + 232 + ], + "spans": [ + { + "bbox": [ + 116, + 89, + 494, + 232 + ], + "score": 0.962, + "type": "image", + "image_path": "f65c1d948657f9b8bf2cf5faeff5b3749c8f5c3f8c2d509c3f217f125dee5dd5.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 89, + 494, + 136.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 136.66666666666666, + 494, + 184.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 184.33333333333331, + 494, + 231.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 504, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 242, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 504, + 254 + ], + "score": 1.0, + "content": "Figure 7: a) The test set accuracy as a function of batch size for a range of momentum coefficients.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "As expected, the peak moves to the right as the momentum coefficient increases. b) The best ob-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 500, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 500, + 276 + ], + "score": 1.0, + "content": "served batch size for a range of momentum coefficients. The green curve exhibits the scaling rule.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "This scaling rule allows us to increase the learning rate with no loss in test accuracy and no increase", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "in computational cost, simply by simultaneously increasing the batch size. We can then exploit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "increased parallelism across multiple GPUs, reducing model training times (Goyal et al., 2017). A", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "similar scaling rule was independently proposed by Jastrzebski et al. (2017) and Chaudhari & Soatto", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "(2017), although neither work identifies the existence of an optimal noise scale. A number of authors", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "have proposed adjusting the batch size adaptively during training (Friedlander & Schmidt, 2012;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "score": 1.0, + "content": "Byrd et al., 2012; De et al., 2017), while Balles et al. (2016) proposed linearly coupling the learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 373, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 390 + ], + "score": 1.0, + "content": "rate and batch size within this framework. In Smith et al. (2017), we show empirically that decaying", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 475, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 475, + 399 + ], + "score": 1.0, + "content": "the learning rate during training and increasing the batch size during training are equivalent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "In figure 6a we exhibit the test set accuracy as a function of batch size, for a range of training set sizes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 180, + 427 + ], + "score": 1.0, + "content": "after 10000 steps", + "type": "text" + }, + { + "bbox": [ + 180, + 415, + 204, + 425 + ], + "score": 0.86, + "content": "\\mathbf { \\epsilon } \\cdot \\mathbf { \\epsilon } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "everywhere). Once again, the peak shifts right as the training set size rises,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "although the generalization gap becomes less pronounced as the training set size increases. In figure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "6b, we plot the best observed batch size as a function of training set size; observing another linear", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 132, + 461 + ], + "score": 1.0, + "content": "trend,", + "type": "text" + }, + { + "bbox": [ + 133, + 447, + 177, + 459 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto N", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 446, + 505, + 461 + ], + "score": 1.0, + "content": ". This scaling rule could be applied to production models, progressively growing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "the batch size as new training data is collected. We expect production datasets to grow considerably", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 470, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 470, + 482 + ], + "score": 1.0, + "content": "over time, and consequently large batch training is likely to become increasingly common.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 188, + 498 + ], + "score": 1.0, + "content": "Finally, in appendix", + "type": "text" + }, + { + "bbox": [ + 188, + 486, + 198, + 496 + ], + "score": 0.4, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "we extend our analysis to SGD with momentum, identifying the noise scale,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 495, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 162, + 511 + ], + "score": 0.95, + "content": "\\begin{array} { r } { g \\approx \\frac { \\epsilon N ^ { \\mathrm { ~ \\scriptsize ~ \\cdots ~ } } } { B ( 1 - m ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 495, + 164, + 511 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 165, + 496, + 194, + 509 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 194, + 498, + 205, + 507 + ], + "score": 0.71, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "denotes the momentum coefficient. Notice that this reduces to the noise", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 228, + 522 + ], + "score": 1.0, + "content": "scale of conventional SGD as", + "type": "text" + }, + { + "bbox": [ + 228, + 510, + 260, + 520 + ], + "score": 0.91, + "content": "m 0", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 510, + 291, + 522 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 292, + 510, + 321, + 520 + ], + "score": 0.89, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 510, + 472, + 522 + ], + "score": 1.0, + "content": ", we obtain an additional scaling rule", + "type": "text" + }, + { + "bbox": [ + 472, + 509, + 505, + 522 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 520, + 151, + 533 + ], + "score": 0.91, + "content": "1 / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 520, + 505, + 533 + ], + "score": 1.0, + "content": ". This scaling rule predicts that the optimal batch size will increase when the momentum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "coefficient is increased. In figure 7a we plot the test set performance as a function of batch size after", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 205, + 555 + ], + "score": 1.0, + "content": "10000 gradient updates (", + "type": "text" + }, + { + "bbox": [ + 205, + 543, + 229, + 553 + ], + "score": 0.88, + "content": "\\mathbf { \\epsilon } _ { \\epsilon } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "everywhere), for a range of momentum coefficients. In figure 7b, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "plot the best observed batch size as a function of the momentum coefficient, and fit our results to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 563, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 505, + 578 + ], + "score": 1.0, + "content": "scaling rule above; obtaining remarkably good agreement. We propose a simple heuristic for tuning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 575, + 388, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 388, + 587 + ], + "score": 1.0, + "content": "the batch size, learning rate and momentum coefficient in appendix E.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 201, + 618 + ], + "lines": [ + { + "bbox": [ + 104, + 604, + 203, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 203, + 622 + ], + "score": 1.0, + "content": "6 CONCLUSIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Just like deep neural networks, linear models which generalize well on informative labels can mem-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "orize random labels of the same inputs. These observations are explained by the Bayesian evidence,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "which is composed of the cost function and an “Occam factor”. The Occam factor penalizes sharp", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "minima but it is invariant to changes in model parameterization. Mini-batch noise drives SGD away", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "from sharp minima, and therefore there is an optimum batch size which maximizes the test accuracy.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "Interpreting SGD as the discretization of a stochastic differential equation, we predict this optimum", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 451, + 713 + ], + "score": 1.0, + "content": "batch size should scale linearly with both the learning rate and the training set size,", + "type": "text" + }, + { + "bbox": [ + 452, + 699, + 501, + 711 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 697, + 506, + 713 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 259, + 723 + ], + "score": 1.0, + "content": "We derive an additional scaling rule,", + "type": "text" + }, + { + "bbox": [ + 259, + 709, + 339, + 722 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto 1 / ( 1 - \\stackrel { \\textstyle - } { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ", between the optimal batch size and the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 490, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 490, + 734 + ], + "score": 1.0, + "content": "momentum coefficient. We verify these scaling rules empirically and discuss their implications.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 89, + 494, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 89, + 494, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 89, + 494, + 232 + ], + "spans": [ + { + "bbox": [ + 116, + 89, + 494, + 232 + ], + "score": 0.962, + "type": "image", + "image_path": "f65c1d948657f9b8bf2cf5faeff5b3749c8f5c3f8c2d509c3f217f125dee5dd5.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 89, + 494, + 136.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 136.66666666666666, + 494, + 184.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 184.33333333333331, + 494, + 231.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 504, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 242, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 504, + 254 + ], + "score": 1.0, + "content": "Figure 7: a) The test set accuracy as a function of batch size for a range of momentum coefficients.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "As expected, the peak moves to the right as the momentum coefficient increases. b) The best ob-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 500, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 500, + 276 + ], + "score": 1.0, + "content": "served batch size for a range of momentum coefficients. The green curve exhibits the scaling rule.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "This scaling rule allows us to increase the learning rate with no loss in test accuracy and no increase", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "in computational cost, simply by simultaneously increasing the batch size. We can then exploit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "increased parallelism across multiple GPUs, reducing model training times (Goyal et al., 2017). A", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "similar scaling rule was independently proposed by Jastrzebski et al. (2017) and Chaudhari & Soatto", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "(2017), although neither work identifies the existence of an optimal noise scale. A number of authors", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "have proposed adjusting the batch size adaptively during training (Friedlander & Schmidt, 2012;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 379 + ], + "score": 1.0, + "content": "Byrd et al., 2012; De et al., 2017), while Balles et al. (2016) proposed linearly coupling the learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 373, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 390 + ], + "score": 1.0, + "content": "rate and batch size within this framework. In Smith et al. (2017), we show empirically that decaying", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 475, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 475, + 399 + ], + "score": 1.0, + "content": "the learning rate during training and increasing the batch size during training are equivalent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 297, + 506, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "In figure 6a we exhibit the test set accuracy as a function of batch size, for a range of training set sizes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 180, + 427 + ], + "score": 1.0, + "content": "after 10000 steps", + "type": "text" + }, + { + "bbox": [ + 180, + 415, + 204, + 425 + ], + "score": 0.86, + "content": "\\mathbf { \\epsilon } \\cdot \\mathbf { \\epsilon } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "everywhere). Once again, the peak shifts right as the training set size rises,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "although the generalization gap becomes less pronounced as the training set size increases. In figure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "6b, we plot the best observed batch size as a function of training set size; observing another linear", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 132, + 461 + ], + "score": 1.0, + "content": "trend,", + "type": "text" + }, + { + "bbox": [ + 133, + 447, + 177, + 459 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto N", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 446, + 505, + 461 + ], + "score": 1.0, + "content": ". This scaling rule could be applied to production models, progressively growing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "the batch size as new training data is collected. We expect production datasets to grow considerably", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 470, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 470, + 482 + ], + "score": 1.0, + "content": "over time, and consequently large batch training is likely to become increasingly common.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 403, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 188, + 498 + ], + "score": 1.0, + "content": "Finally, in appendix", + "type": "text" + }, + { + "bbox": [ + 188, + 486, + 198, + 496 + ], + "score": 0.4, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "we extend our analysis to SGD with momentum, identifying the noise scale,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 495, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 162, + 511 + ], + "score": 0.95, + "content": "\\begin{array} { r } { g \\approx \\frac { \\epsilon N ^ { \\mathrm { ~ \\scriptsize ~ \\cdots ~ } } } { B ( 1 - m ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 495, + 164, + 511 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 165, + 496, + 194, + 509 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 194, + 498, + 205, + 507 + ], + "score": 0.71, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "denotes the momentum coefficient. Notice that this reduces to the noise", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 228, + 522 + ], + "score": 1.0, + "content": "scale of conventional SGD as", + "type": "text" + }, + { + "bbox": [ + 228, + 510, + 260, + 520 + ], + "score": 0.91, + "content": "m 0", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 510, + 291, + 522 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 292, + 510, + 321, + 520 + ], + "score": 0.89, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 510, + 472, + 522 + ], + "score": 1.0, + "content": ", we obtain an additional scaling rule", + "type": "text" + }, + { + "bbox": [ + 472, + 509, + 505, + 522 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 520, + 151, + 533 + ], + "score": 0.91, + "content": "1 / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 520, + 505, + 533 + ], + "score": 1.0, + "content": ". This scaling rule predicts that the optimal batch size will increase when the momentum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "coefficient is increased. In figure 7a we plot the test set performance as a function of batch size after", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 205, + 555 + ], + "score": 1.0, + "content": "10000 gradient updates (", + "type": "text" + }, + { + "bbox": [ + 205, + 543, + 229, + 553 + ], + "score": 0.88, + "content": "\\mathbf { \\epsilon } _ { \\epsilon } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "everywhere), for a range of momentum coefficients. In figure 7b, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "plot the best observed batch size as a function of the momentum coefficient, and fit our results to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 563, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 505, + 578 + ], + "score": 1.0, + "content": "scaling rule above; obtaining remarkably good agreement. We propose a simple heuristic for tuning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 575, + 388, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 388, + 587 + ], + "score": 1.0, + "content": "the batch size, learning rate and momentum coefficient in appendix E.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 484, + 506, + 587 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 201, + 618 + ], + "lines": [ + { + "bbox": [ + 104, + 604, + 203, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 203, + 622 + ], + "score": 1.0, + "content": "6 CONCLUSIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Just like deep neural networks, linear models which generalize well on informative labels can mem-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "orize random labels of the same inputs. These observations are explained by the Bayesian evidence,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "which is composed of the cost function and an “Occam factor”. The Occam factor penalizes sharp", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "minima but it is invariant to changes in model parameterization. Mini-batch noise drives SGD away", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "from sharp minima, and therefore there is an optimum batch size which maximizes the test accuracy.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "Interpreting SGD as the discretization of a stochastic differential equation, we predict this optimum", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 451, + 713 + ], + "score": 1.0, + "content": "batch size should scale linearly with both the learning rate and the training set size,", + "type": "text" + }, + { + "bbox": [ + 452, + 699, + 501, + 711 + ], + "score": 0.91, + "content": "B _ { o p t } \\propto \\epsilon N", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 697, + 506, + 713 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 259, + 723 + ], + "score": 1.0, + "content": "We derive an additional scaling rule,", + "type": "text" + }, + { + "bbox": [ + 259, + 709, + 339, + 722 + ], + "score": 0.92, + "content": "B _ { o p t } \\propto 1 / ( 1 - \\stackrel { \\textstyle - } { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ", between the optimal batch size and the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 490, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 490, + 734 + ], + "score": 1.0, + "content": "momentum coefficient. We verify these scaling rules empirically and discuss their implications.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 633, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 201, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 201, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 201, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 504, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "We thank Pieter-Jan Kindermans, Prajit Ramachandran, Jascha Sohl-Dickstein, Jon Shlens, Kevin", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 113, + 503, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 503, + 126 + ], + "score": 1.0, + "content": "Murphy, Samy Bengio, Yasaman Bahri and Saeed Saremi for helpful comments on the manuscript.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 141, + 176, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 176, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 176, + 154 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 105, + 160, + 504, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "Alessandro Achille and Stefano Soatto. 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The matrix", + "type": "text" + }, + { + "bbox": [ + 299, + 583, + 306, + 592 + ], + "score": 0.77, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 581, + 411, + 595 + ], + "score": 1.0, + "content": "denotes the identity, while", + "type": "text" + }, + { + "bbox": [ + 411, + 582, + 420, + 592 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "is the “temperature”.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "Notice this Langevin equation is extremely similar to the stochastic differential equation of SGD,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 415, + 618 + ], + "score": 1.0, + "content": "discussed in section 5 of the main text. Indeed, if the gradient covariances", + "type": "text" + }, + { + "bbox": [ + 415, + 604, + 438, + 616 + ], + "score": 0.91, + "content": "F ( \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "were stationary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "and proportional to the identity, then the SGD would integrate an overdamped Langevin equation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 624, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 320, + 640 + ], + "score": 1.0, + "content": "with temperature proportional to the SGD noise scale", + "type": "text" + }, + { + "bbox": [ + 320, + 627, + 327, + 637 + ], + "score": 0.74, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 624, + 344, + 640 + ], + "score": 1.0, + "content": ". 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Understanding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 396, + 475, + 409 + ], + "spans": [ + { + "bbox": [ + 115, + 396, + 475, + 409 + ], + "score": 1.0, + "content": "deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 384, + 506, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 441, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 443, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 443, + 444 + ], + "score": 1.0, + "content": "A BAYESIAN POSTERIOR SAMPLING AND LANGEVIN DYNAMICS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "Instead of minimizing the cost function, Bayesian usually prefer to sample parameter values from", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 465, + 230, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 230, + 478 + ], + "score": 1.0, + "content": "the posterior (MacKay, 1992),", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 453, + 505, + 478 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 482, + 362, + 497 + ], + "lines": [ + { + "bbox": [ + 249, + 482, + 362, + 497 + ], + "spans": [ + { + "bbox": [ + 249, + 482, + 362, + 497 + ], + "score": 0.92, + "content": "P ( \\omega | y , x ; M ) \\propto e ^ { - C ( \\omega ; M ) } ,", + "type": "interline_equation", + "image_path": "25ecff15d1d1ee78f02b54607b9bb88464578a2200b1baf5b139a154417faa00.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 249, + 482, + 362, + 497 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 133, + 515 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 502, + 172, + 514 + ], + "score": 0.93, + "content": "C ( \\omega ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "is the regularized summed cost function, as shown in section 2 of the main text. 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The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 184, + 138 + ], + "score": 1.0, + "content": "larger the step size", + "type": "text" + }, + { + "bbox": [ + 184, + 129, + 190, + 137 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 127, + 358, + 138 + ], + "score": 1.0, + "content": ", the greater the discretization error, but if", + "type": "text" + }, + { + "bbox": [ + 358, + 128, + 364, + 136 + ], + "score": 0.75, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "is sufficiently small and we iterate", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 447, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 447, + 150 + ], + "score": 1.0, + "content": "equation 17 sufficiently many times, we will obtain valid samples from the posterior.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 354, + 168 + ], + "score": 1.0, + "content": "Since the probability of sampling any given parameter vector", + "type": "text" + }, + { + "bbox": [ + 354, + 156, + 362, + 164 + ], + "score": 0.76, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "is proportional to the posterior, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "probability of sampling a parameter vector belonging to any given local minimum is proportional to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 466, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 325, + 189 + ], + "score": 1.0, + "content": "the integral of the posterior over the bowl of attraction", + "type": "text" + }, + { + "bbox": [ + 325, + 177, + 335, + 186 + ], + "score": 0.84, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 176, + 466, + 189 + ], + "score": 1.0, + "content": "which surrounds that minimum.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 190, + 387, + 217 + ], + "lines": [ + { + "bbox": [ + 223, + 190, + 387, + 217 + ], + "spans": [ + { + "bbox": [ + 223, + 190, + 387, + 217 + ], + "score": 0.95, + "content": "P ( \\omega \\in D , t \\infty ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .", + "type": "interline_equation", + "image_path": "a67abd61022557b8527f0ceadb80a46befc91282e1cd1c40bfff4965dfd908a0.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 190, + 387, + 203.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 223, + 203.5, + 387, + 217.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 218, + 504, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "Meanwhile we showed in section 2 of the main text that the evidence in favor of a model is propor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 354, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 354, + 242 + ], + "score": 1.0, + "content": "tional to the integral of the posterior over all parameter space.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 243, + 371, + 269 + ], + "lines": [ + { + "bbox": [ + 240, + 243, + 371, + 269 + ], + "spans": [ + { + "bbox": [ + 240, + 243, + 371, + 269 + ], + "score": 0.94, + "content": "P ( y | x ; 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Meanwhile, in non-convex models, there are many such minima, and so we can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 445, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 445, + 317 + ], + "score": 1.0, + "content": "instead define the evidence as a sum over local evidences in favor of each minimum,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 318, + 384, + 344 + ], + "lines": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "spans": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "score": 0.95, + "content": "P ( y | x ; M ) = \\sum _ { i } P ( y | x , \\omega \\in D _ { i } ; M ) ,", + "type": "interline_equation", + "image_path": "9d21a0bc2f36dbe12c5a5f6c1e18aa3112145d4a431dcd517692d713c252e91b.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 503, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "where we define the evidence in favor of a minimum as the integral over the local bowl of attraction,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 365, + 388, + 391 + ], + "lines": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "spans": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "score": 0.95, + "content": "P ( y | x , \\omega \\in D ; M ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .", + "type": "interline_equation", + "image_path": "1585219cdde94fb4528524bae5b3410dec703bc1860283c24f667d7be7f7137f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 405 + ], + "score": 1.0, + "content": "Since the combined bowls of attraction of all the minima perfectly tile the entire parameter space,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "equations 19 and 20 are equivalent. Meanwhile, equating equations 18 and 21 we find that, when one", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "performs Bayesian posterior sampling, the probability of sampling the parameter vector from a local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "minimum is proportional to the evidence in favor of that minimum. This demonstrates that bayesian", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "posterior samples are biased in favor of local minima whose evidence is large, which explains why", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 446, + 489, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 208, + 459 + ], + "score": 1.0, + "content": "a single posterior sample", + "type": "text" + }, + { + "bbox": [ + 208, + 447, + 221, + 459 + ], + "score": 0.87, + "content": "\\omega _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 446, + 474, + 459 + ], + "score": 1.0, + "content": "often achieves lower test error than the cost function minimum", + "type": "text" + }, + { + "bbox": [ + 474, + 450, + 486, + 457 + ], + "score": 0.72, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 446, + 489, + 459 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 105, + 474, + 460, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 462, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 462, + 488 + ], + "score": 1.0, + "content": "B THE EFFECT OF REGULARIZATION ON THE GENERALIZATION GAP", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 497, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 435, + 510 + ], + "score": 1.0, + "content": "In the experiments of section 4 of the main text, the L2 regularization coefficient", + "type": "text" + }, + { + "bbox": [ + 435, + 498, + 462, + 508 + ], + "score": 0.89, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 497, + 505, + 510 + ], + "score": 1.0, + "content": ". In figure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 321, + 520 + ], + "score": 1.0, + "content": "8, we plot the evolution of the training curves when", + "type": "text" + }, + { + "bbox": [ + 321, + 509, + 357, + 519 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 509, + 506, + 520 + ], + "score": 1.0, + "content": ", for both small batch and full batch", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "image", + "bbox": [ + 117, + 545, + 492, + 687 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 545, + 492, + 687 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 545, + 492, + 687 + ], + "spans": [ + { + "bbox": [ + 117, + 545, + 492, + 687 + ], + "score": 0.965, + "type": "image", + "image_path": "9b725df1b2851f06eebf187dcc70cd8ce8c558a8b2aed25f92d9127ad2a42bfe.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 117, + 545, + 492, + 592.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 117, + 592.3333333333334, + 492, + 639.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 117, + 639.6666666666667, + 492, + 687.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 696, + 506, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "score": 1.0, + "content": "Figure 8: The mean test accuracy (a) and the mean test cross-entropy (b) of a regularized model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "score": 1.0, + "content": "during training. 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The noise inherent in small batch training causes the performance to fluctuate.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + } + ], + "index": 32.5 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 134, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 84, + 142, + 92 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 82, + 342, + 95 + ], + "score": 1.0, + "content": "denotes a Gaussian random variable with mean", + "type": "text" + }, + { + "bbox": [ + 342, + 82, + 380, + 95 + ], + "score": 0.94, + "content": "\\langle \\alpha \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 82, + 437, + 95 + ], + "score": 1.0, + "content": "and variance", + "type": "text" + }, + { + "bbox": [ + 437, + 82, + 501, + 95 + ], + "score": 0.93, + "content": "\\langle \\alpha ^ { 2 } \\rangle \\ : = \\ : 2 \\epsilon I / N", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "which introduces isotropic noise to the gradient update as described in section 4 of the main text.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 171, + 117 + ], + "score": 1.0, + "content": "Note that, since", + "type": "text" + }, + { + "bbox": [ + 172, + 105, + 210, + 116 + ], + "score": 0.92, + "content": "C ( \\omega ; M )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "denotes the summed cost function, we chose to scale our step size by the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 174, + 127 + ], + "score": 1.0, + "content": "training set size", + "type": "text" + }, + { + "bbox": [ + 174, + 116, + 185, + 125 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 115, + 505, + 127 + ], + "score": 1.0, + "content": ". This also matches our treatment of SGD in section 5 of the main text. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 184, + 138 + ], + "score": 1.0, + "content": "larger the step size", + "type": "text" + }, + { + "bbox": [ + 184, + 129, + 190, + 137 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 127, + 358, + 138 + ], + "score": 1.0, + "content": ", the greater the discretization error, but if", + "type": "text" + }, + { + "bbox": [ + 358, + 128, + 364, + 136 + ], + "score": 0.75, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "is sufficiently small and we iterate", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 447, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 447, + 150 + ], + "score": 1.0, + "content": "equation 17 sufficiently many times, we will obtain valid samples from the posterior.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 505, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 354, + 168 + ], + "score": 1.0, + "content": "Since the probability of sampling any given parameter vector", + "type": "text" + }, + { + "bbox": [ + 354, + 156, + 362, + 164 + ], + "score": 0.76, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "is proportional to the posterior, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "probability of sampling a parameter vector belonging to any given local minimum is proportional to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 466, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 325, + 189 + ], + "score": 1.0, + "content": "the integral of the posterior over the bowl of attraction", + "type": "text" + }, + { + "bbox": [ + 325, + 177, + 335, + 186 + ], + "score": 0.84, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 176, + 466, + 189 + ], + "score": 1.0, + "content": "which surrounds that minimum.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 153, + 506, + 189 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 190, + 387, + 217 + ], + "lines": [ + { + "bbox": [ + 223, + 190, + 387, + 217 + ], + "spans": [ + { + "bbox": [ + 223, + 190, + 387, + 217 + ], + "score": 0.95, + "content": "P ( \\omega \\in D , t \\infty ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .", + "type": "interline_equation", + "image_path": "a67abd61022557b8527f0ceadb80a46befc91282e1cd1c40bfff4965dfd908a0.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 190, + 387, + 203.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 223, + 203.5, + 387, + 217.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 218, + 504, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "Meanwhile we showed in section 2 of the main text that the evidence in favor of a model is propor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 354, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 354, + 242 + ], + "score": 1.0, + "content": "tional to the integral of the posterior over all parameter space.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 217, + 505, + 242 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 243, + 371, + 269 + ], + "lines": [ + { + "bbox": [ + 240, + 243, + 371, + 269 + ], + "spans": [ + { + "bbox": [ + 240, + 243, + 371, + 269 + ], + "score": 0.94, + "content": "P ( y | x ; M ) \\propto \\int d \\omega e ^ { - C ( \\omega ; M ) } .", + "type": "interline_equation", + "image_path": "645eab6b826d2ff9c83f869bf1f7b717a71bcdf29443928aef6d566e2f195ad3.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 240, + 243, + 371, + 269 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "As we discussed, this evidence is dominated by the contributions to the integral near local minima.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "In a convex model, there is only one such minimum; which allows us to accurately estimate the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 294, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 504, + 306 + ], + "score": 1.0, + "content": "model evidence. Meanwhile, in non-convex models, there are many such minima, and so we can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 445, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 445, + 317 + ], + "score": 1.0, + "content": "instead define the evidence as a sum over local evidences in favor of each minimum,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 271, + 505, + 317 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 318, + 384, + 344 + ], + "lines": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "spans": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "score": 0.95, + "content": "P ( y | x ; M ) = \\sum _ { i } P ( y | x , \\omega \\in D _ { i } ; M ) ,", + "type": "interline_equation", + "image_path": "9d21a0bc2f36dbe12c5a5f6c1e18aa3112145d4a431dcd517692d713c252e91b.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 226, + 318, + 384, + 344 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 503, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "where we define the evidence in favor of a minimum as the integral over the local bowl of attraction,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 345, + 505, + 360 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 365, + 388, + 391 + ], + "lines": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "spans": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "score": 0.95, + "content": "P ( y | x , \\omega \\in D ; M ) \\propto \\int _ { D } d \\omega \\ e ^ { - C ( \\omega ; M ) } .", + "type": "interline_equation", + "image_path": "1585219cdde94fb4528524bae5b3410dec703bc1860283c24f667d7be7f7137f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 223, + 365, + 388, + 391 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 405 + ], + "score": 1.0, + "content": "Since the combined bowls of attraction of all the minima perfectly tile the entire parameter space,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "equations 19 and 20 are equivalent. Meanwhile, equating equations 18 and 21 we find that, when one", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "performs Bayesian posterior sampling, the probability of sampling the parameter vector from a local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "minimum is proportional to the evidence in favor of that minimum. This demonstrates that bayesian", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "posterior samples are biased in favor of local minima whose evidence is large, which explains why", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 446, + 489, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 208, + 459 + ], + "score": 1.0, + "content": "a single posterior sample", + "type": "text" + }, + { + "bbox": [ + 208, + 447, + 221, + 459 + ], + "score": 0.87, + "content": "\\omega _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 446, + 474, + 459 + ], + "score": 1.0, + "content": "often achieves lower test error than the cost function minimum", + "type": "text" + }, + { + "bbox": [ + 474, + 450, + 486, + 457 + ], + "score": 0.72, + "content": "\\omega _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 446, + 489, + 459 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 390, + 506, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 474, + 460, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 462, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 462, + 488 + ], + "score": 1.0, + "content": "B THE EFFECT OF REGULARIZATION ON THE GENERALIZATION GAP", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 105, + 497, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 435, + 510 + ], + "score": 1.0, + "content": "In the experiments of section 4 of the main text, the L2 regularization coefficient", + "type": "text" + }, + { + "bbox": [ + 435, + 498, + 462, + 508 + ], + "score": 0.89, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 497, + 505, + 510 + ], + "score": 1.0, + "content": ". In figure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 321, + 520 + ], + "score": 1.0, + "content": "8, we plot the evolution of the training curves when", + "type": "text" + }, + { + "bbox": [ + 321, + 509, + 357, + 519 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 509, + 506, + 520 + ], + "score": 1.0, + "content": ", for both small batch and full batch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "training. Excluding the regularization parameter, these experiments are identical to figure 3. To our", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "surprise, regularized full batch training took longer to converge than small batch training. In another", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "surprise, regularization significantly reduced the size of the generalization gap. While large batch", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "regularized training achieves slightly lower test set accuracy than unregularized small batch training,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "it also achieves lower test cross-entropy. The test cross-entropy of our regularized models does not", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 335, + 408, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 408, + 349 + ], + "score": 1.0, + "content": "degrade after many gradient updates, removing the need for early stopping.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 497, + 506, + 520 + ] + }, + { + "type": "image", + "bbox": [ + 117, + 545, + 492, + 687 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 545, + 492, + 687 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 545, + 492, + 687 + ], + "spans": [ + { + "bbox": [ + 117, + 545, + 492, + 687 + ], + "score": 0.965, + "type": "image", + "image_path": "9b725df1b2851f06eebf187dcc70cd8ce8c558a8b2aed25f92d9127ad2a42bfe.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 117, + 545, + 492, + 592.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 117, + 592.3333333333334, + 492, + 639.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 117, + 639.6666666666667, + 492, + 687.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 696, + 506, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "score": 1.0, + "content": "Figure 8: The mean test accuracy (a) and the mean test cross-entropy (b) of a regularized model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 720 + ], + "score": 1.0, + "content": "during training. While full batch training takes longer to converge, it achieves similar performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 719, + 477, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 477, + 731 + ], + "score": 1.0, + "content": "at long times. 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Excluding the regularization parameter, these experiments are identical to figure 3. To our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "surprise, regularized full batch training took longer to converge than small batch training. In another", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "surprise, regularization significantly reduced the size of the generalization gap. While large batch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "regularized training achieves slightly lower test set accuracy than unregularized small batch training,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "it also achieves lower test cross-entropy. The test cross-entropy of our regularized models does not", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 335, + 408, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 408, + 349 + ], + "score": 1.0, + "content": "degrade after many gradient updates, removing the need for early stopping.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 365, + 438, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 441, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 441, + 380 + ], + "score": 1.0, + "content": "C THE GAUSSIAN APPROXIMATION TO THE MINI-BATCH ERROR", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "In section 5 of the main text, we approximated the difference between the full batch gradient and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 395, + 511, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 243, + 416 + ], + "score": 1.0, + "content": "the mini-batch gradient estimate,", + "type": "text" + }, + { + "bbox": [ + 243, + 400, + 312, + 416 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 395, + 511, + 425 + ], + "score": 1.0, + "content": ", by a Gaussian random variable. This enabled", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "us to derive the scaling rules, which we verified empirically. We motivated this assumption by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "reference to the central limit theorem, which states that the gradient error will tend towards Gaussian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 143, + 448 + ], + "score": 1.0, + "content": "noise as", + "type": "text" + }, + { + "bbox": [ + 143, + 436, + 274, + 448 + ], + "score": 0.91, + "content": "\\{ N \\to \\infty , B \\to \\infty , B \\ll N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 437, + 505, + 448 + ], + "score": 1.0, + "content": ", so long as the distribution of gradients over individual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 360, + 460 + ], + "score": 1.0, + "content": "training examples does not have heavy tails. In practice neither", + "type": "text" + }, + { + "bbox": [ + 361, + 448, + 370, + 457 + ], + "score": 0.26, + "content": "\\mathbf { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "nor B is infinite, and the gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "distribution may be heavy tailed, especially when gradients are sparse. Nonetheless the central limit", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 469, + 445, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 445, + 482 + ], + "score": 1.0, + "content": "theorem tends to be surprisingly robust in practice, and is consequently widely used.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "It is beyond the scope of this work to perform a thorough study of the gradient noise distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "in deep networks. However as a brief proof of principle, we present the distribution of the gradient", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "immediately after random initialization in figure 9, for the shallow neural network discussed in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "sections 4 and 5 of the main text. In figure 9a, we present the distribution over the individual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "training examples, of the gradient of a single matrix element in the softmax output layer, chosen", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "randomly. The distribution is double peaked and clearly not Gaussian. However in figure 7b, we plot", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the distribution of the gradient of the same matrix element, when averaged over randomly sampled", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "mini-batches of 30 images (without replacement). A single peak emerges, and while the distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "is still slightly skewed, it is clearly already approaching the Gaussian limit. We conclude that the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 584, + 457, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 457, + 597 + ], + "score": 1.0, + "content": "Gaussian approximation is likely to be reasonable for commonly used mini-batch sizes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 437, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 439, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 439, + 628 + ], + "score": 1.0, + "content": "D DERIVING THE SCALING RULES FOR SGD WITH MOMENTUM", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 446, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 448, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 448, + 652 + ], + "score": 1.0, + "content": "Momentum simulates a generalized Langevin equation (with structured fluctuations),", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 656, + 366, + 682 + ], + "lines": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "spans": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "score": 0.94, + "content": "\\frac { d ^ { 2 } \\omega } { d t ^ { 2 } } = - \\lambda \\frac { d \\omega } { d t } - \\frac { d C } { d \\omega } + \\eta ( t ) .", + "type": "interline_equation", + "image_path": "16e299b6d82ae6a50ef8986f4a0a56d4aff5be1686a4f726b75f607a014315fa.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 114, + 698 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 687, + 249, + 700 + ], + "score": 1.0, + "content": "is the “damping coefficient” and", + "type": "text" + }, + { + "bbox": [ + 249, + 687, + 267, + 700 + ], + "score": 0.92, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 687, + 441, + 700 + ], + "score": 1.0, + "content": "describes Gaussian noise, whose statistics", + "type": "text" + }, + { + "bbox": [ + 441, + 687, + 487, + 700 + ], + "score": 0.93, + "content": "\\langle \\eta ( t ) \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 234, + 711 + ], + "score": 0.91, + "content": "\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = \\bar { g } \\lambda \\bar { F } ( w ) \\delta ( t - t ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 699, + 343, + 712 + ], + "score": 1.0, + "content": ". As before, the coefficient", + "type": "text" + }, + { + "bbox": [ + 343, + 701, + 349, + 710 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "describes the scale of random fluctua-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 212, + 722 + ], + "score": 1.0, + "content": "tions in the dynamics, and", + "type": "text" + }, + { + "bbox": [ + 212, + 711, + 235, + 722 + ], + "score": 0.9, + "content": "F ( \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "describes the gradient covariances between parameters. We include", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 152, + 732 + ], + "score": 1.0, + "content": "a factor of", + "type": "text" + }, + { + "bbox": [ + 152, + 721, + 159, + 730 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "in the noise variance to satisfy the fluctuation-dissipation theorem, which states that", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 117, + 84, + 495, + 227 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 84, + 495, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 84, + 495, + 227 + ], + "spans": [ + { + "bbox": [ + 117, + 84, + 495, + 227 + ], + "score": 0.94, + "type": "image", + "image_path": "0e9f4bc9e0527e3032e6c0929c574be057d50f2e1d13b90d5cc6350b4fcdf353.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 117, + 84, + 495, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 117, + 131.66666666666666, + 495, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 117, + 179.33333333333331, + 495, + 226.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 237, + 504, + 260 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "Figure 9: The gradient distribution of a randomly selected parameter in the softmax layer, when", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "score": 1.0, + "content": "measured over a single training example (a), and when averaged over mini-batches of 30 images (b).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 347 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 281, + 506, + 349 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 365, + 438, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 441, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 441, + 380 + ], + "score": 1.0, + "content": "C THE GAUSSIAN APPROXIMATION TO THE MINI-BATCH ERROR", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "In section 5 of the main text, we approximated the difference between the full batch gradient and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 395, + 511, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 243, + 416 + ], + "score": 1.0, + "content": "the mini-batch gradient estimate,", + "type": "text" + }, + { + "bbox": [ + 243, + 400, + 312, + 416 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha = ( \\frac { d \\hat { C } } { d \\omega } - \\frac { d C } { d \\omega } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 395, + 511, + 425 + ], + "score": 1.0, + "content": ", by a Gaussian random variable. This enabled", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "us to derive the scaling rules, which we verified empirically. We motivated this assumption by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "reference to the central limit theorem, which states that the gradient error will tend towards Gaussian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 143, + 448 + ], + "score": 1.0, + "content": "noise as", + "type": "text" + }, + { + "bbox": [ + 143, + 436, + 274, + 448 + ], + "score": 0.91, + "content": "\\{ N \\to \\infty , B \\to \\infty , B \\ll N \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 437, + 505, + 448 + ], + "score": 1.0, + "content": ", so long as the distribution of gradients over individual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 360, + 460 + ], + "score": 1.0, + "content": "training examples does not have heavy tails. In practice neither", + "type": "text" + }, + { + "bbox": [ + 361, + 448, + 370, + 457 + ], + "score": 0.26, + "content": "\\mathbf { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "nor B is infinite, and the gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "distribution may be heavy tailed, especially when gradients are sparse. Nonetheless the central limit", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 469, + 445, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 445, + 482 + ], + "score": 1.0, + "content": "theorem tends to be surprisingly robust in practice, and is consequently widely used.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 389, + 511, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "It is beyond the scope of this work to perform a thorough study of the gradient noise distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "in deep networks. However as a brief proof of principle, we present the distribution of the gradient", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "immediately after random initialization in figure 9, for the shallow neural network discussed in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "sections 4 and 5 of the main text. In figure 9a, we present the distribution over the individual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "training examples, of the gradient of a single matrix element in the softmax output layer, chosen", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "randomly. The distribution is double peaked and clearly not Gaussian. However in figure 7b, we plot", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the distribution of the gradient of the same matrix element, when averaged over randomly sampled", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "mini-batches of 30 images (without replacement). A single peak emerges, and while the distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "is still slightly skewed, it is clearly already approaching the Gaussian limit. We conclude that the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 584, + 457, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 457, + 597 + ], + "score": 1.0, + "content": "Gaussian approximation is likely to be reasonable for commonly used mini-batch sizes.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 486, + 506, + 597 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 437, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 439, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 439, + 628 + ], + "score": 1.0, + "content": "D DERIVING THE SCALING RULES FOR SGD WITH MOMENTUM", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 446, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 448, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 448, + 652 + ], + "score": 1.0, + "content": "Momentum simulates a generalized Langevin equation (with structured fluctuations),", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 638, + 448, + 652 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 656, + 366, + 682 + ], + "lines": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "spans": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "score": 0.94, + "content": "\\frac { d ^ { 2 } \\omega } { d t ^ { 2 } } = - \\lambda \\frac { d \\omega } { d t } - \\frac { d C } { d \\omega } + \\eta ( t ) .", + "type": "interline_equation", + "image_path": "16e299b6d82ae6a50ef8986f4a0a56d4aff5be1686a4f726b75f607a014315fa.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 245, + 656, + 366, + 682 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 114, + 698 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 687, + 249, + 700 + ], + "score": 1.0, + "content": "is the “damping coefficient” and", + "type": "text" + }, + { + "bbox": [ + 249, + 687, + 267, + 700 + ], + "score": 0.92, + "content": "\\eta ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 687, + 441, + 700 + ], + "score": 1.0, + "content": "describes Gaussian noise, whose statistics", + "type": "text" + }, + { + "bbox": [ + 441, + 687, + 487, + 700 + ], + "score": 0.93, + "content": "\\langle \\eta ( t ) \\rangle = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 234, + 711 + ], + "score": 0.91, + "content": "\\langle \\eta ( t ) \\eta ( t ^ { \\prime } ) \\rangle = \\bar { g } \\lambda \\bar { F } ( w ) \\delta ( t - t ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 699, + 343, + 712 + ], + "score": 1.0, + "content": ". As before, the coefficient", + "type": "text" + }, + { + "bbox": [ + 343, + 701, + 349, + 710 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "describes the scale of random fluctua-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 212, + 722 + ], + "score": 1.0, + "content": "tions in the dynamics, and", + "type": "text" + }, + { + "bbox": [ + 212, + 711, + 235, + 722 + ], + "score": 0.9, + "content": "F ( \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "describes the gradient covariances between parameters. We include", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 152, + 732 + ], + "score": 1.0, + "content": "a factor of", + "type": "text" + }, + { + "bbox": [ + 152, + 721, + 159, + 730 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "in the noise variance to satisfy the fluctuation-dissipation theorem, which states that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "we can vary the damping coefficient without changing the probability of sampling any particular", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 493, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 265, + 107 + ], + "score": 1.0, + "content": "configuration of parameters in the limit", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 265, + 94, + 295, + 104 + ], + "score": 0.9, + "content": "t \\to \\infty", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 296, + 93, + 493, + 107 + ], + "score": 1.0, + "content": ", if we proportionally increase the noise variance.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "we can vary the damping coefficient without changing the probability of sampling any particular", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 493, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 265, + 107 + ], + "score": 1.0, + "content": "configuration of parameters in the limit", + "type": "text" + }, + { + "bbox": [ + 265, + 94, + 295, + 104 + ], + "score": 0.9, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 93, + 493, + 107 + ], + "score": 1.0, + "content": ", if we proportionally increase the noise variance.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 503, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "score": 1.0, + "content": "To relate this Langevin equation to the usual momentum equations, we first re-express it as two", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 272, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 272, + 134 + ], + "score": 1.0, + "content": "coupled first order differential equations,", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 134, + 365, + 184 + ], + "lines": [ + { + "bbox": [ + 243, + 134, + 365, + 184 + ], + "spans": [ + { + "bbox": [ + 243, + 134, + 365, + 184 + ], + "score": 0.92, + "content": "\\begin{array} { r c l } { \\displaystyle \\frac { d p } { d t } } & { = } & { \\displaystyle - 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We also predict an additional scaling relation between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 294, + 504 + ], + "score": 1.0, + "content": "the batch size and the momentum parameter,", + "type": "text" + }, + { + "bbox": [ + 294, + 492, + 365, + 504 + ], + "score": 0.93, + "content": "B \\propto \\mathsf { \\bar { 1 } } / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 491, + 505, + 504 + ], + "score": 1.0, + "content": ". Note that one can also interpret", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 309, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 181, + 515 + ], + "score": 0.93, + "content": "\\epsilon _ { e f f } = \\epsilon / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 503, + 309, + 515 + ], + "score": 1.0, + "content": "as the “effective learning rate”.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 530, + 348, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 349, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 349, + 544 + ], + "score": 1.0, + "content": "E HOW TO ACHIEVE LARGE BATCH TRAINING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "Here we propose a simple heuristic for tuning the batch size, learning rate and momentum parame-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "ter; in order to maximize both test accuracy and batch size (enabling parallel training across many", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 468, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 468, + 589 + ], + "score": 1.0, + "content": "machines). Note that this is only worthwhile if one expects to retrain a model many times.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 129, + 596, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 129, + 594, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 129, + 594, + 505, + 610 + ], + "score": 1.0, + "content": "1. Set the learning rate to 0.1 and the momentum coefficient to 0.9. Run experiments at a range", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 142, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "of batch sizes on a logarithmic scale, and identify the optimal batch size which maximizes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 618, + 493, + 631 + ], + "spans": [ + { + "bbox": [ + 141, + 618, + 493, + 631 + ], + "score": 1.0, + "content": "the validation set accuracy. If training is not stable, reduce the learning rate, and repeat.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 128, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 128, + 631, + 473, + 645 + ], + "score": 1.0, + "content": "2. Repeatedly increase the batch size by a factor of 3, while scaling the learning rate", + "type": "text" + }, + { + "bbox": [ + 474, + 632, + 501, + 642 + ], + "score": 0.88, + "content": "\\epsilon \\propto B", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 631, + 505, + 645 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 141, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "until the validation set accuracy starts to fall. Then repeatedly increase the batch size by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 141, + 654, + 367, + 666 + ], + "score": 1.0, + "content": "a factor of 3, while scaling the momentum coefficient", + "type": "text" + }, + { + "bbox": [ + 367, + 654, + 437, + 666 + ], + "score": 0.93, + "content": "( 1 - \\dot { m } ) \\propto 1 / B", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 654, + 505, + 666 + ], + "score": 1.0, + "content": ", until either the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 664, + 468, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 664, + 468, + 678 + ], + "score": 1.0, + "content": "validation set accuracy falls or the batch size reaches the limits of your hardware.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 129, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 129, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "3. 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We also predict an additional scaling relation between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 294, + 504 + ], + "score": 1.0, + "content": "the batch size and the momentum parameter,", + "type": "text" + }, + { + "bbox": [ + 294, + 492, + 365, + 504 + ], + "score": 0.93, + "content": "B \\propto \\mathsf { \\bar { 1 } } / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 491, + 505, + 504 + ], + "score": 1.0, + "content": ". Note that one can also interpret", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 309, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 181, + 515 + ], + "score": 0.93, + "content": "\\epsilon _ { e f f } = \\epsilon / ( 1 - m )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 503, + 309, + 515 + ], + "score": 1.0, + "content": "as the “effective learning rate”.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 470, + 505, + 515 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 530, + 348, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 349, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 349, + 544 + ], + "score": 1.0, + "content": "E HOW TO ACHIEVE LARGE BATCH TRAINING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 553, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "Here we propose a simple heuristic for tuning the batch size, learning rate and momentum parame-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 579 + ], + "score": 1.0, + "content": "ter; in order to maximize both test accuracy and batch size (enabling parallel training across many", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 468, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 468, + 589 + ], + "score": 1.0, + "content": "machines). Note that this is only worthwhile if one expects to retrain a model many times.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 554, + 505, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 596, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 129, + 594, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 129, + 594, + 505, + 610 + ], + "score": 1.0, + "content": "1. Set the learning rate to 0.1 and the momentum coefficient to 0.9. Run experiments at a range", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 142, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "of batch sizes on a logarithmic scale, and identify the optimal batch size which maximizes", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 618, + 493, + 631 + ], + "spans": [ + { + "bbox": [ + 141, + 618, + 493, + 631 + ], + "score": 1.0, + "content": "the validation set accuracy. If training is not stable, reduce the learning rate, and repeat.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 128, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 128, + 631, + 473, + 645 + ], + "score": 1.0, + "content": "2. Repeatedly increase the batch size by a factor of 3, while scaling the learning rate", + "type": "text" + }, + { + "bbox": [ + 474, + 632, + 501, + 642 + ], + "score": 0.88, + "content": "\\epsilon \\propto B", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 631, + 505, + 645 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 141, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "until the validation set accuracy starts to fall. Then repeatedly increase the batch size by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 141, + 654, + 367, + 666 + ], + "score": 1.0, + "content": "a factor of 3, while scaling the momentum coefficient", + "type": "text" + }, + { + "bbox": [ + 367, + 654, + 437, + 666 + ], + "score": 0.93, + "content": "( 1 - \\dot { m } ) \\propto 1 / B", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 654, + 505, + 666 + ], + "score": 1.0, + "content": ", until either the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 664, + 468, + 678 + ], + "spans": [ + { + "bbox": [ + 141, + 664, + 468, + 678 + ], + "score": 1.0, + "content": "validation set accuracy falls or the batch size reaches the limits of your hardware.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 129, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 129, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "3. Having identified the final learning rate and momentum parameter, retune the batch size on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 691, + 402, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 691, + 402, + 702 + ], + "score": 1.0, + "content": "a linear scale in the local neighborhood of the current batch size.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 128, + 594, + 505, + 702 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "We believe that this simple procedure will increase the test accuracy, reduce the cost of tuning hyper-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "parameters, and significantly 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+ } + } +] \ No newline at end of file diff --git a/parse/train/Byl1W1rtvH/Byl1W1rtvH.md b/parse/train/Byl1W1rtvH/Byl1W1rtvH.md new file mode 100644 index 0000000000000000000000000000000000000000..1c0373b34acc2a4eff33afe783d6b5be417d8d44 --- /dev/null +++ b/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. + +
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·
foriter=1,2,. do
Randomly select a mini-batch of m documents consisting of J sentences to form a subset X =
{di,1:J,Si,1:J}m1; m,J,L
Draw random noise {e',j }i=1.j=1,l=1 from uniform distribution;
Calculate VΩL (S2, Φl,II'; X,e,j) according to (6),and update Ω; Sample 0’,j from (7) and (8) via Ω
to update {II′}=1 and {Φ′}𝑖=1, will be described in Appendix C; end for
+ +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. + +
ModelLSTM SizeTopic SizePerplexity
APNEWSIMDBBNC
LCLM(Tian & Cho,2016)60054.1867.7896.50
900-900一50.6367.8687.77
LDA+LSTM (Lau et al., 2017)60010055.5269.6496.50
900-90010050.7563.0487.77
TopicRNN (Dieng et al.,2017)60010054.5467.8393.57
900-90010050.2461.5984.62
TDLM (Lau et al., 2017)60010052.7563.4585.99
900-90010048.9759.0481.83
TCNLM (Wang et al.,2018)60010052.6362.6486.44
TGVAE (Wang et al.,2019)900-90010047.8156.3880.14
6005048.7357.1187.86
basic-LSTM(Hochreiter & Schmidhuber,1997)600一64.1372.14102.89
900-90058.8966.4794.23
GBN-RNN900-900-900一60.1365.1695.73
600 600-51210047.4257.0186.39
100-8044.6455.4282.95
600-512-256100-80-5044.3554.5380.25
60010046.3555.7681.94
rGBN-RNN600-512100-8043.2653.8280.25
600-512-256100-80-5042.7151.3679.13
+ +![](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. 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In ICLR, 2018. +He Zhao, Lan Du, Wray Buntine, and Mingyuan Zhou. Dirichlet belief networks for topic structure learning. In Neural Information Processing Systems, pp. 7955–7966, 2018. +Mingyuan Zhou and Lawrence Carin. Negative binomial process count and mixture modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(2):307–320, 2015. +Mingyuan Zhou, Yulai Cong, and Bo Chen. Augmentable gamma belief networks. J. Mach. Learn. Res., 17(163):1–44, 2016. +Yaoming Zhu, Sidi Lu, Lei Zheng, Jiaxian Guo, Weinan Zhang, Jun Wang, and Yong Yu. Texygen: A benchmarking platform for text generation models. SIGIR, 2018. + +# 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. + +
DatasetVocubalryTrainingValidationTesting
LMTMDocsSentsTokensDocsSentsTokensDocsSentsTokens
APNEWS342313216950K0.8M15M2K33K0.6M2K32K0.6M
IMDB360093492575K1.1M20M12.5K0.18M0.3M12.5K0.18M0.3M
BNC437034155215K1M18M1K57K1M1K66K1M
+ +# 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. + +
Component ParamLanguage ModelTopic Model
LSTMword in (3)glinBΦin (2)II in (2)RNNsent in (8)f in (8)f in (8)
Layer1O(4×(E+H) ×H")O(3×(K1+H)×H1)O(Vc × K1)O(K1 × K1)O(Vc+H𝑖) × HiO(Hi)O(K1 x Hi)
Layer2O(4×(H+H)×H²)O(3 × (K2+H) × H)O(K1 × K2)O(K2 ×K2)O((Hi +H²)×H)O(H)O(K2 ×H)
Layer3O(4×(H+H)×H))1O(3×(K3+H)×H)[O(K2×K3)|O(K3 × K3)|O(H+H) × H)O(H)O(K ×H)
+ +Table 4: Comparison of the number of parameters of different models when used for language generation. + +
ModelLSTM SizeTopic Size# LM Param#TMParam# All Param
TDLM (Lau et al., 2017)600 900-900100 1003.35M 13.38M0.019M 0.019M3.37M 13.40M
basic-LSTM (Hochreiter & Schmidhuber,1997)600 900-900 900-900-9002.16M 10.80M 17.68M2.16M 10.80M 17.68M
GBN-RNN600 600-512 600-512-256100 100-80 100-80-503.40M 6.50M 7.20M0.02M 0.04M 0.05M3.42M 6.54M 7.25M
rGBN-RNN600 600-512 600-512-256100 100-80 100-80-503.40M 6.50M 7.20M0.03M 0.06M 0.07M3.43M 6.56M 7.27M
+ +![](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. \ No newline at end of file diff --git a/parse/train/HJMHpjC9Ym/HJMHpjC9Ym.md b/parse/train/HJMHpjC9Ym/HJMHpjC9Ym.md new file mode 100644 index 0000000000000000000000000000000000000000..110c7c5f8dca9b62fab5d1e1e2fed48e7d53748c --- /dev/null +++ b/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. + +
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α = 2, β = 2)22.72%2.91 (1.41×)26.97
bL-ResNet-50(α = 2, β= 4)22.69%2.85 (1.44×)26.69
bL-ResNet-50 (α = 4, β = 2)23.20%2.49 (1.64×)26.31
bL-ResNet-50 (α= 4, β= 4)23.15%2.48 (1.65×)26.24
+ +Table 2: Performance comparison for bL-ResNet, bL-ResNeXt and bL-SEResNeXt. + +
ModelTop-1 ErrorFLOPs (109)Params (106)Speed (ms/batch)t
ResNet-10121.95%7.8044.54186
bL-ResNet-101 (α= 2,β= 4)21.80%3.89 (2.01×)41.85140 (1.33×)
bL-ResNet-101@256(α = 2,β= 4)21.04%5.08 (1.54×)41.85162 (1.15×)
ResNet-15221.51%11.5160.19266
bL-ResNet-152(α = 2,β =4)21.16%5.04 (2.28×)57.36178 (1.49×)
bL-ResNet-152@256 (α= 2, β=4)20.34%6.58 (1.75x)57.36205 (1.30×)
ResNeXt-50 (32×4d)22.20%4.2325.03157
bL-ResNeXt-50 (32×4d) (α =2,β= 4)21.60%3.03 (1.40×)26.19125 (1.26×)
bL-ResNeXt-50@256(α = 2, β= 4)20.96%3.95 (1.08×)26.19153 (1.03×)
ResNeXt-101 (32×4d)21.20%7.9744.17269
bL-ResNeXt-101 (32×4d) (α =2,β =4)21.08%4.08 (1.95×)41.51169 (1.59×)
bL-ResNeXt-101@256(α = 2, β=4)20.48%5.33 (1.50x)41.51203 (1.33×)
ResNeXt-101(64×4d)20.73%15.4683.46485
bL-ResNeXt-101 (64×4d) (α = 2,β =4)20.48%7.14 (2.17×)77.36263 (1.98×)
bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)19.65%9.32 (1.66x)77.36318 (1.53×)
SEResNeXt-50 (32×4d)21.78%4.2327.56216
bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)21.44%3.03 (1.40×)28.77163 (1.33×)
20.74%3.95 (1.08×)28.77192 (1.03×)
SEResNeXt-101 (32×4d)21.00%7.9748.96376
bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)20.87%4.08 (1.95×)45.88235 (1.60×)
bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)19.87%5.33 (1.50×)45.88270 (1.39×)
+ +†: 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. + +
ModelFLOPs (109)Params (106)WER AvgHub5Hub5 CH
1Baseline: ResNet-221.113.0214.67%11.15%18.17%
2bL-ResNet-22 (α = 4,β = 1)0.68 (1.63×)3.1514.72%11.24%18.18%
3bL-ResNet-22 (α = 4,β = 2)0.66 (1.68×)3.1114.47%10.95%17.95%
4bL-ResNet-22(α = 4,β= 3)0.65 (1.70×)3.1014.66%11.25%18.05%
5bL-ResNet-22 (α = 2,β = 3)0.77 (1.43×)3.0714.46%11.10%17.80%
6bL-ResNet-22 (α = 4,β=1) cat0.70 (1.58×)3.1814.67%11.31%18.00%
7bL-PYR-ResNet-22 (α = 4, β =1)0.98 (1.13×)3.3214.50%11.05%17.92%
+ +# 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. + +# REFERENCES + +Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. + +# 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. + +
LayersbL-ResNet-50ResNet-50
Convolution7 × 7,64, s2 (112 × 112)
bL-module3×3,323 × 3,64,s2 3×3,32,s2(56 × 56)(1×1,64MaxPooling (56 × 56)
bL-moduleResBlockg,256 ×2 ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)ResBlock,256 ×3, s1(56 × 56)
bL-moduleResBlockp,512 ×3 ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)ResBlock,512 ×4, s2(28× 28)
bL-moduleResBlockb,1024 ×5 ResBlockL,512 ×1ResBlock,1024 (14 × 14)ResBlock,1024 ×6, s2(14 × 14)
ResBlockResBlock,2048 ×3, s2 (7 × 7)
Average pool7 ×7 average pooling
FC, softmax1000
+ +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 $\beta = 4$ + +
LayersOutput SizebL-ResNet-101 bL-ResNet-152
Convolution112 × 1127 × 7,64,s2
bL-module56× 56（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64
bL-module56×56/1×1,64 3×3,64 ×3(2) (1×1,256) B/1×1,32 3×3,32 （1×1,128） L×1/1×1,64 3×3,64 × 4(2) （1×1,256) B/1×1,32 3×3,32 （1×1,128)×1 L
transition layer28×28/1×1,64 3×3,64,s2 ×1 （1×1,256
bL-module28×28(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B×1 （1×1,256） L(1×1,128) 3×3,128 (1×1,512)× 11(7)/1×1,64 3×3,64 ×2 (1×1,256) L
transition layer14 × 14/1×1,128 3×3,128,s2 ×1 (1×1,512B
bL-module14 × 14/1×1,256 3×3,256 × 17(22) （1×1,1024）B(1×1,128) 3×3,128 （1×1,512 L×3/1×1,256 3×3,256 (1×1,1024) B× 29(35)(1×1,128) 3×3,128 （1×1,512）×6 L
transition layer14 × 14/1×1,256 3×3,256 ×1 （1×1,1024）
ResBlock7×7/1×1,512 3 ×3,512,s2 ×3
Average pool1×1（1×1,2048 7×7 average pooling
FC, softmax1000
+ +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. + +
NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
ResNet-50-lowres26.10%1.29 (3.17×)25.60
ResNet-10121.95%7.8044.54
ResNet-101-lowres24.80%2.22 (3.51x)44.57
+ +# 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. + +
NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (addition,K = 2)22.69 %2.85 (1.43×)26.69
bL-ResNet-50 (concatenation,K = 2)24.04%2.01 ( (2.03x)20.57
bL-ResNet-50 (addition,K = 3)24.12%3.91 (1.04×)27.23
+ +Table 8: Different number of merges in bL-ResNet. $m$ : number of merges. $( \alpha = 2 , \beta = 4 )$ + +
ModelTop-1 ErrorFLOPs (109)Params (106)
bL-ResNet-50 (m = 4) (baseline)22.69%2.8526.69
bL-ResNet-50 (m = 2)23.48%2.7426.66
bL-ResNet-50 (m = 1)24.57%2.6426.64
bL-ResNet-101 (m= 4) (baseline)21.80%3.8941.85
bL-ResNet-101 (m = 7)21.85%5.2144.44
+ +# 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) $( \alpha = 2 , \beta = 2 ) .$ + +
Model|Model widthTop-1 ErrorFLOPs (106)Params (106)
ShuffleNetV21×30.60%1462.30
1.5×28.03%2993.51
2×26.85%5887.40
bL-ShuffleNetV2 @ 2561×30.84%1502.33
1.5×27.83%2984.80
2×26.38%5907.60
+ +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. + +
NetworkmAP@[IoU=0.5] (bbox)FLOPst (109)
ResNet-10181.5137.70
bL-ResNet-101 (α=2, β=4)81.489.21 (1.54×)
bL-ResNet-101 (α=2, β=2)81.593.95 (1.47×)
+ +Detection performance on the MS COCO val2017 dataset. + +
NetworkmAP@[IoU=0.50:0.95](bbox)|FLOPs‡ (109)
ResNet-10139.2234.85
bL-ResNet-101 (α=2, β=4)39.5151.11 (1.55×)
bL-ResNet-101 (α=2, β=2)40.4160.21 (1.47×)
+ +†: 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. + +
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α =1, β=1)21.75%5.6534.12
bL-ResNet-50 (α =1, β=2）22.11%4.3430.14
bL-ResNet-50(α= 2,) β=2)22.72%2.9126.97
bL-ResNet-50(α =2,) β=4)22.69%2.8526.69
bL-ResNet-50(α = 4, β = 2)23.20%2.4926.31
bL-ResNet-50 (α =4, β= 4)23.15%2.4826.24
ResNet-10121.95%7.8044.54
bL-ResNet-101 (α=1,β=1)20.31%10.2963.32
bL-ResNet-101 (α = 2, β= 2)21.40%4.2743.39
bL-ResNet-101 (α = 2,β=4)21.80%3.8941.85
+ +# 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. + +
LayersOutput Size 一bL-ResNet-22 (α=4,β=1)bL-ResNet-22 (α= 2,β=3)
Convolution32×T T=49→455×5,64,s2
bL-module32×T T=45→35（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1(3×3,16) （3×3,16）L (3 ×3,16), × 1×2（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1（3×3,32） （3×3,32)×1
transition layer16×T T=35→33(3×3,64,s2)×1
bL-module16×T T= 33→23（3×3,128) ×2 （3×3,128） B（3×3,32) （3×3,32）×2 L（3×3,128) ×2 3×3,128 B（3×3,64) 3×3,64）×1 L
transition layer8×T T=23→21（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1
bL-module8×T T=21→11（3×3,256) ×2 3×3,256 B(3×3,64) （3×3,64） L×2（3×3,256) ×2 3×3,256 B（3×3,128) 3×3,128）×1 L
transition layer4×T(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1
bL-moduleT=11→9 4×T（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2
ConvolutionT=9→1 1×1（3×3,512） （3×3,128） （3×3,512) B L
Convolution1×14×1,512 1 × 1,32k
+ +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 $\alpha = 4 , \beta = 1$ ). \ No newline at end of file diff --git a/parse/train/HJMHpjC9Ym/HJMHpjC9Ym_content_list.json b/parse/train/HJMHpjC9Ym/HJMHpjC9Ym_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..fa4520b079e42bbaefa5a8cbd399fe17ab0fd81c --- /dev/null +++ b/parse/train/HJMHpjC9Ym/HJMHpjC9Ym_content_list.json @@ -0,0 +1,2175 @@ +[ + { + "type": "text", + "text": "BIG-LITTLE NET: AN EFFICIENT MULTI-SCALE FEATURE REPRESENTATION FOR VISUAL AND SPEECH RECOGNITION ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 170 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 183, + 194, + 821, + 237 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 273, + 544, + 289 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 233, + 306, + 766, + 513 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 542, + 334, + 559 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 575, + 825, + 659 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 666, + 825, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/0574022df5e50adb5747f9dd6217fb99fafb4314cbc906cbae2227b65290d2a4.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 235, + 99, + 758, + 217 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 342, + 823, + 371 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 377, + 825, + 488 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contributions of our paper are summarized as follows: ", + "bbox": [ + 174, + 496, + 593, + 510 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose an efficient and effective multi-scale CNN architecture for object and speech recognition. \n• 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. \nWe validate the proposed method on a speech recognition task, where we also achieve better word error rates while reducing the number of FLOPs by $30 \\%$ . ", + "bbox": [ + 215, + 522, + 825, + 655 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 675, + 344, + 691 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 708, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 236, + 825, + 291 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 OUR APPROACH", + "text_level": 1, + "bbox": [ + 176, + 311, + 341, + 328 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 343, + 825, + 441 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 BIG-LITTLE NET ", + "text_level": 1, + "bbox": [ + 174, + 457, + 333, + 472 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 174, + 483, + 825, + 569 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/38b4f2f7819bca1f03c5c82724c77c42dc7a5db52263169ef9963f47f82c4437.jpg", + "text": "$$\n\\mathbf { x } _ { i + 1 } = F \\left( \\sum _ { k = 0 } ^ { K - 1 } c ^ { k } S ^ { k } \\left( f _ { k } \\left( \\mathbf { x } _ { i } ^ { k } \\right) \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 574, + 617, + 618 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 623, + 825, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 700, + 825, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 790, + 825, + 875 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 214 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 222, + 825, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 NETWORK MERGING ", + "text_level": 1, + "bbox": [ + 176, + 391, + 361, + 405 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 417, + 825, + 570 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 578, + 825, + 689 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 709, + 418, + 726 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 741, + 825, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 OBJECT RECOGNITION ", + "text_level": 1, + "bbox": [ + 176, + 828, + 374, + 842 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/3bc54325361fb652f3594cfd1795aeeaaf24386f1e1be2957aaf8663dc1b8927.jpg", + "table_caption": [ + "Table 1: Complexity study of the Little-Branch $\\scriptstyle { \\alpha }$ and $\\beta$ ) for bL-ResNet-50. " + ], + "table_footnote": [], + "table_body": "
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α = 2, β = 2)22.72%2.91 (1.41×)26.97
bL-ResNet-50(α = 2, β= 4)22.69%2.85 (1.44×)26.69
bL-ResNet-50 (α = 4, β = 2)23.20%2.49 (1.64×)26.31
bL-ResNet-50 (α= 4, β= 4)23.15%2.48 (1.65×)26.24
", + "bbox": [ + 272, + 127, + 722, + 210 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/2e2890abf33d01d59ce186599437a1b288d85424966123cd2fd1e51d3c3ece55.jpg", + "table_caption": [ + "Table 2: Performance comparison for bL-ResNet, bL-ResNeXt and bL-SEResNeXt. " + ], + "table_footnote": [ + "†: 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. " + ], + "table_body": "
ModelTop-1 ErrorFLOPs (109)Params (106)Speed (ms/batch)t
ResNet-10121.95%7.8044.54186
bL-ResNet-101 (α= 2,β= 4)21.80%3.89 (2.01×)41.85140 (1.33×)
bL-ResNet-101@256(α = 2,β= 4)21.04%5.08 (1.54×)41.85162 (1.15×)
ResNet-15221.51%11.5160.19266
bL-ResNet-152(α = 2,β =4)21.16%5.04 (2.28×)57.36178 (1.49×)
bL-ResNet-152@256 (α= 2, β=4)20.34%6.58 (1.75x)57.36205 (1.30×)
ResNeXt-50 (32×4d)22.20%4.2325.03157
bL-ResNeXt-50 (32×4d) (α =2,β= 4)21.60%3.03 (1.40×)26.19125 (1.26×)
bL-ResNeXt-50@256(α = 2, β= 4)20.96%3.95 (1.08×)26.19153 (1.03×)
ResNeXt-101 (32×4d)21.20%7.9744.17269
bL-ResNeXt-101 (32×4d) (α =2,β =4)21.08%4.08 (1.95×)41.51169 (1.59×)
bL-ResNeXt-101@256(α = 2, β=4)20.48%5.33 (1.50x)41.51203 (1.33×)
ResNeXt-101(64×4d)20.73%15.4683.46485
bL-ResNeXt-101 (64×4d) (α = 2,β =4)20.48%7.14 (2.17×)77.36263 (1.98×)
bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)19.65%9.32 (1.66x)77.36318 (1.53×)
SEResNeXt-50 (32×4d)21.78%4.2327.56216
bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)21.44%3.03 (1.40×)28.77163 (1.33×)
20.74%3.95 (1.08×)28.77192 (1.03×)
SEResNeXt-101 (32×4d)21.00%7.9748.96376
bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)20.87%4.08 (1.95×)45.88235 (1.60×)
bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)19.87%5.33 (1.50×)45.88270 (1.39×)
", + "bbox": [ + 176, + 236, + 821, + 526 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 568, + 825, + 625 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 631, + 826, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 736, + 825, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 840, + 825, + 922 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/a6d6b75cec7913fe409b728001e78bc9467fe9e9c10ca41f278de9c663887d7f.jpg", + "image_caption": [ + "Figure 2: Comparison with the ResNet and ResNeXt related works. " + ], + "image_footnote": [], + "bbox": [ + 352, + 121, + 625, + 280 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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 $0 . 1 3 \\%$ . ", + "bbox": [ + 174, + 318, + 825, + 387 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 395, + 825, + 506 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1.1 COMPARISON WITH RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 527, + 482, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 554, + 823, + 583 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 589, + 825, + 742 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d00d2c0fac8445379485651a3bc5072b0003cb35fce602124c3b6ac1929e4ff2.jpg", + "image_caption": [ + "Figure 3: Comparison performance among other types of networks. (a) FLOPs. (b) GPU Speed. " + ], + "image_footnote": [], + "bbox": [ + 199, + 122, + 779, + 301 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 329, + 823, + 358 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 176, + 364, + 825, + 420 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 428, + 825, + 511 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 176, + 518, + 825, + 560 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 SPEECH RECOGNITION ", + "text_level": 1, + "bbox": [ + 176, + 578, + 374, + 592 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 603, + 825, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 736, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f62d4c092458680f4d4c84ce38f356efedea97e77400f4db20af9d4d87061474.jpg", + "table_caption": [ + "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_footnote": [], + "table_body": "
ModelFLOPs (109)Params (106)WER AvgHub5Hub5 CH
1Baseline: ResNet-221.113.0214.67%11.15%18.17%
2bL-ResNet-22 (α = 4,β = 1)0.68 (1.63×)3.1514.72%11.24%18.18%
3bL-ResNet-22 (α = 4,β = 2)0.66 (1.68×)3.1114.47%10.95%17.95%
4bL-ResNet-22(α = 4,β= 3)0.65 (1.70×)3.1014.66%11.25%18.05%
5bL-ResNet-22 (α = 2,β = 3)0.77 (1.43×)3.0714.46%11.10%17.80%
6bL-ResNet-22 (α = 4,β=1) cat0.70 (1.58×)3.1814.67%11.31%18.00%
7bL-PYR-ResNet-22 (α = 4, β =1)0.98 (1.13×)3.3214.50%11.05%17.92%
", + "bbox": [ + 176, + 141, + 821, + 263 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 282, + 825, + 449 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 DISCUSSION ON bL-Net ", + "text_level": 1, + "bbox": [ + 176, + 472, + 377, + 487 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 501, + 825, + 640 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 666, + 318, + 683 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 702, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The authors would like to thank Dr. Paul Crumley and Dr. I-Hsin Chung for their help with the hardware infrastructure setup. ", + "bbox": [ + 174, + 127, + 823, + 156 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 162, + 825, + 233 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 253, + 285, + 268 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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", + "bbox": [ + 173, + 424, + 823, + 463 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. ", + "bbox": [ + 174, + 472, + 821, + 498 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 174, + 99, + 292, + 122 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The appendix illustrates the details of our experiments on object recognition and speech recognition and more ablation study. ", + "bbox": [ + 174, + 127, + 823, + 156 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A bL-Net FOR OBJECT RECOGNITION ", + "text_level": 1, + "bbox": [ + 176, + 176, + 500, + 193 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 208, + 379, + 222 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 234, + 826, + 387 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/daa98a8098783ad23ddd1613c134e943733e72d710d3825625f041f529fa5569.jpg", + "table_caption": [ + "Table 4: Network configurations of bL-ResNet-50. Output size is illustrated in the parenthesis. " + ], + "table_footnote": [ + "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. " + ], + "table_body": "
LayersbL-ResNet-50ResNet-50
Convolution7 × 7,64, s2 (112 × 112)
bL-module3×3,323 × 3,64,s2 3×3,32,s2(56 × 56)(1×1,64MaxPooling (56 × 56)
bL-moduleResBlockg,256 ×2 ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)ResBlock,256 ×3, s1(56 × 56)
bL-moduleResBlockp,512 ×3 ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)ResBlock,512 ×4, s2(28× 28)
bL-moduleResBlockb,1024 ×5 ResBlockL,512 ×1ResBlock,1024 (14 × 14)ResBlock,1024 ×6, s2(14 × 14)
ResBlockResBlock,2048 ×3, s2 (7 × 7)
Average pool7 ×7 average pooling
FC, softmax1000
", + "bbox": [ + 181, + 428, + 816, + 679 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 NETWORK STRUCTURE ", + "text_level": 1, + "bbox": [ + 176, + 746, + 380, + 761 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 772, + 825, + 843 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 849, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 138, + 825, + 250 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/caff856ca5859ab4a7dd26fe7d3e44c4bc2bb66e5bfe8c4384020382983fa6b0.jpg", + "table_caption": [ + "Table 5: Network configurations of bL-ResNets, and $\\alpha = 2$ and $\\beta = 4$ " + ], + "table_footnote": [ + "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. " + ], + "table_body": "
LayersOutput SizebL-ResNet-101 bL-ResNet-152
Convolution112 × 1127 × 7,64,s2
bL-module56× 56（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64
bL-module56×56/1×1,64 3×3,64 ×3(2) (1×1,256) B/1×1,32 3×3,32 （1×1,128） L×1/1×1,64 3×3,64 × 4(2) （1×1,256) B/1×1,32 3×3,32 （1×1,128)×1 L
transition layer28×28/1×1,64 3×3,64,s2 ×1 （1×1,256
bL-module28×28(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B×1 （1×1,256） L(1×1,128) 3×3,128 (1×1,512)× 11(7)/1×1,64 3×3,64 ×2 (1×1,256) L
transition layer14 × 14/1×1,128 3×3,128,s2 ×1 (1×1,512B
bL-module14 × 14/1×1,256 3×3,256 × 17(22) （1×1,1024）B(1×1,128) 3×3,128 （1×1,512 L×3/1×1,256 3×3,256 (1×1,1024) B× 29(35)(1×1,128) 3×3,128 （1×1,512）×6 L
transition layer14 × 14/1×1,256 3×3,256 ×1 （1×1,1024）
ResBlock7×7/1×1,512 3 ×3,512,s2 ×3
Average pool1×1（1×1,2048 7×7 average pooling
FC, softmax1000
", + "bbox": [ + 174, + 297, + 823, + 614 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 PERFORMANCE ON LOW RESOLUTION INPUT ", + "text_level": 1, + "bbox": [ + 178, + 704, + 529, + 718 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 734, + 825, + 833 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/bf948a2c1696a33afbd37fc5e16cb47be047e9384139f171958c590313dc30a9.jpg", + "image_caption": [ + "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)). " + ], + "image_footnote": [], + "bbox": [ + 272, + 101, + 725, + 287 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/fb28e4e8d27205eda372c39a4e3c63ca31e28eed8348b009f72dc98a06a3e615.jpg", + "table_caption": [ + "Table 6: Performance of ResNets at different input resolutions. " + ], + "table_footnote": [], + "table_body": "
NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
ResNet-50-lowres26.10%1.29 (3.17×)25.60
ResNet-10121.95%7.8044.54
ResNet-101-lowres24.80%2.22 (3.51x)44.57
", + "bbox": [ + 269, + 401, + 727, + 492 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.4 ABLATION STUDY ON NETWORK MERGING AND MULTI-BRANCH ", + "text_level": 1, + "bbox": [ + 176, + 518, + 673, + 534 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 545, + 826, + 657 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 662, + 825, + 775 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/28c2d9f94c55dfa98f8434b3d6c8da798ade4b751255a0a4038859c961298a48.jpg", + "table_caption": [ + "Table 7: Different scales and merging schemes on bL-ResNet. " + ], + "table_footnote": [], + "table_body": "
NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (addition,K = 2)22.69 %2.85 (1.43×)26.69
bL-ResNet-50 (concatenation,K = 2)24.04%2.01 ( (2.03x)20.57
bL-ResNet-50 (addition,K = 3)24.12%3.91 (1.04×)27.23
", + "bbox": [ + 209, + 815, + 789, + 914 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/fc2be9be041d19908fe7a93a78382119796c237cfd9a547ddd42a556c87ec0de.jpg", + "table_caption": [ + "Table 8: Different number of merges in bL-ResNet. $m$ : number of merges. $( \\alpha = 2 , \\beta = 4 )$ " + ], + "table_footnote": [], + "table_body": "
ModelTop-1 ErrorFLOPs (109)Params (106)
bL-ResNet-50 (m = 4) (baseline)22.69%2.8526.69
bL-ResNet-50 (m = 2)23.48%2.7426.66
bL-ResNet-50 (m = 1)24.57%2.6426.64
bL-ResNet-101 (m= 4) (baseline)21.80%3.8941.85
bL-ResNet-101 (m = 7)21.85%5.2144.44
", + "bbox": [ + 240, + 127, + 756, + 224 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Number of Merges in bL-Net ", + "text_level": 1, + "bbox": [ + 174, + 256, + 377, + 271 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 277, + 825, + 333 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 340, + 825, + 424 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 431, + 825, + 556 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.5 bL-Net FOR HIGH-EFFICIENCY NETWORK ", + "text_level": 1, + "bbox": [ + 174, + 580, + 504, + 594 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 609, + 825, + 666 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/0c537ce4c089b03840c7381094c1c3411f36b679512e81cdc50ed80016497127.jpg", + "table_caption": [ + "Table 9: Comparison with ShuffleNetV2 (Ma et al., 2018) $( \\alpha = 2 , \\beta = 2 ) .$ " + ], + "table_footnote": [ + "All models are trained by ourselves. " + ], + "table_body": "
Model|Model widthTop-1 ErrorFLOPs (106)Params (106)
ShuffleNetV21×30.60%1462.30
1.5×28.03%2993.51
2×26.85%5887.40
bL-ShuffleNetV2 @ 2561×30.84%1502.33
1.5×27.83%2984.80
2×26.38%5907.60
", + "bbox": [ + 272, + 713, + 725, + 803 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.6 bL-Net FOR OBJECT DETECTION ", + "text_level": 1, + "bbox": [ + 176, + 852, + 441, + 867 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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 ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/83aa27c29e1168def3585ab6f05177f29097e81914b5f0dec44f260ea4c795af.jpg", + "table_caption": [ + "Table 10: Objection detection results. Detection performance on the PASCAL VOC 2007test dataset. " + ], + "table_footnote": [], + "table_body": "
NetworkmAP@[IoU=0.5] (bbox)FLOPst (109)
ResNet-10181.5137.70
bL-ResNet-101 (α=2, β=4)81.489.21 (1.54×)
bL-ResNet-101 (α=2, β=2)81.593.95 (1.47×)
", + "bbox": [ + 223, + 195, + 769, + 266 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/dd64b8bf1f542d42c09cc8d4d375e16ed5dfdbf7ccde7c19e265c0176851c472.jpg", + "table_caption": [ + "Detection performance on the MS COCO val2017 dataset. " + ], + "table_footnote": [ + "†: 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. " + ], + "table_body": "
NetworkmAP@[IoU=0.50:0.95](bbox)|FLOPs‡ (109)
ResNet-10139.2234.85
bL-ResNet-101 (α=2, β=4)39.5151.11 (1.55×)
bL-ResNet-101 (α=2, β=2)40.4160.21 (1.47×)
", + "bbox": [ + 223, + 294, + 769, + 363 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.7 COMPARISON OF MEMORY REQUIREMENT ", + "text_level": 1, + "bbox": [ + 176, + 438, + 514, + 453 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 468, + 825, + 539 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 545, + 825, + 616 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/da4e5aec7cfa4b70d0b83baf77d263cacc8fcabedbed9c1b250d87d538f80742.jpg", + "image_caption": [ + "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.) " + ], + "image_footnote": [], + "bbox": [ + 218, + 724, + 769, + 876 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/c44b6b75071334dc3ac51d75693383281d4e0f1ed4a6654e6c2ef6c2970e7408.jpg", + "table_caption": [ + "Table 11: Complexity study of the Little-Branch $\\scriptstyle { \\alpha }$ and $\\beta$ ) for bL-ResNet-50 and bL-ResNet-101. " + ], + "table_footnote": [], + "table_body": "
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α =1, β=1)21.75%5.6534.12
bL-ResNet-50 (α =1, β=2）22.11%4.3430.14
bL-ResNet-50(α= 2,) β=2)22.72%2.9126.97
bL-ResNet-50(α =2,) β=4)22.69%2.8526.69
bL-ResNet-50(α = 4, β = 2)23.20%2.4926.31
bL-ResNet-50 (α =4, β= 4)23.15%2.4826.24
ResNet-10121.95%7.8044.54
bL-ResNet-101 (α=1,β=1)20.31%10.2963.32
bL-ResNet-101 (α = 2, β= 2)21.40%4.2743.39
bL-ResNet-101 (α = 2,β=4)21.80%3.8941.85
", + "bbox": [ + 271, + 128, + 722, + 287 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B bL-Net FOR SPEECH RECOGNITION ", + "text_level": 1, + "bbox": [ + 173, + 292, + 500, + 308 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 323, + 379, + 338 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 349, + 825, + 476 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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 $ ). ", + "bbox": [ + 174, + 481, + 825, + 526 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 NETWORK STRUCTURES ", + "text_level": 1, + "bbox": [ + 174, + 541, + 387, + 556 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 568, + 825, + 707 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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 $( 1 \\times 1 , 3 2 \\mathbf { k } )$ . ", + "bbox": [ + 176, + 714, + 823, + 756 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 762, + 825, + 847 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/35a78c641734758cee5a21f2327ed10ac66e506131a0a17cb9cc0161bf8cb6e9.jpg", + "table_caption": [ + "Table 12: Network configurations of bL-ResNets applied to speech for acoustic modeling. " + ], + "table_footnote": [ + "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. " + ], + "table_body": "
LayersOutput Size 一bL-ResNet-22 (α=4,β=1)bL-ResNet-22 (α= 2,β=3)
Convolution32×T T=49→455×5,64,s2
bL-module32×T T=45→35（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1(3×3,16) （3×3,16）L (3 ×3,16), × 1×2（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1（3×3,32） （3×3,32)×1
transition layer16×T T=35→33(3×3,64,s2)×1
bL-module16×T T= 33→23（3×3,128) ×2 （3×3,128） B（3×3,32) （3×3,32）×2 L（3×3,128) ×2 3×3,128 B（3×3,64) 3×3,64）×1 L
transition layer8×T T=23→21（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1
bL-module8×T T=21→11（3×3,256) ×2 3×3,256 B(3×3,64) （3×3,64） L×2（3×3,256) ×2 3×3,256 B（3×3,128) 3×3,128）×1 L
transition layer4×T(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1
bL-moduleT=11→9 4×T（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2
ConvolutionT=9→1 1×1（3×3,512） （3×3,128） （3×3,512) B L
Convolution1×14×1,512 1 × 1,32k
", + "bbox": [ + 176, + 127, + 823, + 462 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 540, + 825, + 611 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "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 $\\lceil 4 / 3 \\rceil = \\lceil 4 / 2 \\rceil = 2$ . ", + "bbox": [ + 173, + 617, + 826, + 799 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 804, + 825, + 888 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "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 $\\alpha = 4 , \\beta = 1$ ). 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For object recognition, our approach reduces", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 205, + 354 + ], + "score": 1.0, + "content": "computation by", + "type": "text" + }, + { + "bbox": [ + 206, + 342, + 222, + 354 + ], + "score": 0.46, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 342, + 397, + 354 + ], + "score": 1.0, + "content": "while improving accuracy significantly over", + "type": "text" + }, + { + "bbox": [ + 398, + 342, + 412, + 352 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 342, + 469, + 354 + ], + "score": 1.0, + "content": "point than the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 384, + 365 + ], + "score": 1.0, + "content": "baselines, and the computational savings can be higher up to", + "type": "text" + }, + { + "bbox": [ + 385, + 353, + 401, + 365 + ], + "score": 0.49, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "without compro-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 364, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 469, + 375 + ], + "score": 1.0, + "content": "mising the accuracy. Our model also surpasses state-of-the-art CNN acceleration", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 375, + 470, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 470, + 387 + ], + "score": 1.0, + "content": "approaches by a large margin in terms of accuracy and FLOPs. On the task of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 385, + 470, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 367, + 398 + ], + "score": 1.0, + "content": "speech recognition, our proposed multi-scale CNNs save", + "type": "text" + }, + { + "bbox": [ + 367, + 386, + 387, + 396 + ], + "score": 0.85, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 385, + 470, + 398 + ], + "score": 1.0, + "content": "FLOPs with slightly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 397, + 419, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 419, + 408 + ], + "score": 1.0, + "content": "better word error rates, showing good generalization across domains.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 243, + 471, + 408 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 430, + 205, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 208, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 208, + 446 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "Deep Convolutional Neural Network (CNN) models have achieved substantial performance gains", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "in many computer vision and speech recognition tasks (He et al., 2016; 2017; Vinyals et al., 2017;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "Sercu & Goel, 2016). However, the accuracy obtained by these models usually grows proportionally", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "with their complexity and computational cost. This poses an issue for deploying these models", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "in applications that require real-time inferencing and low-memory footprint, such as self-driving", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 386, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 386, + 523 + ], + "score": 1.0, + "content": "vehicles, human-machine interaction on mobile devices, and robotics.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 456, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "Motivated by these applications, many methods have been proposed for model compression and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "acceleration, including techniques such as pruning (Dong et al., 2017; Li et al., 2017; Han et al., 2015),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "score": 1.0, + "content": "quantization (Hubara et al., 2016; Li & Liu, 2016), and low-rank factorization (Wen et al., 2017;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "Ioannou et al., 2015; Zhang et al., 2016). Most of these methods have been applied to single-scale", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "inputs, without considering multi-resolution processing. More recently, another line of work applies", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "dynamic routing to allocate different workloads in the networks according to image complexity (Wu", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "et al., 2018; Wang et al., 2018; Figurnov et al., 2017; Veit & Belongie, 2018). Multi-scale feature", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "representations have proven successful for many vision and speech recognition tasks compared to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 507, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 507, + 630 + ], + "score": 1.0, + "content": "single-scale methods (Nah et al., 2017; Chen et al., 2017a; Tóth, 2017; Farabet et al., 2013); however,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 444, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 444, + 639 + ], + "score": 1.0, + "content": "the computational complexity has not been addressed much in multi-scale networks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 529, + 507, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "The computational cost of a CNN model has much to do with the input image size. A model, if", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 425, + 667 + ], + "score": 1.0, + "content": "running at half of the image size, can gain a remarkable computational saving of", + "type": "text" + }, + { + "bbox": [ + 426, + 655, + 445, + 666 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ". Based on this", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "fact, we propose an efficient network architecture by combining image information at different scales", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "through a multi-branch network. As shown in Fig. 1, our key idea is to use a high-complexity branch", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(accurate but costly) for low-scale feature representation and low-complexity branch (efficient but", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "less accurate) for high-scale feature representation. The two types of features are frequently merged", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "together to complement and enrich each other, leading to a stronger feature representation than either", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "of them individually. We refer to the deeper branch operating at low image resolution as Big-Branch", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "and the shallower one as Little-Branch to reflect their differences in computation. The new network", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 281, + 399, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 399, + 295 + ], + "score": 1.0, + "content": "architecture is thus called Big-Little Net or bL-Net for short in this paper.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 144, + 79, + 464, + 172 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 144, + 79, + 464, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 79, + 464, + 172 + ], + "spans": [ + { + "bbox": [ + 144, + 79, + 464, + 172 + ], + "score": 0.971, + "type": "image", + "image_path": "0574022df5e50adb5747f9dd6217fb99fafb4314cbc906cbae2227b65290d2a4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 144, + 79, + 464, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 144, + 110.0, + 464, + 141.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 144, + 141.0, + 464, + 172.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 177, + 505, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "Figure 1: Our proposed Big-Little Net (bL-Net) for efficient multi-scale feature representations. (a)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 189, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 390, + 200 + ], + "score": 1.0, + "content": "The bL-Net stacks several Big-Little Modules. 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(b) Our implementation of the Big-Little Module includes two branches. The Big-Branch", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "has the same structure as the baseline model while the Little-Branch reduces the convolutional layers", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 191, + 246 + ], + "score": 1.0, + "content": "and feature maps by", + "type": "text" + }, + { + "bbox": [ + 192, + 234, + 200, + 243 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 232, + 218, + 246 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 233, + 226, + 244 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 232, + 353, + 246 + ], + "score": 1.0, + "content": ", respectively. Larger values of", + "type": "text" + }, + { + "bbox": [ + 353, + 234, + 361, + 243 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 232, + 379, + 246 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 233, + 387, + 244 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "lead to lower computational", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 244, + 225, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 225, + 256 + ], + "score": 1.0, + "content": "complexity in Big-Little Net.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "and the shallower one as Little-Branch to reflect their differences in computation. The new network", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 281, + 399, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 399, + 295 + ], + "score": 1.0, + "content": "architecture is thus called Big-Little Net or bL-Net for short in this paper.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "While being structurally simple, our approach is quite effective. We demonstrate later that when", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "bL-Net is integrated into state-of-the-art CNNs such as ResNet, ResNeXt and SEResNeXt, it yields", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 120, + 332 + ], + "score": 0.84, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "computational savings over the baselines without losing any accuracy. 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Network pruning (Dong et al., 2017; Li et al., 2017) and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "quantization (Hubara et al., 2016; Li & Liu, 2016) are popular techniques to remove model redundancy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "and save computational cost. Another thread of work consists of training a sparse model directly,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "such as IGCv2 (Xie et al., 2018) and SCConv (Fan et al., 2017). Efficient network architectures like", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "MobileNetV2 (Sandler et al., 2018) and ShuffleNetV2 (Ma et al., 2018) have also been explored for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "training compact deep networks. 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Network pruning (Dong et al., 2017; Li et al., 2017) and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "quantization (Hubara et al., 2016; Li & Liu, 2016) are popular techniques to remove model redundancy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "and save computational cost. Another thread of work consists of training a sparse model directly,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "such as IGCv2 (Xie et al., 2018) and SCConv (Fan et al., 2017). Efficient network architectures like", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 616 + ], + "score": 1.0, + "content": "MobileNetV2 (Sandler et al., 2018) and ShuffleNetV2 (Ma et al., 2018) have also been explored for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "training compact deep networks. Other methods include knowledge distillation (Hinton et al., 2015),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "compression with structured matrices (Cheng et al., 2015; Sindhwani et al., 2015), and hashing (Chen", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "et al., 2015). Dynamic routing (Wu et al., 2018; Wang et al., 2018; Figurnov et al., 2017; Veit &", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "Belongie, 2018) has also been explored in residual networks to improve efficiency. These methods", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "operate with single-resolution inputs, while our approach processes multi-resolution data. It could be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 670, + 367, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 367, + 684 + ], + "score": 1.0, + "content": "used in tandem with these methods to further improve efficiency.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 560, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Multi-Resolution Feature Representations. The notion of multi-scale feature representation can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "be dated back to image pyramids (Adelson et al., 1984) and scale-space theory (Lindeberg &", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "ter Haar Romeny, 1994). More recently, several methods have proposed multi-scale CNN-based", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "architectures for object detection and recognition. MSCNN (Cai et al., 2016), DAG-CNN (Yang", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "& Ramanan, 2015) and FPN (Lin et al., 2017) use features at different layers to form multi-scale", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "features. Hourglass networks (Newell et al., 2016) use a hierarchical multi-resolution model for", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "human pose estimation. However, this approach induces a heavy workload as the complexity of each", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "sub-network in their model is equal. Nah et al. (Nah et al., 2017) and Eigen et al. (Eigen & Fergus,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "2015) combine the features from multiple networks working on different resolutions to generate", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "multi-scale features. The overall computational cost grows along with the number of scales, leading", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "to inefficient models. In contrast to existing methods, our approach uses different network capacities", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "for different scales, and yields more powerful multi-scale features by fusing them at multiple levels", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 196, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 196, + 181 + ], + "score": 1.0, + "content": "of the network model.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "& Ramanan, 2015) and FPN (Lin et al., 2017) use features at different layers to form multi-scale", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "features. Hourglass networks (Newell et al., 2016) use a hierarchical multi-resolution model for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "human pose estimation. However, this approach induces a heavy workload as the complexity of each", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "sub-network in their model is equal. Nah et al. (Nah et al., 2017) and Eigen et al. (Eigen & Fergus,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "2015) combine the features from multiple networks working on different resolutions to generate", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "multi-scale features. The overall computational cost grows along with the number of scales, leading", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "to inefficient models. In contrast to existing methods, our approach uses different network capacities", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "for different scales, and yields more powerful multi-scale features by fusing them at multiple levels", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 196, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 196, + 181 + ], + "score": 1.0, + "content": "of the network model.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "Closely related to our work, approaches such as (Huang et al., 2018; Saxena & Verbeek, 2016), apply", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "multiple branches at multi-scales while aiming at reducing computational complexity. In contrast to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "our work, their computational gain mostly comes from early exit depending on the input image. Our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 393, + 234 + ], + "score": 1.0, + "content": "approach speeds up a network constantly regardless of the input image.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 209, + 260 + ], + "lines": [ + { + "bbox": [ + 104, + 245, + 212, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 212, + 262 + ], + "score": 1.0, + "content": "3 OUR APPROACH", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "We develop a simple, easy-to-implement, yet very efficient and effective network architecture. It", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "learns multi-scale feature representations by fusing multiple branches with different image scales", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "and computational complexity. As shown in Fig. 1, we design a multi-scale feature module with the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "following principles: (I) each branch corresponds to a single unique image scale (or resolution); (II)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 330 + ], + "score": 1.0, + "content": "the computational cost of a branch is inversely proportional to the scale. 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Within a Big-Little Module, assume we have", + "type": "text" + }, + { + "bbox": [ + 360, + 395, + 371, + 404 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 394, + 465, + 407 + ], + "score": 1.0, + "content": "branches working on", + "type": "text" + }, + { + "bbox": [ + 465, + 395, + 476, + 404 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "scales", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 402, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 405, + 212, + 417 + ], + "score": 0.91, + "content": "[ 1 , \\bar { 1 } / 2 , 1 / 4 , . . . , 1 / 2 ^ { K - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 402, + 321, + 420 + ], + "score": 1.0, + "content": ". 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Following the principles above, we propose a multi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "network architecture that integrates two branches for multi-scale feature representation. Figure 1 (b)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 682, + 292, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 292, + 694 + ], + "score": 1.0, + "content": "shows an example Big-Little Net architecture.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "The module includes two branches, each of which represents a separate network block from a deep", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "model (accurate but costly) and a less deep counterpart (efficient but less accurate). 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In contrast to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "our work, their computational gain mostly comes from early exit depending on the input image. Our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 393, + 234 + ], + "score": 1.0, + "content": "approach speeds up a network constantly regardless of the input image.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 186, + 505, + 234 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 247, + 209, + 260 + ], + "lines": [ + { + "bbox": [ + 104, + 245, + 212, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 212, + 262 + ], + "score": 1.0, + "content": "3 OUR APPROACH", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "We develop a simple, easy-to-implement, yet very efficient and effective network architecture. 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As shown in Fig. 1, we design a multi-scale feature module with the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "following principles: (I) each branch corresponds to a single unique image scale (or resolution); (II)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 330 + ], + "score": 1.0, + "content": "the computational cost of a branch is inversely proportional to the scale. Note that the principle (II)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "implies that we use high-complexity networks at lower resolutions and low-complexity networks at", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 284, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 284, + 351 + ], + "score": 1.0, + "content": "higher resolutions for the sake of efficiency.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 272, + 506, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 204, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 206, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 206, + 376 + ], + "score": 1.0, + "content": "3.1 BIG-LITTLE NET", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 428, + 397 + ], + "score": 1.0, + "content": "Big-Little Net is a sequence of Big-Little Modules, each one taking input", + "type": "text" + }, + { + "bbox": [ + 428, + 385, + 439, + 394 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 382, + 506, + 397 + ], + "score": 1.0, + "content": "and producing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 136, + 407 + ], + "score": 1.0, + "content": "output", + "type": "text" + }, + { + "bbox": [ + 136, + 396, + 157, + 406 + ], + "score": 0.88, + "content": "\\mathbf { x } _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 394, + 360, + 407 + ], + "score": 1.0, + "content": ". Within a Big-Little Module, assume we have", + "type": "text" + }, + { + "bbox": [ + 360, + 395, + 371, + 404 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 394, + 465, + 407 + ], + "score": 1.0, + "content": "branches working on", + "type": "text" + }, + { + "bbox": [ + 465, + 395, + 476, + 404 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "scales", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 402, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 405, + 212, + 417 + ], + "score": 0.91, + "content": "[ 1 , \\bar { 1 } / 2 , 1 / 4 , . . . , 1 / 2 ^ { K - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 402, + 321, + 420 + ], + "score": 1.0, + "content": ". We denote a feature map", + "type": "text" + }, + { + "bbox": [ + 321, + 407, + 333, + 416 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 402, + 367, + 420 + ], + "score": 1.0, + "content": "at scale", + "type": "text" + }, + { + "bbox": [ + 367, + 405, + 388, + 417 + ], + "score": 0.92, + "content": "1 / 2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 402, + 401, + 420 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 401, + 405, + 413, + 417 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { i } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 402, + 506, + 420 + ], + "score": 1.0, + "content": ", indicating the spatial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 135, + 430 + ], + "score": 1.0, + "content": "size of", + "type": "text" + }, + { + "bbox": [ + 135, + 419, + 146, + 428 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 416, + 217, + 430 + ], + "score": 1.0, + "content": "downsampled by", + "type": "text" + }, + { + "bbox": [ + 217, + 416, + 228, + 428 + ], + "score": 0.86, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "with respect to the original input dimension. We use a weighted sum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 356, + 441 + ], + "score": 1.0, + "content": "to combine all branches into a feature representation at scale", + "type": "text" + }, + { + "bbox": [ + 356, + 428, + 376, + 441 + ], + "score": 0.9, + "content": "1 / 2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 428, + 505, + 441 + ], + "score": 1.0, + "content": ". Mathematically, the module’s", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 239, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 134, + 452 + ], + "score": 1.0, + "content": "output", + "type": "text" + }, + { + "bbox": [ + 135, + 441, + 155, + 451 + ], + "score": 0.89, + "content": "\\mathbf x _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 439, + 239, + 452 + ], + "score": 1.0, + "content": "can be expressed by", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 382, + 506, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 455, + 378, + 490 + ], + "lines": [ + { + "bbox": [ + 232, + 455, + 378, + 490 + ], + "spans": [ + { + "bbox": [ + 232, + 455, + 378, + 490 + ], + "score": 0.94, + "content": "\\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) ,", + "type": "interline_equation", + "image_path": "38b4f2f7819bca1f03c5c82724c77c42dc7a5db52263169ef9963f47f82c4437.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 455, + 378, + 472.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 232, + 472.5, + 378, + 490.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 133, + 506 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 494, + 155, + 506 + ], + "score": 0.91, + "content": "f _ { k } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 495, + 419, + 506 + ], + "score": 1.0, + "content": "denotes a sequence of convolutional layers. Typically for higher", + "type": "text" + }, + { + "bbox": [ + 419, + 495, + 426, + 505 + ], + "score": 0.69, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 495, + 429, + 506 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 430, + 495, + 441, + 506 + ], + "score": 0.81, + "content": "f _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "will have more", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 295, + 518 + ], + "score": 1.0, + "content": "convolutional layers having more feature maps.", + "type": "text" + }, + { + "bbox": [ + 296, + 505, + 318, + 518 + ], + "score": 0.91, + "content": "S ^ { k } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "is the operation that matches the output size of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 369, + 529 + ], + "score": 1.0, + "content": "the branches, either: (1) increasing the number feature maps with a", + "type": "text" + }, + { + "bbox": [ + 370, + 517, + 391, + 527 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "convolution, (2) upsampling", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 228, + 540 + ], + "score": 1.0, + "content": "to match the output size of the", + "type": "text" + }, + { + "bbox": [ + 228, + 528, + 253, + 537 + ], + "score": 0.9, + "content": "k = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 527, + 320, + 540 + ], + "score": 1.0, + "content": "branch, or both.", + "type": "text" + }, + { + "bbox": [ + 321, + 527, + 330, + 537 + ], + "score": 0.86, + "content": "c ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "indicates the weighting coefficients of each", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 537, + 457, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 205, + 552 + ], + "score": 1.0, + "content": "scale in the merge while", + "type": "text" + }, + { + "bbox": [ + 206, + 538, + 225, + 550 + ], + "score": 0.91, + "content": "F ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 537, + 457, + 552 + ], + "score": 1.0, + "content": "is an optional final fusion layer like a convolutional layer.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 494, + 505, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "Note that branches are merged at the end of every Big-Little Module and merging the branch", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "outputs happens at the highest resolution and highest number of feature maps between the branches.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 575, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 505, + 591 + ], + "score": 1.0, + "content": "Maintaining these large intermediate states avoids information loss. A crucial aspect of this design", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "is that through consecutive merging and downsampling, the expensive branches operating at low", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "resolution still have access to the high resolution information, processed by the cheaper branches in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 610, + 192, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 192, + 622 + ], + "score": 1.0, + "content": "the previous module.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 555, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "While our design is suitable for any number of networks, in this work we primarily focus on the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 221, + 650 + ], + "score": 1.0, + "content": "case of two networks, i.e.,", + "type": "text" + }, + { + "bbox": [ + 222, + 638, + 254, + 648 + ], + "score": 0.89, + "content": "K = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 637, + 380, + 650 + ], + "score": 1.0, + "content": ". We also experimented with", + "type": "text" + }, + { + "bbox": [ + 380, + 638, + 414, + 649 + ], + "score": 0.9, + "content": "K > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "in object and speech", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 209, + 662 + ], + "score": 1.0, + "content": "recognition; however, the", + "type": "text" + }, + { + "bbox": [ + 210, + 649, + 239, + 659 + ], + "score": 0.9, + "content": "K = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "case provided the best balance between accuracy and computation", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "score": 1.0, + "content": "(See Appendix A.4 and Section 4.2 for details). Following the principles above, we propose a multi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "network architecture that integrates two branches for multi-scale feature representation. Figure 1 (b)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 682, + 292, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 292, + 694 + ], + "score": 1.0, + "content": "shows an example Big-Little Net architecture.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 627, + 507, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "The module includes two branches, each of which represents a separate network block from a deep", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "model (accurate but costly) and a less deep counterpart (efficient but less accurate). The two branches", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 383, + 733 + ], + "score": 1.0, + "content": "are fused at the end through linear combination with unit weights (i.e.,", + "type": "text" + }, + { + "bbox": [ + 383, + 720, + 441, + 731 + ], + "score": 0.9, + "content": "c ^ { 0 } = c ^ { 1 } = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "). Before fusion,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the low resolution feature maps are upsampled using bilinear interpolation to spatially match the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 228, + 106 + ], + "score": 1.0, + "content": "higher-resolution counterparts", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 229, + 93, + 263, + 105 + ], + "score": 0.94, + "content": "\\bar { ( = { S ^ { 1 } ( \\cdot ) } ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 263, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ". Similarly, the high resolution feature map has an additional", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 129, + 115 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 130, + 104, + 374, + 117 + ], + "score": 1.0, + "content": "convolutional layer to increase the number of output channels", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 374, + 104, + 408, + 116 + ], + "score": 0.91, + "content": "( = S ^ { 0 } ( \\cdot ) )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 409, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ". Furthermore, since our", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 485, + 128 + ], + "score": 1.0, + "content": "design is based on ResNet, we add a residual block to further fuse the combined features (i.e.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 486, + 115, + 505, + 127 + ], + "score": 0.89, + "content": "F ( \\cdot )", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "is a residual block). For convenience, we refer to these two branches as Big-Branch (more layers", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "and channels at low resolution) and Little-Branch (fewer layers and channels at high resolution),", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "respectively. We also denote the module as Big-Little Module and the entire architecture as Big-Little", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 166, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 166, + 171 + ], + "score": 1.0, + "content": "Net or bL-Net.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the low resolution feature maps are upsampled using bilinear interpolation to spatially match the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 228, + 106 + ], + "score": 1.0, + "content": "higher-resolution counterparts", + "type": "text" + }, + { + "bbox": [ + 229, + 93, + 263, + 105 + ], + "score": 0.94, + "content": "\\bar { ( = { S ^ { 1 } ( \\cdot ) } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ". Similarly, the high resolution feature map has an additional", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 129, + 115 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 104, + 374, + 117 + ], + "score": 1.0, + "content": "convolutional layer to increase the number of output channels", + "type": "text" + }, + { + "bbox": [ + 374, + 104, + 408, + 116 + ], + "score": 0.91, + "content": "( = S ^ { 0 } ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ". Furthermore, since our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 485, + 128 + ], + "score": 1.0, + "content": "design is based on ResNet, we add a residual block to further fuse the combined features (i.e.,", + "type": "text" + }, + { + "bbox": [ + 486, + 115, + 505, + 127 + ], + "score": 0.89, + "content": "F ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "is a residual block). For convenience, we refer to these two branches as Big-Branch (more layers", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "and channels at low resolution) and Little-Branch (fewer layers and channels at high resolution),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "respectively. We also denote the module as Big-Little Module and the entire architecture as Big-Little", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 166, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 166, + 171 + ], + "score": 1.0, + "content": "Net or bL-Net.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 104, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "To control the complexity of bL-Net, we introduce two parameters to specify the complexity of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "Little-Branch with respect to the Big-Branch. The Big-Branch typically follows the structure of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "the original network, but the Little-Branch needs to be heavily slimmed and shortened to reduce", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 413, + 221 + ], + "score": 1.0, + "content": "computation as it operates on high resolution. Here we use two parameters", + "type": "text" + }, + { + "bbox": [ + 413, + 211, + 421, + 219 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 209, + 439, + 221 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 210, + 447, + 221 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "to control the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 404, + 232 + ], + "score": 1.0, + "content": "width and depth of the Little-Branch, respectively. As shown in Fig. 1 (b),", + "type": "text" + }, + { + "bbox": [ + 405, + 222, + 412, + 230 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "specifies the reduction", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "factor of the number of channels in the convolutional layers of Little-Branch with respect to that of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 216, + 255 + ], + "score": 1.0, + "content": "the original network while", + "type": "text" + }, + { + "bbox": [ + 217, + 243, + 224, + 253 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "is the reduction factor of the number of convolutional layers. Larger", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 146, + 266 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 254, + 154, + 263 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 252, + 172, + 266 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 253, + 180, + 264 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "lead to lower complexity in bL-Net. As demonstrated later, with an appropriate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 146, + 276 + ], + "score": 1.0, + "content": "choice of", + "type": "text" + }, + { + "bbox": [ + 146, + 266, + 154, + 274 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 263, + 172, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 264, + 180, + 275 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 263, + 431, + 276 + ], + "score": 1.0, + "content": "(See Table 1 and Table 3), the cost of the Little-Branch can be", + "type": "text" + }, + { + "bbox": [ + 432, + 264, + 448, + 276 + ], + "score": 0.64, + "content": "1 / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 263, + 474, + 276 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 475, + 264, + 487, + 274 + ], + "score": 0.3, + "content": "b L", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "-Net", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 123, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 275, + 145, + 286 + ], + "score": 0.64, + "content": "1 / 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "of the original network while still providing sufficient complementary information to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 158, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 158, + 299 + ], + "score": 1.0, + "content": "Big-Branch.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 310, + 221, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 222, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 222, + 323 + ], + "score": 1.0, + "content": "3.2 NETWORK MERGING", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "We consider two options for merging the outputs of the branches. The first option is a linear", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "combination, which joins features from two networks by addition (Newell et al., 2016). The alternative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 481, + 366 + ], + "score": 1.0, + "content": "concatenates the outputs of the two networks along the channel dimension, and if needed, a", + "type": "text" + }, + { + "bbox": [ + 481, + 353, + 505, + 364 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "score": 1.0, + "content": "convolution can be subsequently applied to reduce the number of feature maps (Szegedy et al., 2015).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "Both merging approaches have their pros and cons. With linear combination, the branches can easily", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 398 + ], + "score": 1.0, + "content": "compensate each other, meaning each branch can activate output neurons not activated by the other.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "score": 1.0, + "content": "However, additional cost is added as both the size of feature maps and the number of channels in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "the two branches need to be adjusted to be the same before addition. On the other hand, merging by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "concatenation only needs to spatially align the feature maps. On the other hand, concatenation only", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 331, + 443 + ], + "score": 1.0, + "content": "needs to align the feature map size, however requires a", + "type": "text" + }, + { + "bbox": [ + 332, + 430, + 356, + 441 + ], + "score": 0.9, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "convolution reducing the number of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 485, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 485, + 454 + ], + "score": 1.0, + "content": "channels after concatenation, which is a more expensive operation than the pointwise addition.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "While linear combination provides an immediate exchange of the activations of both branches,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "concatenation relies on the following layers for this exchange. This delay in exchange could possibly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "be problematic if the information from each branch is destructively altered before merging. For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "score": 1.0, + "content": "example, a nonlinearity such as ReLU would discard all activations less than zero, effectively ignoring", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "negative features in both branches before merging. Since linear combination does not cause too", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "much overhead and provides better accuracy, we chose linear combination as our merging approach.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "In Appendix A.4, we empirically show that the linear combination approach performs better than", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 535, + 251, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 251, + 547 + ], + "score": 1.0, + "content": "concatenation in object recognition.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 256, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 258, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 258, + 576 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "We conducted extensive experiments, as discussed below, to validate the effectiveness of our proposed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "bL-Net on object and speech recognition tasks. bL-Net can be easily integrated with many modern", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "CNNs and here we chose ResNet (He et al., 2016) as the primary architecture to evaluate our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 503, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 493, + 633 + ], + "score": 1.0, + "content": "approach. For simplicity, from now on, we denote by bL-M the bL-Net using a backbone network", + "type": "text" + }, + { + "bbox": [ + 493, + 621, + 503, + 630 + ], + "score": 0.48, + "content": "M", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 386, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 386, + 644 + ], + "score": 1.0, + "content": "For example, bL-ResNet-50 is the Big-Little net based on ResNet-50.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "title", + "bbox": [ + 108, + 656, + 229, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 231, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 231, + 669 + ], + "score": 1.0, + "content": "4.1 OBJECT RECOGNITION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We used the ImageNet dataset (Russakovsky et al., 2015) for all the experiments below on object", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "recognition. This dataset is a common benchmark for object recognition, which contains 1.28 million", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 191, + 712 + ], + "score": 1.0, + "content": "training images and", + "type": "text" + }, + { + "bbox": [ + 191, + 699, + 208, + 709 + ], + "score": 0.3, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "validation images with labels from 1000 categories. The details of our", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "experimental setup and the network structures for bL-ResNet-50, 101 and 152 can be found in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 168, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 168, + 733 + ], + "score": 1.0, + "content": "Appendix A.1.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 170 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 171 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 104, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "To control the complexity of bL-Net, we introduce two parameters to specify the complexity of the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "Little-Branch with respect to the Big-Branch. The Big-Branch typically follows the structure of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "the original network, but the Little-Branch needs to be heavily slimmed and shortened to reduce", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 413, + 221 + ], + "score": 1.0, + "content": "computation as it operates on high resolution. Here we use two parameters", + "type": "text" + }, + { + "bbox": [ + 413, + 211, + 421, + 219 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 209, + 439, + 221 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 210, + 447, + 221 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "to control the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 404, + 232 + ], + "score": 1.0, + "content": "width and depth of the Little-Branch, respectively. As shown in Fig. 1 (b),", + "type": "text" + }, + { + "bbox": [ + 405, + 222, + 412, + 230 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "specifies the reduction", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "factor of the number of channels in the convolutional layers of Little-Branch with respect to that of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 216, + 255 + ], + "score": 1.0, + "content": "the original network while", + "type": "text" + }, + { + "bbox": [ + 217, + 243, + 224, + 253 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "is the reduction factor of the number of convolutional layers. Larger", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 146, + 266 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 254, + 154, + 263 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 252, + 172, + 266 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 253, + 180, + 264 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "lead to lower complexity in bL-Net. As demonstrated later, with an appropriate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 146, + 276 + ], + "score": 1.0, + "content": "choice of", + "type": "text" + }, + { + "bbox": [ + 146, + 266, + 154, + 274 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 263, + 172, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 264, + 180, + 275 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 263, + 431, + 276 + ], + "score": 1.0, + "content": "(See Table 1 and Table 3), the cost of the Little-Branch can be", + "type": "text" + }, + { + "bbox": [ + 432, + 264, + 448, + 276 + ], + "score": 0.64, + "content": "1 / 6", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 263, + 474, + 276 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 475, + 264, + 487, + 274 + ], + "score": 0.3, + "content": "b L", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "-Net", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 123, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 275, + 145, + 286 + ], + "score": 0.64, + "content": "1 / 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "of the original network while still providing sufficient complementary information to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 158, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 158, + 299 + ], + "score": 1.0, + "content": "Big-Branch.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 175, + 506, + 299 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 310, + 221, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 310, + 222, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 222, + 323 + ], + "score": 1.0, + "content": "3.2 NETWORK MERGING", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "We consider two options for merging the outputs of the branches. The first option is a linear", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "combination, which joins features from two networks by addition (Newell et al., 2016). The alternative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 481, + 366 + ], + "score": 1.0, + "content": "concatenates the outputs of the two networks along the channel dimension, and if needed, a", + "type": "text" + }, + { + "bbox": [ + 481, + 353, + 505, + 364 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 377 + ], + "score": 1.0, + "content": "convolution can be subsequently applied to reduce the number of feature maps (Szegedy et al., 2015).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "Both merging approaches have their pros and cons. With linear combination, the branches can easily", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 398 + ], + "score": 1.0, + "content": "compensate each other, meaning each branch can activate output neurons not activated by the other.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "score": 1.0, + "content": "However, additional cost is added as both the size of feature maps and the number of channels in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "the two branches need to be adjusted to be the same before addition. On the other hand, merging by", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "concatenation only needs to spatially align the feature maps. On the other hand, concatenation only", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 331, + 443 + ], + "score": 1.0, + "content": "needs to align the feature map size, however requires a", + "type": "text" + }, + { + "bbox": [ + 332, + 430, + 356, + 441 + ], + "score": 0.9, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "convolution reducing the number of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 485, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 485, + 454 + ], + "score": 1.0, + "content": "channels after concatenation, which is a more expensive operation than the pointwise addition.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 331, + 506, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "While linear combination provides an immediate exchange of the activations of both branches,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "concatenation relies on the following layers for this exchange. This delay in exchange could possibly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "be problematic if the information from each branch is destructively altered before merging. For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "score": 1.0, + "content": "example, a nonlinearity such as ReLU would discard all activations less than zero, effectively ignoring", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "negative features in both branches before merging. Since linear combination does not cause too", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "much overhead and provides better accuracy, we chose linear combination as our merging approach.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "In Appendix A.4, we empirically show that the linear combination approach performs better than", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 535, + 251, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 251, + 547 + ], + "score": 1.0, + "content": "concatenation in object recognition.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 457, + 506, + 547 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 256, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 258, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 258, + 576 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "We conducted extensive experiments, as discussed below, to validate the effectiveness of our proposed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "bL-Net on object and speech recognition tasks. bL-Net can be easily integrated with many modern", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "CNNs and here we chose ResNet (He et al., 2016) as the primary architecture to evaluate our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 503, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 493, + 633 + ], + "score": 1.0, + "content": "approach. For simplicity, from now on, we denote by bL-M the bL-Net using a backbone network", + "type": "text" + }, + { + "bbox": [ + 493, + 621, + 503, + 630 + ], + "score": 0.48, + "content": "M", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 386, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 386, + 644 + ], + "score": 1.0, + "content": "For example, bL-ResNet-50 is the Big-Little net based on ResNet-50.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 587, + 506, + 644 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 656, + 229, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 231, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 231, + 669 + ], + "score": 1.0, + "content": "4.1 OBJECT RECOGNITION", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We used the ImageNet dataset (Russakovsky et al., 2015) for all the experiments below on object", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "recognition. 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The details of our", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "experimental setup and the network structures for bL-ResNet-50, 101 and 152 can be found in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 168, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 168, + 733 + ], + "score": 1.0, + "content": "Appendix A.1.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 167, + 101, + 442, + 167 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 89, + 457, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 88, + 459, + 102 + ], + "spans": [ + { + "bbox": [ + 151, + 88, + 345, + 102 + ], + "score": 1.0, + "content": "Table 1: Complexity study of the Little-Branch", + "type": "text" + }, + { + "bbox": [ + 346, + 91, + 354, + 99 + ], + "score": 0.61, + "content": "\\scriptstyle { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 88, + 371, + 102 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 90, + 379, + 101 + ], + "score": 0.73, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 88, + 459, + 102 + ], + "score": 1.0, + "content": ") for bL-ResNet-50.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 167, + 101, + 442, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 101, + 442, + 167 + ], + "spans": [ + { + "bbox": [ + 167, + 101, + 442, + 167 + ], + "score": 0.971, + "html": "
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α = 2, β = 2)22.72%2.91 (1.41×)26.97
bL-ResNet-50(α = 2, β= 4)22.69%2.85 (1.44×)26.69
bL-ResNet-50 (α = 4, β = 2)23.20%2.49 (1.64×)26.31
bL-ResNet-50 (α= 4, β= 4)23.15%2.48 (1.65×)26.24
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ModelTop-1 ErrorFLOPs (109)Params (106)Speed (ms/batch)t
ResNet-10121.95%7.8044.54186
bL-ResNet-101 (α= 2,β= 4)21.80%3.89 (2.01×)41.85140 (1.33×)
bL-ResNet-101@256(α = 2,β= 4)21.04%5.08 (1.54×)41.85162 (1.15×)
ResNet-15221.51%11.5160.19266
bL-ResNet-152(α = 2,β =4)21.16%5.04 (2.28×)57.36178 (1.49×)
bL-ResNet-152@256 (α= 2, β=4)20.34%6.58 (1.75x)57.36205 (1.30×)
ResNeXt-50 (32×4d)22.20%4.2325.03157
bL-ResNeXt-50 (32×4d) (α =2,β= 4)21.60%3.03 (1.40×)26.19125 (1.26×)
bL-ResNeXt-50@256(α = 2, β= 4)20.96%3.95 (1.08×)26.19153 (1.03×)
ResNeXt-101 (32×4d)21.20%7.9744.17269
bL-ResNeXt-101 (32×4d) (α =2,β =4)21.08%4.08 (1.95×)41.51169 (1.59×)
bL-ResNeXt-101@256(α = 2, β=4)20.48%5.33 (1.50x)41.51203 (1.33×)
ResNeXt-101(64×4d)20.73%15.4683.46485
bL-ResNeXt-101 (64×4d) (α = 2,β =4)20.48%7.14 (2.17×)77.36263 (1.98×)
bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)19.65%9.32 (1.66x)77.36318 (1.53×)
SEResNeXt-50 (32×4d)21.78%4.2327.56216
bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)21.44%3.03 (1.40×)28.77163 (1.33×)
20.74%3.95 (1.08×)28.77192 (1.03×)
SEResNeXt-101 (32×4d)21.00%7.9748.96376
bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)20.87%4.08 (1.95×)45.88235 (1.60×)
bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)19.87%5.33 (1.50×)45.88270 (1.39×)
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Except for", + "type": "text" + }, + { + "bbox": [ + 340, + 417, + 358, + 425 + ], + "score": 0.76, + "content": "@ 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 416, + 464, + 426 + ], + "score": 1.0, + "content": ", speed is evaluated under image size", + "type": "text" + }, + { + "bbox": [ + 464, + 417, + 492, + 425 + ], + "score": 0.87, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 416, + 495, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 112, + 424, + 478, + 435 + ], + "spans": [ + { + "bbox": [ + 112, + 424, + 478, + 435 + ], + "score": 1.0, + "content": "We trained all the ResNet, ResNeXt and SEResNeXt models by ourselves, so the accuracy is slightly different from the papers.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 496, + 462 + ], + "score": 1.0, + "content": "ResNet as the backbone network. We experimented with different complexity control factors (", + "type": "text" + }, + { + "bbox": [ + 496, + 452, + 504, + 461 + ], + "score": 0.58, + "content": "\\overset { \\cdot } { \\alpha }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 124, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 462, + 132, + 473 + ], + "score": 0.76, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 461, + 343, + 474 + ], + "score": 1.0, + "content": ") to better understand their effects on performance.", + "type": "text" + }, + { + "bbox": [ + 343, + 464, + 351, + 472 + ], + "score": 0.71, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 461, + 369, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 462, + 378, + 473 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "control both the structural and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 473, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 484 + ], + "score": 1.0, + "content": "computational complexity of the Little-Branch, which determines the overall computational cost of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 482, + 141, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 141, + 496 + ], + "score": 1.0, + "content": "bL-Net.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "As can be seen in Table 1, all the models based on ResNet-50 yield better performance over the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "baseline with less computation, clearly demonstrating the advantage of combining low- and high-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "complexity networks to balance between speed and accuracy. In addition, the small performance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 423, + 547 + ], + "score": 1.0, + "content": "gaps between these models suggest that a computationally light Little-Branch (", + "type": "text" + }, + { + "bbox": [ + 423, + 533, + 453, + 544 + ], + "score": 0.89, + "content": "\\textless 1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "of the entire", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 544, + 507, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 507, + 558 + ], + "score": 1.0, + "content": "network) can compensate well for the low resolution representation by providing finer image details.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 159, + 568 + ], + "score": 1.0, + "content": "We consider", + "type": "text" + }, + { + "bbox": [ + 159, + 556, + 187, + 565 + ], + "score": 0.9, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 555, + 205, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 555, + 233, + 567 + ], + "score": 0.91, + "content": "\\beta = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "as the default setting for the following experiments. Furthermore,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 566, + 415, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 415, + 578 + ], + "score": 1.0, + "content": "there are more ablation studies on the design of bL-Net in the Appendix A.4.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "We further evaluated our approach on deeper models by using ResNet-101 and ResNet-152 as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "backbone networks. We see from Table 2 that bL-ResNet-101 and bL-ResNet-152 behave similarly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "to bL-ResNet-50. As expected, both of them produce better results against the baseline models and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "achieving notable computational gains. Interestingly, our approach computationally favors deeper", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 444, + 639 + ], + "score": 1.0, + "content": "models, as evidenced by the fact that more speedups are observed on bL-ResNet-152", + "type": "text" + }, + { + "bbox": [ + 445, + 627, + 472, + 638 + ], + "score": 0.84, + "content": "( 2 . 3 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "than on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 170, + 652 + ], + "score": 1.0, + "content": "bL-ResNet-101", + "type": "text" + }, + { + "bbox": [ + 170, + 638, + 197, + 649 + ], + "score": 0.82, + "content": "( 2 . 0 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 637, + 273, + 652 + ], + "score": 1.0, + "content": "and bL-ResNet-50", + "type": "text" + }, + { + "bbox": [ + 274, + 638, + 301, + 649 + ], + "score": 0.84, + "content": "( 1 . 4 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 637, + 505, + 652 + ], + "score": 1.0, + "content": ". This is mainly because the Little Branch operating", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 649, + 354, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 354, + 661 + ], + "score": 1.0, + "content": "on low resolution spends less computation in a deeper model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 507, + 678 + ], + "score": 1.0, + "content": "ResNeXt and SEResNeXt as the backbone network. We extended bL-Net to ResNeXt and SERes-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "NeXt, two of the more accurate yet compact network architectures. We also experimented with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 311, + 699 + ], + "score": 1.0, + "content": "(SE)ResNeXt-50 and (SE)ResNeXt-101 using the", + "type": "text" + }, + { + "bbox": [ + 311, + 688, + 341, + 698 + ], + "score": 0.9, + "content": "3 2 \\times 4 \\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 688, + 359, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 688, + 389, + 698 + ], + "score": 0.89, + "content": "6 4 \\times 4 \\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "setting (Xie et al., 2017; Hu", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "et al., 2018). In our case, the Big-Branch follows the same setting of (SE)ResNeXt; however, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "changed the cardinality of the Little-Branch to align with the input channels of a group convolution", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 330, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 330, + 732 + ], + "score": 1.0, + "content": "in the Big-Branch. All the results are shown in Table 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 167, + 101, + 442, + 167 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 89, + 457, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 88, + 459, + 102 + ], + "spans": [ + { + "bbox": [ + 151, + 88, + 345, + 102 + ], + "score": 1.0, + "content": "Table 1: Complexity study of the Little-Branch", + "type": "text" + }, + { + "bbox": [ + 346, + 91, + 354, + 99 + ], + "score": 0.61, + "content": "\\scriptstyle { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 88, + 371, + 102 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 90, + 379, + 101 + ], + "score": 0.73, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 88, + 459, + 102 + ], + "score": 1.0, + "content": ") for bL-ResNet-50.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 167, + 101, + 442, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 101, + 442, + 167 + ], + "spans": [ + { + "bbox": [ + 167, + 101, + 442, + 167 + ], + "score": 0.971, + "html": "
ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α = 2, β = 2)22.72%2.91 (1.41×)26.97
bL-ResNet-50(α = 2, β= 4)22.69%2.85 (1.44×)26.69
bL-ResNet-50 (α = 4, β = 2)23.20%2.49 (1.64×)26.31
bL-ResNet-50 (α= 4, β= 4)23.15%2.48 (1.65×)26.24
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ModelTop-1 ErrorFLOPs (109)Params (106)Speed (ms/batch)t
ResNet-10121.95%7.8044.54186
bL-ResNet-101 (α= 2,β= 4)21.80%3.89 (2.01×)41.85140 (1.33×)
bL-ResNet-101@256(α = 2,β= 4)21.04%5.08 (1.54×)41.85162 (1.15×)
ResNet-15221.51%11.5160.19266
bL-ResNet-152(α = 2,β =4)21.16%5.04 (2.28×)57.36178 (1.49×)
bL-ResNet-152@256 (α= 2, β=4)20.34%6.58 (1.75x)57.36205 (1.30×)
ResNeXt-50 (32×4d)22.20%4.2325.03157
bL-ResNeXt-50 (32×4d) (α =2,β= 4)21.60%3.03 (1.40×)26.19125 (1.26×)
bL-ResNeXt-50@256(α = 2, β= 4)20.96%3.95 (1.08×)26.19153 (1.03×)
ResNeXt-101 (32×4d)21.20%7.9744.17269
bL-ResNeXt-101 (32×4d) (α =2,β =4)21.08%4.08 (1.95×)41.51169 (1.59×)
bL-ResNeXt-101@256(α = 2, β=4)20.48%5.33 (1.50x)41.51203 (1.33×)
ResNeXt-101(64×4d)20.73%15.4683.46485
bL-ResNeXt-101 (64×4d) (α = 2,β =4)20.48%7.14 (2.17×)77.36263 (1.98×)
bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)19.65%9.32 (1.66x)77.36318 (1.53×)
SEResNeXt-50 (32×4d)21.78%4.2327.56216
bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)21.44%3.03 (1.40×)28.77163 (1.33×)
20.74%3.95 (1.08×)28.77192 (1.03×)
SEResNeXt-101 (32×4d)21.00%7.9748.96376
bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)20.87%4.08 (1.95×)45.88235 (1.60×)
bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)19.87%5.33 (1.50×)45.88270 (1.39×)
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Except for", + "type": "text" + }, + { + "bbox": [ + 340, + 417, + 358, + 425 + ], + "score": 0.76, + "content": "@ 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 416, + 464, + 426 + ], + "score": 1.0, + "content": ", speed is evaluated under image size", + "type": "text" + }, + { + "bbox": [ + 464, + 417, + 492, + 425 + ], + "score": 0.87, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 416, + 495, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 112, + 424, + 478, + 435 + ], + "spans": [ + { + "bbox": [ + 112, + 424, + 478, + 435 + ], + "score": 1.0, + "content": "We trained all the ResNet, ResNeXt and SEResNeXt models by ourselves, so the accuracy is slightly different from the papers.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 496, + 462 + ], + "score": 1.0, + "content": "ResNet as the backbone network. We experimented with different complexity control factors (", + "type": "text" + }, + { + "bbox": [ + 496, + 452, + 504, + 461 + ], + "score": 0.58, + "content": "\\overset { \\cdot } { \\alpha }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 124, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 462, + 132, + 473 + ], + "score": 0.76, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 461, + 343, + 474 + ], + "score": 1.0, + "content": ") to better understand their effects on performance.", + "type": "text" + }, + { + "bbox": [ + 343, + 464, + 351, + 472 + ], + "score": 0.71, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 461, + 369, + 474 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 462, + 378, + 473 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "control both the structural and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 473, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 484 + ], + "score": 1.0, + "content": "computational complexity of the Little-Branch, which determines the overall computational cost of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 482, + 141, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 141, + 496 + ], + "score": 1.0, + "content": "bL-Net.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 450, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "As can be seen in Table 1, all the models based on ResNet-50 yield better performance over the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "baseline with less computation, clearly demonstrating the advantage of combining low- and high-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "complexity networks to balance between speed and accuracy. In addition, the small performance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 423, + 547 + ], + "score": 1.0, + "content": "gaps between these models suggest that a computationally light Little-Branch (", + "type": "text" + }, + { + "bbox": [ + 423, + 533, + 453, + 544 + ], + "score": 0.89, + "content": "\\textless 1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "of the entire", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 544, + 507, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 507, + 558 + ], + "score": 1.0, + "content": "network) can compensate well for the low resolution representation by providing finer image details.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 159, + 568 + ], + "score": 1.0, + "content": "We consider", + "type": "text" + }, + { + "bbox": [ + 159, + 556, + 187, + 565 + ], + "score": 0.9, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 555, + 205, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 555, + 233, + 567 + ], + "score": 0.91, + "content": "\\beta = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "as the default setting for the following experiments. Furthermore,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 566, + 415, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 415, + 578 + ], + "score": 1.0, + "content": "there are more ablation studies on the design of bL-Net in the Appendix A.4.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 501, + 507, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "We further evaluated our approach on deeper models by using ResNet-101 and ResNet-152 as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "backbone networks. We see from Table 2 that bL-ResNet-101 and bL-ResNet-152 behave similarly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "to bL-ResNet-50. As expected, both of them produce better results against the baseline models and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "achieving notable computational gains. Interestingly, our approach computationally favors deeper", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 444, + 639 + ], + "score": 1.0, + "content": "models, as evidenced by the fact that more speedups are observed on bL-ResNet-152", + "type": "text" + }, + { + "bbox": [ + 445, + 627, + 472, + 638 + ], + "score": 0.84, + "content": "( 2 . 3 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "than on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 170, + 652 + ], + "score": 1.0, + "content": "bL-ResNet-101", + "type": "text" + }, + { + "bbox": [ + 170, + 638, + 197, + 649 + ], + "score": 0.82, + "content": "( 2 . 0 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 637, + 273, + 652 + ], + "score": 1.0, + "content": "and bL-ResNet-50", + "type": "text" + }, + { + "bbox": [ + 274, + 638, + 301, + 649 + ], + "score": 0.84, + "content": "( 1 . 4 \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 637, + 505, + 652 + ], + "score": 1.0, + "content": ". This is mainly because the Little Branch operating", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 649, + 354, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 354, + 661 + ], + "score": 1.0, + "content": "on low resolution spends less computation in a deeper model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 582, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 507, + 678 + ], + "score": 1.0, + "content": "ResNeXt and SEResNeXt as the backbone network. We extended bL-Net to ResNeXt and SERes-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "NeXt, two of the more accurate yet compact network architectures. We also experimented with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 311, + 699 + ], + "score": 1.0, + "content": "(SE)ResNeXt-50 and (SE)ResNeXt-101 using the", + "type": "text" + }, + { + "bbox": [ + 311, + 688, + 341, + 698 + ], + "score": 0.9, + "content": "3 2 \\times 4 \\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 688, + 359, + 699 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 688, + 389, + 698 + ], + "score": 0.89, + "content": "6 4 \\times 4 \\mathrm { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "setting (Xie et al., 2017; Hu", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "et al., 2018). In our case, the Big-Branch follows the same setting of (SE)ResNeXt; however, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "changed the cardinality of the Little-Branch to align with the input channels of a group convolution", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 720, + 330, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 330, + 732 + ], + "score": 1.0, + "content": "in the Big-Branch. 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However, bL-ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 264, + 264, + 298, + 275 + ], + "score": 0.87, + "content": "( 3 2 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 263, + 472, + 276 + ], + "score": 1.0, + "content": "gains a much more substantial speedup of", + "type": "text" + }, + { + "bbox": [ + 472, + 264, + 487, + 274 + ], + "score": 0.87, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 134, + 287 + ], + "score": 1.0, + "content": "seeing", + "type": "text" + }, + { + "bbox": [ + 135, + 275, + 162, + 285 + ], + "score": 0.87, + "content": "0 . 1 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 274, + 468, + 287 + ], + "score": 1.0, + "content": "improvement in accuracy. The same trend can be seen on bL-ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 468, + 275, + 502, + 285 + ], + "score": 0.86, + "content": "( 6 4 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 274, + 506, + 287 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "On the other hand, our bL-SEResNeXts also produce better performance while reducing more FLOPs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 495, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 303, + 308 + ], + "score": 1.0, + "content": "than SEResNeXts. bL-SEResNeXt-101 achieves", + "type": "text" + }, + { + "bbox": [ + 303, + 297, + 318, + 307 + ], + "score": 0.87, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 296, + 464, + 308 + ], + "score": 1.0, + "content": "speedups and improves accuracy by", + "type": "text" + }, + { + "bbox": [ + 464, + 296, + 491, + 307 + ], + "score": 0.86, + "content": "0 . 1 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 296, + 495, + 308 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "Since our bL-Net saves more budget in computation, we can evaluate a model at a larger image", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 248, + 337 + ], + "score": 1.0, + "content": "scale for better performance, e.g.,", + "type": "text" + }, + { + "bbox": [ + 249, + 324, + 293, + 335 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 324, + 506, + 337 + ], + "score": 1.0, + "content": ". As illustrated in Table 2, our models evaluated at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 150, + 345 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "is consistently better than their corresponding baselines while still using fewer FLOPs.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 442, + 359 + ], + "score": 1.0, + "content": "The advantage becomes more pronounced with deeper models. For instance, with", + "type": "text" + }, + { + "bbox": [ + 443, + 346, + 463, + 356 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "reduction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 357, + 504, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 223, + 369 + ], + "score": 1.0, + "content": "on FLOPs, bL-ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 224, + 357, + 249, + 367 + ], + "score": 0.57, + "content": "\\textcircled { \\scriptsize { a } } 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 357, + 285, + 367 + ], + "score": 0.84, + "content": "( 6 4 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 357, + 420, + 369 + ], + "score": 1.0, + "content": "boosts the top-1 performance by", + "type": "text" + }, + { + "bbox": [ + 420, + 357, + 442, + 367 + ], + "score": 0.87, + "content": "1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 357, + 504, + 369 + ], + "score": 1.0, + "content": ", which is quite", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 253, + 380 + ], + "score": 1.0, + "content": "impressive given that ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 253, + 369, + 287, + 379 + ], + "score": 0.85, + "content": "( 6 4 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 368, + 506, + 380 + ], + "score": 1.0, + "content": "is a very competitive baseline. Table 2 also shows the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "running times of bL-Net on GPU, which indicate the practical speedups of these models are consistent", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 340, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 340, + 402 + ], + "score": 1.0, + "content": "with the theoretical FLOP reductions reported in Table 2.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 418, + 295, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 298, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 298, + 430 + ], + "score": 1.0, + "content": "4.1.1 COMPARISON WITH RELATED WORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 453 + ], + "score": 1.0, + "content": "We first compared our method with the approaches that aim to accelerate ResNets or ResNeXts using", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 502, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 502, + 463 + ], + "score": 1.0, + "content": "techniques such as network pruning and adaptive computations. The results are shown in Figure 2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 507, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 507, + 480 + ], + "score": 1.0, + "content": "Our bL-Net significantly outperforms all related works regarding FLOPs reduction and accuracy.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 200, + 492 + ], + "score": 1.0, + "content": "Our bL-ResNet-101 is", + "type": "text" + }, + { + "bbox": [ + 200, + 478, + 226, + 489 + ], + "score": 0.89, + "content": "\\sim 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 477, + 507, + 492 + ], + "score": 1.0, + "content": "better than the network pruning approaches such as PFEC (Li et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "2017) and LCCL (Dong et al., 2017), but still using less computation. When compared to SACT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 425, + 513 + ], + "score": 1.0, + "content": "and ACT (Figurnov et al., 2017), our bL-ResNet-101 improves the accuracy by", + "type": "text" + }, + { + "bbox": [ + 425, + 501, + 441, + 511 + ], + "score": 0.84, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "while using the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "same number of FLOPs. On the other hand, our bL-ResNet-101 outperforms some of the most recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "works including BlockDrop (Wu et al., 2018), SkipNet (Wang et al., 2018) and ConvNet-AIG (Veit", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 200, + 546 + ], + "score": 1.0, + "content": "& Belongie, 2018) by", + "type": "text" + }, + { + "bbox": [ + 200, + 533, + 222, + 544 + ], + "score": 0.84, + "content": "3 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 533, + 227, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 227, + 533, + 250, + 544 + ], + "score": 0.83, + "content": "2 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 533, + 254, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 533, + 277, + 544 + ], + "score": 0.84, + "content": "1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "top-1 accuracy at the same FLOPs, respectively. This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "clearly demonstrates the advantages of a simple fusion of two branches at different scales over the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "more sophisticated dynamic routing techniques developed in these approaches. 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These models introduces attention", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "mechanisms to enhance feature representations in either the channel or the spatial domain. From", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "Figure 2, it can be seen that our bL-Net outperforms all of them in both FLOPs and accuracy. Our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 380, + 650 + ], + "score": 1.0, + "content": "bL-Net achieves better performance than ResAttNeXt while saving", + "type": "text" + }, + { + "bbox": [ + 380, + 638, + 432, + 649 + ], + "score": 0.53, + "content": "1 . 5 \\times \\mathrm { F L O P s }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 637, + 506, + 650 + ], + "score": 1.0, + "content": ". It also surpasses", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 129, + 659 + ], + "score": 0.87, + "content": "{ \\sim } 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 648, + 362, + 661 + ], + "score": 1.0, + "content": "point with similar FLOPs or the similar accuracy with", + "type": "text" + }, + { + "bbox": [ + 363, + 649, + 393, + 659 + ], + "score": 0.88, + "content": "{ \\sim } 1 . 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "FLOPs reduction for both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "SEResNeXt and CBAM. 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As illustrated in Table 2, our models evaluated at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 150, + 345 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "is consistently better than their corresponding baselines while still using fewer FLOPs.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 442, + 359 + ], + "score": 1.0, + "content": "The advantage becomes more pronounced with deeper models. For instance, with", + "type": "text" + }, + { + "bbox": [ + 443, + 346, + 463, + 356 + ], + "score": 0.87, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "reduction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 357, + 504, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 223, + 369 + ], + "score": 1.0, + "content": "on FLOPs, bL-ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 224, + 357, + 249, + 367 + ], + "score": 0.57, + "content": "\\textcircled { \\scriptsize { a } } 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 357, + 285, + 367 + ], + "score": 0.84, + "content": "( 6 4 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 357, + 420, + 369 + ], + "score": 1.0, + "content": "boosts the top-1 performance by", + "type": "text" + }, + { + "bbox": [ + 420, + 357, + 442, + 367 + ], + "score": 0.87, + "content": "1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 357, + 504, + 369 + ], + "score": 1.0, + "content": ", which is quite", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 253, + 380 + ], + "score": 1.0, + "content": "impressive given that ResNeXt-101", + "type": "text" + }, + { + "bbox": [ + 253, + 369, + 287, + 379 + ], + "score": 0.85, + "content": "( 6 4 \\times 4 \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 368, + 506, + 380 + ], + "score": 1.0, + "content": "is a very competitive baseline. Table 2 also shows the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "running times of bL-Net on GPU, which indicate the practical speedups of these models are consistent", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 340, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 340, + 402 + ], + "score": 1.0, + "content": "with the theoretical FLOP reductions reported in Table 2.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 312, + 506, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 418, + 295, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 298, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 298, + 430 + ], + "score": 1.0, + "content": "4.1.1 COMPARISON WITH RELATED WORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 453 + ], + "score": 1.0, + "content": "We first compared our method with the approaches that aim to accelerate ResNets or ResNeXts using", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 502, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 502, + 463 + ], + "score": 1.0, + "content": "techniques such as network pruning and adaptive computations. The results are shown in Figure 2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 437, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 507, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 507, + 480 + ], + "score": 1.0, + "content": "Our bL-Net significantly outperforms all related works regarding FLOPs reduction and accuracy.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 200, + 492 + ], + "score": 1.0, + "content": "Our bL-ResNet-101 is", + "type": "text" + }, + { + "bbox": [ + 200, + 478, + 226, + 489 + ], + "score": 0.89, + "content": "\\sim 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 477, + 507, + 492 + ], + "score": 1.0, + "content": "better than the network pruning approaches such as PFEC (Li et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "2017) and LCCL (Dong et al., 2017), but still using less computation. When compared to SACT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 425, + 513 + ], + "score": 1.0, + "content": "and ACT (Figurnov et al., 2017), our bL-ResNet-101 improves the accuracy by", + "type": "text" + }, + { + "bbox": [ + 425, + 501, + 441, + 511 + ], + "score": 0.84, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "while using the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "same number of FLOPs. On the other hand, our bL-ResNet-101 outperforms some of the most recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "works including BlockDrop (Wu et al., 2018), SkipNet (Wang et al., 2018) and ConvNet-AIG (Veit", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 200, + 546 + ], + "score": 1.0, + "content": "& Belongie, 2018) by", + "type": "text" + }, + { + "bbox": [ + 200, + 533, + 222, + 544 + ], + "score": 0.84, + "content": "3 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 533, + 227, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 227, + 533, + 250, + 544 + ], + "score": 0.83, + "content": "2 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 533, + 254, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 533, + 277, + 544 + ], + "score": 0.84, + "content": "1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "top-1 accuracy at the same FLOPs, respectively. This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "clearly demonstrates the advantages of a simple fusion of two branches at different scales over the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "more sophisticated dynamic routing techniques developed in these approaches. In comparison to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 470, + 578 + ], + "score": 1.0, + "content": "SPPoint (Kuen et al., 2018) under the same FLOPs, our bL-ResNet-101 surpasses it by", + "type": "text" + }, + { + "bbox": [ + 470, + 566, + 493, + 577 + ], + "score": 0.85, + "content": "2 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 146, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 146, + 590 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 467, + 507, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "We also compared bL-Net with the variants of ResNets, like ResAttNe(X)t (Wang et al., 2017),", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "SEResNeXt (Hu et al., 2018) and CBAM (Woo et al., 2018). These models introduces attention", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "mechanisms to enhance feature representations in either the channel or the spatial domain. From", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "Figure 2, it can be seen that our bL-Net outperforms all of them in both FLOPs and accuracy. Our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 380, + 650 + ], + "score": 1.0, + "content": "bL-Net achieves better performance than ResAttNeXt while saving", + "type": "text" + }, + { + "bbox": [ + 380, + 638, + 432, + 649 + ], + "score": 0.53, + "content": "1 . 5 \\times \\mathrm { F L O P s }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 637, + 506, + 650 + ], + "score": 1.0, + "content": ". It also surpasses", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 129, + 659 + ], + "score": 0.87, + "content": "{ \\sim } 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 648, + 362, + 661 + ], + "score": 1.0, + "content": "point with similar FLOPs or the similar accuracy with", + "type": "text" + }, + { + "bbox": [ + 363, + 649, + 393, + 659 + ], + "score": 0.88, + "content": "{ \\sim } 1 . 7 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "FLOPs reduction for both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "SEResNeXt and CBAM. It’s worth noting that bL-Net can be potentially integrated with these models", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 300, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 300, + 684 + ], + "score": 1.0, + "content": "to further improve their accuracy and efficiency.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 594, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 296, + 699 + ], + "score": 1.0, + "content": "Finally, we ran a benchmark test on bL-ResNeXt", + "type": "text" + }, + { + "bbox": [ + 296, + 688, + 320, + 698 + ], + "score": 0.3, + "content": "@ 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "under the PyTorch framework with a batch size", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "of 16 on a K80 GPU, and compared against various models that are publicly available. These models", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "include Inception-V3 (Szegedy et al., 2016a), Inception-V4 (Szegedy et al., 2017), Inception-ResNet-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "V2 (Szegedy et al., 2017), PolyNet (Zhang et al., 2017), NASNet (Zoph et al., 2018), PNASNet (Liu", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2018), DualPathNet (Chen et al., 2017b) and DenseNet (Huang et al., 2017b). 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(a) FLOPs. (b) GPU Speed.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 105, + 261, + 504, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2018), DualPathNet (Chen et al., 2017b) and DenseNet (Huang et al., 2017b). Among them,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 271, + 345, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 345, + 285 + ], + "score": 1.0, + "content": "NASNet currently achieves the best accuracy on ImageNet.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "From Fig. 3, we can see that overall, our bL-ResNeXt gains a better tradeoff between efficiency and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "accuracy. 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Gains from this setting are typically maintained in the standard", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 566, + 489, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 489, + 579 + ], + "score": 1.0, + "content": "further pipelines like fine-tuning with sequence training, using more complex language models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 479, + 506, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Appendix B gives a thorough overview of the architecture of the speech acoustic models. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "main difference speech acoustic models have compared to image classification networks, is that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "striding or pooling only happens along the frequency axis, while along in the time direction we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "need to output dense predictions per frame (Sercu & Goel, 2016). This means that the branches at", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "different resolutions have a fundamentally different view of the signal as it is propagating through", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "the network; the ratio of resolution in frequency (downsampled in the Big-Branch) vs resolution", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "in time (same between branches) is different. We can think about this as the convolutional kernels", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "having different “aspect ratios” between branches. Therefore we not only expect FLOP reductions in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "bL-Net, but expect to have increased representational power. In addition, similar to the case in object", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "recognition (Table 2), we could process the speech signal at higher frequency resolution than what is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 693, + 309, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 309, + 705 + ], + "score": 1.0, + "content": "computationally feasible for the baseline ResNets.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 583, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Table 3 shows the results for the different architectures described in Appendix B. Most results are in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "line with the observations in the object recognition bL-Net. When comparing the baseline ResNet-22", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "(line 1) to the best bL-ResNet-22 (line 5), we see not only a reduction in FLOPs, but also a modest", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 424, + 248 + ], + "score": 1.0, + "content": "gain in Word Error Rate (WER). Comparing lines 2-4, we see that increasing", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 424, + 236, + 432, + 246 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 433, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "(i.e. shorter little", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "branches at full resolution) causes no WER degradation, while reducing the number of FLOPs. 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Line 7", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "(compare to line 2) shows an experiment with additional branches on the lower layers (See Appendix", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "B). 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ModelFLOPs (109)Params (106)WER AvgHub5Hub5 CH
1Baseline: ResNet-221.113.0214.67%11.15%18.17%
2bL-ResNet-22 (α = 4,β = 1)0.68 (1.63×)3.1514.72%11.24%18.18%
3bL-ResNet-22 (α = 4,β = 2)0.66 (1.68×)3.1114.47%10.95%17.95%
4bL-ResNet-22(α = 4,β= 3)0.65 (1.70×)3.1014.66%11.25%18.05%
5bL-ResNet-22 (α = 2,β = 3)0.77 (1.43×)3.0714.46%11.10%17.80%
6bL-ResNet-22 (α = 4,β=1) cat0.70 (1.58×)3.1814.67%11.31%18.00%
7bL-PYR-ResNet-22 (α = 4, β =1)0.98 (1.13×)3.3214.50%11.05%17.92%
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From line 2 vs. line 6, the concatenation merge mode", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "performs similar to the default additive merging, while increasing the number of FLOPs. Line 7", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "(compare to line 2) shows an experiment with additional branches on the lower layers (See Appendix", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "B). Although there is some gain in WER, the added parameters and compute on the lower layers may", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 345, + 257, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 257, + 356 + ], + "score": 1.0, + "content": "not make this a worthwhile trade-off.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 231, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 232, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 232, + 388 + ], + "score": 1.0, + "content": "4.3 DISCUSSION ON bL-Net", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "From the results of both tasks, we observe the following common insights, which enable us to design", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "an efficient multi-scale network with competitive performance: (I) The Little-Branch can be very", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 479, + 431 + ], + "score": 1.0, + "content": "light-weight, (II) bL-Net performs better when the Little-Branch is wide and shallow (smaller", + "type": "text" + }, + { + "bbox": [ + 479, + 421, + 487, + 429 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 429, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 131, + 444 + ], + "score": 1.0, + "content": "larger", + "type": "text" + }, + { + "bbox": [ + 131, + 430, + 138, + 441 + ], + "score": 0.77, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 429, + 506, + 444 + ], + "score": 1.0, + "content": "), (III) merging is effective when the feature dimension has changed, and (IV) branch merging", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 452 + ], + "score": 1.0, + "content": "by addition is more effective than concatenation. (I) is because the Big-Branch can extract essential", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "information, a light Little-Branch is good enough to provide sufficient information the Big-Branch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "lacks. Regarding (II), wider networks have been shown to perform better than deep networks while", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 471, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 506, + 489 + ], + "score": 1.0, + "content": "using a similar number of parameters. (III) is well-discussed in Appendix A.4. 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ModelFLOPs (109)Params (106)WER AvgHub5Hub5 CH
1Baseline: ResNet-221.113.0214.67%11.15%18.17%
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7bL-PYR-ResNet-22 (α = 4, β =1)0.98 (1.13×)3.3214.50%11.05%17.92%
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In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 374, + 505, + 396 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 179, + 97 + ], + "lines": [ + { + "bbox": [ + 103, + 77, + 182, + 100 + ], + "spans": [ + { + "bbox": [ + 103, + 77, + 182, + 100 + ], + "score": 1.0, + "content": "Appendix", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 504, + 124 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 115 + ], + "score": 1.0, + "content": "The appendix illustrates the details of our experiments on object recognition and speech recognition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 113, + 206, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 206, + 124 + ], + "score": 1.0, + "content": "and more ablation study.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 140, + 306, + 153 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 307, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 307, + 155 + ], + "score": 1.0, + "content": "A bL-Net FOR OBJECT RECOGNITION", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 108, + 165, + 232, + 176 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 233, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 233, + 177 + ], + "score": 1.0, + "content": "A.1 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 506, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "We used the ImageNet dataset for all experiments. We trained all the models by Tensorpack (Wu,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "2017), a higher-level wrapper for Tensorflow (Abadi et al., 2015). 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We deployed the popular augmentation technique in (Szegedy et al., 2016b; Gross & Wilber,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 240, + 507, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 367, + 254 + ], + "score": 1.0, + "content": "2016) to increase the variety of training data, and randomly crop a", + "type": "text" + }, + { + "bbox": [ + 367, + 241, + 410, + 252 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 240, + 507, + 254 + ], + "score": 1.0, + "content": "patch as training image.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 265 + ], + "score": 1.0, + "content": "The validation error is evaluated by resizing the shorter side of an image to 256 and then crop a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 150, + 273 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "from the center. 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LayersbL-ResNet-50ResNet-50
Convolution7 × 7,64, s2 (112 × 112)
bL-module3×3,323 × 3,64,s2 3×3,32,s2(56 × 56)(1×1,64MaxPooling (56 × 56)
bL-moduleResBlockg,256 ×2 ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)ResBlock,256 ×3, s1(56 × 56)
bL-moduleResBlockp,512 ×3 ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)ResBlock,512 ×4, s2(28× 28)
bL-moduleResBlockb,1024 ×5 ResBlockL,512 ×1ResBlock,1024 (14 × 14)ResBlock,1024 ×6, s2(14 × 14)
ResBlockResBlock,2048 ×3, s2 (7 × 7)
Average pool7 ×7 average pooling
FC, softmax1000
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LayersbL-ResNet-50ResNet-50
Convolution7 × 7,64, s2 (112 × 112)
bL-module3×3,323 × 3,64,s2 3×3,32,s2(56 × 56)(1×1,64MaxPooling (56 × 56)
bL-moduleResBlockg,256 ×2 ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)ResBlock,256 ×3, s1(56 × 56)
bL-moduleResBlockp,512 ×3 ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)ResBlock,512 ×4, s2(28× 28)
bL-moduleResBlockb,1024 ×5 ResBlockL,512 ×1ResBlock,1024 (14 × 14)ResBlock,1024 ×6, s2(14 × 14)
ResBlockResBlock,2048 ×3, s2 (7 × 7)
Average pool7 ×7 average pooling
FC, softmax1000
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LayersOutput SizebL-ResNet-101 bL-ResNet-152
Convolution112 × 1127 × 7,64,s2
bL-module56× 56（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64
bL-module56×56/1×1,64 3×3,64 ×3(2) (1×1,256) B/1×1,32 3×3,32 （1×1,128） L×1/1×1,64 3×3,64 × 4(2) （1×1,256) B/1×1,32 3×3,32 （1×1,128)×1 L
transition layer28×28/1×1,64 3×3,64,s2 ×1 （1×1,256
bL-module28×28(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B×1 （1×1,256） L(1×1,128) 3×3,128 (1×1,512)× 11(7)/1×1,64 3×3,64 ×2 (1×1,256) L
transition layer14 × 14/1×1,128 3×3,128,s2 ×1 (1×1,512B
bL-module14 × 14/1×1,256 3×3,256 × 17(22) （1×1,1024）B(1×1,128) 3×3,128 （1×1,512 L×3/1×1,256 3×3,256 (1×1,1024) B× 29(35)(1×1,128) 3×3,128 （1×1,512）×6 L
transition layer14 × 14/1×1,256 3×3,256 ×1 （1×1,1024）
ResBlock7×7/1×1,512 3 ×3,512,s2 ×3
Average pool1×1（1×1,2048 7×7 average pooling
FC, softmax1000
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As shown in Table 6, ResNet-50-lowres reduces lots of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 395, + 617 + ], + "score": 1.0, + "content": "computations but its accuracy is not acceptable; however, bL-ResNet-50", + "type": "text" + }, + { + "bbox": [ + 395, + 605, + 421, + 615 + ], + "score": 0.87, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 605, + 439, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 605, + 465, + 616 + ], + "score": 0.91, + "content": "\\beta = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ") achieves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "a better balance between accuracy and performance. 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While such performance is unsatisfying compared to the state of the art,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 316, + 650 + ], + "score": 1.0, + "content": "it is quite reasonable and expected given that almost", + "type": "text" + }, + { + "bbox": [ + 316, + 638, + 349, + 649 + ], + "score": 0.89, + "content": "3 \\sim 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "reduction of computation are achieved", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 648, + 167, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 167, + 661 + ], + "score": 1.0, + "content": "in such a case.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Figure 4 shows the prediction results from bL-ResNet-50 and ResNet-50-lowres. When both models", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "predict correctly (4 (a) and (b)), the bL-ResNet-50 provides better confidence for the prediction.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Because the object only occupies a small portion of an image, the Little-Branch can still capture the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "object clearly. On the other hand, when the key features of an object is small, like the shape of beak", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of a bird (c) and the spots of a ladybug (d), bL-ResNet-50 can easily retain that key feature to predict", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 721, + 373, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 373, + 733 + ], + "score": 1.0, + "content": "correctly while ResNet-50-lowres provides wrong predicted label.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "score": 1.0, + "content": "For bL-ResNet-101 and bL-ResNet-152, we redistributed the residual blocks to different stages to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 506, + 133 + ], + "score": 1.0, + "content": "balance the residual blocks at each stage. A ResNet model has 5 stages, and each stage has the same", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 131, + 504, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 489, + 145 + ], + "score": 1.0, + "content": "spatial size. These two models accumulate most of the convolutions (or computations) on the", + "type": "text" + }, + { + "bbox": [ + 489, + 131, + 504, + 142 + ], + "score": 0.86, + "content": "4 ^ { t h }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 504, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 265, + 155 + ], + "score": 1.0, + "content": "stage, where the size of feature maps is", + "type": "text" + }, + { + "bbox": [ + 266, + 144, + 299, + 154 + ], + "score": 0.91, + "content": "1 4 \\times 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 144, + 374, + 155 + ], + "score": 1.0, + "content": "when input size is", + "type": "text" + }, + { + "bbox": [ + 374, + 144, + 417, + 154 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 144, + 504, + 155 + ], + "score": 1.0, + "content": ". While such a design", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "may be suitable for a very deep model, it likely limits the ability of Big-Branch to learn information", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 466, + 179 + ], + "score": 1.0, + "content": "at large scales, which mostly resides at earlier stages. Thus, we move some blocks in the", + "type": "text" + }, + { + "bbox": [ + 466, + 164, + 480, + 176 + ], + "score": 0.88, + "content": "4 ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "stage", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 218, + 188 + ], + "score": 1.0, + "content": "of these two models to the", + "type": "text" + }, + { + "bbox": [ + 219, + 176, + 235, + 186 + ], + "score": 0.88, + "content": "{ \\dot { \\mathbf { \\zeta } } } _ { 2 ^ { n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 175, + 254, + 188 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 176, + 269, + 186 + ], + "score": 0.88, + "content": "3 ^ { r d }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "stages. Table 5 shows the details of bL-ResNet-101 and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 169, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 169, + 198 + ], + "score": 1.0, + "content": "bLResNet-152.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 109, + 506, + 198 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 236, + 504, + 487 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 163, + 225, + 447, + 236 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 162, + 224, + 445, + 237 + ], + "spans": [ + { + "bbox": [ + 162, + 224, + 375, + 237 + ], + "score": 1.0, + "content": "Table 5: Network configurations of bL-ResNets, and", + "type": "text" + }, + { + "bbox": [ + 375, + 226, + 401, + 235 + ], + "score": 0.89, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 224, + 419, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 419, + 225, + 445, + 237 + ], + "score": 0.89, + "content": "\\beta = 4", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 107, + 236, + 504, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 236, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 107, + 236, + 504, + 487 + ], + "score": 0.985, + "html": "
LayersOutput SizebL-ResNet-101 bL-ResNet-152
Convolution112 × 1127 × 7,64,s2
bL-module56× 56（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64
bL-module56×56/1×1,64 3×3,64 ×3(2) (1×1,256) B/1×1,32 3×3,32 （1×1,128） L×1/1×1,64 3×3,64 × 4(2) （1×1,256) B/1×1,32 3×3,32 （1×1,128)×1 L
transition layer28×28/1×1,64 3×3,64,s2 ×1 （1×1,256
bL-module28×28(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B×1 （1×1,256） L(1×1,128) 3×3,128 (1×1,512)× 11(7)/1×1,64 3×3,64 ×2 (1×1,256) L
transition layer14 × 14/1×1,128 3×3,128,s2 ×1 (1×1,512B
bL-module14 × 14/1×1,256 3×3,256 × 17(22) （1×1,1024）B(1×1,128) 3×3,128 （1×1,512 L×3/1×1,256 3×3,256 (1×1,1024) B× 29(35)(1×1,128) 3×3,128 （1×1,512）×6 L
transition layer14 × 14/1×1,256 3×3,256 ×1 （1×1,1024）
ResBlock7×7/1×1,512 3 ×3,512,s2 ×3
Average pool1×1（1×1,2048 7×7 average pooling
FC, softmax1000
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As shown in Table 6, ResNet-50-lowres reduces lots of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 395, + 617 + ], + "score": 1.0, + "content": "computations but its accuracy is not acceptable; however, bL-ResNet-50", + "type": "text" + }, + { + "bbox": [ + 395, + 605, + 421, + 615 + ], + "score": 0.87, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 605, + 439, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 605, + 465, + 616 + ], + "score": 0.91, + "content": "\\beta = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ") achieves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "a better balance between accuracy and performance. A similar trend is also observed on a deeper", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "model ResNet-101-lowres. While such performance is unsatisfying compared to the state of the art,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 316, + 650 + ], + "score": 1.0, + "content": "it is quite reasonable and expected given that almost", + "type": "text" + }, + { + "bbox": [ + 316, + 638, + 349, + 649 + ], + "score": 0.89, + "content": "3 \\sim 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "reduction of computation are achieved", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 648, + 167, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 167, + 661 + ], + "score": 1.0, + "content": "in such a case.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 583, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Figure 4 shows the prediction results from bL-ResNet-50 and ResNet-50-lowres. When both models", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "predict correctly (4 (a) and (b)), the bL-ResNet-50 provides better confidence for the prediction.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Because the object only occupies a small portion of an image, the Little-Branch can still capture the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "object clearly. On the other hand, when the key features of an object is small, like the shape of beak", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "of a bird (c) and the spots of a ladybug (d), bL-ResNet-50 can easily retain that key feature to predict", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 721, + 373, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 373, + 733 + ], + "score": 1.0, + "content": "correctly while ResNet-50-lowres provides wrong predicted label.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 167, + 80, + 444, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 167, + 80, + 444, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 80, + 444, + 228 + ], + "spans": [ + { + "bbox": [ + 167, + 80, + 444, + 228 + ], + "score": 0.792, + "type": "image", + "image_path": "bf948a2c1696a33afbd37fc5e16cb47be047e9384139f171958c590313dc30a9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 167, + 80, + 444, + 129.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 167, + 129.33333333333334, + 444, + 178.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 167, + 178.66666666666669, + 444, + 228.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 506, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "Figure 4: Prediction results for bL-ResNet-50 and ResNet-50-lowres. True labels, predicted labels", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "and their probability are listed in the table. When both models predicts correctly ((a) and (b)),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "bL-ResNet-50 achieves much higher probability; on the other hand, bL-ResNet-50 captures the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 275, + 351, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 351, + 287 + ], + "score": 1.0, + "content": "details on the object and then predicts correctly ((c) and (d)).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "table", + "bbox": [ + 165, + 318, + 445, + 390 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 178, + 306, + 432, + 317 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 178, + 304, + 432, + 320 + ], + "spans": [ + { + "bbox": [ + 178, + 304, + 432, + 320 + ], + "score": 1.0, + "content": "Table 6: Performance of ResNets at different input resolutions.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table_body", + "bbox": [ + 165, + 318, + 445, + 390 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 318, + 445, + 390 + ], + "spans": [ + { + "bbox": [ + 165, + 318, + 445, + 390 + ], + "score": 0.979, + "html": "
NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
ResNet-50-lowres26.10%1.29 (3.17×)25.60
ResNet-10121.95%7.8044.54
ResNet-101-lowres24.80%2.22 (3.51x)44.57
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NetworkTop-1 ErrorFLOPs (109)Params (106)
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NetworkTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
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NetworkTop-1 ErrorFLOPs (109)Params (106)
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ModelTop-1 ErrorFLOPs (109)Params (106)
bL-ResNet-50 (m = 4) (baseline)22.69%2.8526.69
bL-ResNet-50 (m = 2)23.48%2.7426.66
bL-ResNet-50 (m = 1)24.57%2.6426.64
bL-ResNet-101 (m= 4) (baseline)21.80%3.8941.85
bL-ResNet-101 (m = 7)21.85%5.2144.44
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Model|Model widthTop-1 ErrorFLOPs (106)Params (106)
ShuffleNetV21×30.60%1462.30
1.5×28.03%2993.51
2×26.85%5887.40
bL-ShuffleNetV2 @ 2561×30.84%1502.33
1.5×27.83%2984.80
2×26.38%5907.60
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ModelTop-1 ErrorFLOPs (109)Params (106)
bL-ResNet-50 (m = 4) (baseline)22.69%2.8526.69
bL-ResNet-50 (m = 2)23.48%2.7426.66
bL-ResNet-50 (m = 1)24.57%2.6426.64
bL-ResNet-101 (m= 4) (baseline)21.80%3.8941.85
bL-ResNet-101 (m = 7)21.85%5.2144.44
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Below we provide an explanation of why more information exchange is encouraged in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 254, + 372, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 372, + 266 + ], + "score": 1.0, + "content": "our approach and when is the best moment for merging operation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 220, + 506, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "In the above bL-Net, we merged branches before the feature dimension changes, except for the first", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 285, + 293 + ], + "score": 1.0, + "content": "stride convolution; thus, we used 4 merges", + "type": "text" + }, + { + "bbox": [ + 285, + 282, + 315, + 292 + ], + "score": 0.86, + "content": "m = 4 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "). We experimented with a different number of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "merges for bL-ResNet-50 and bL-ResNet-101, and the results are shown in Table 8. Since there", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "are fewer layers in bL-ResNet-50, we reduce the number of merges to show their importance; on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "the other hand, there are more layers in bL-ResNet-101, so we add more merges to show that those", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 325, + 403, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 403, + 338 + ], + "score": 1.0, + "content": "additional merges would similarly not improve the performance anymore.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 270, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 408, + 354 + ], + "score": 1.0, + "content": "The accuracy of the bL-ResNet-50 models with less number of merges (", + "type": "text" + }, + { + "bbox": [ + 408, + 343, + 439, + 352 + ], + "score": 0.88, + "content": "m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 342, + 459, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 343, + 491, + 352 + ], + "score": 0.87, + "content": "m = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 342, + 505, + 354 + ], + "score": 1.0, + "content": ") is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 253, + 365 + ], + "score": 1.0, + "content": "significantly worse than with more", + "type": "text" + }, + { + "bbox": [ + 254, + 353, + 284, + 363 + ], + "score": 0.85, + "content": "\\textlangle m = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 353, + 506, + 365 + ], + "score": 1.0, + "content": ") and they do not save many FLOPs and parameters at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "all. This justifies frequent information exchange improves the performance. On the other hand,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 174, + 388 + ], + "score": 1.0, + "content": "bL-ResNet-101", + "type": "text" + }, + { + "bbox": [ + 174, + 375, + 204, + 385 + ], + "score": 0.85, + "content": "m = 7 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 374, + 506, + 388 + ], + "score": 1.0, + "content": ") uses more merges; however, it also does not improve the performance and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "requires more FLOPs, which comes from more merges. This is because the original setting for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "amount of merging happened when either the channel number or feature map size is changed, so", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "extra merges happened at the feature dimension. Thus, those extra merges could be redundant since", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "merging at identical dimension could be reduced to one merging. Hence, it empirically proves that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 344, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 344, + 443 + ], + "score": 1.0, + "content": "merging before dimension is changed is the most effective.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 342, + 506, + 443 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 460, + 309, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 311, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 311, + 473 + ], + "score": 1.0, + "content": "A.5 bL-Net FOR HIGH-EFFICIENCY NETWORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "To demenstrate bL-Net can be applied on different types of network, we deploy bL-Net on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "score": 1.0, + "content": "high-efficiency network, ShuffleNetV2 (Ma et al., 2018), and results are shown in Table 9. Our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 344, + 518 + ], + "score": 1.0, + "content": "bL-ShuffleNetV2@256 outperforms ShuffleNetV2 by up to", + "type": "text" + }, + { + "bbox": [ + 344, + 506, + 367, + 516 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "point under similar FLOPs, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 404, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 404, + 529 + ], + "score": 1.0, + "content": "suggests that our approach can also improve the high-efficiency networks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 483, + 506, + 529 + ] + }, + { + "type": "table", + "bbox": [ + 167, + 565, + 444, + 636 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 156, + 552, + 453, + 564 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 155, + 551, + 452, + 566 + ], + "spans": [ + { + "bbox": [ + 155, + 551, + 390, + 566 + ], + "score": 1.0, + "content": "Table 9: Comparison with ShuffleNetV2 (Ma et al., 2018)", + "type": "text" + }, + { + "bbox": [ + 391, + 553, + 452, + 564 + ], + "score": 0.34, + "content": "( \\alpha = 2 , \\beta = 2 ) .", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "table_body", + "bbox": [ + 167, + 565, + 444, + 636 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 167, + 565, + 444, + 636 + ], + "spans": [ + { + "bbox": [ + 167, + 565, + 444, + 636 + ], + "score": 0.979, + "html": "
Model|Model widthTop-1 ErrorFLOPs (106)Params (106)
ShuffleNetV21×30.60%1462.30
1.5×28.03%2993.51
2×26.85%5887.40
bL-ShuffleNetV2 @ 2561×30.84%1502.33
1.5×27.83%2984.80
2×26.38%5907.60
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NetworkmAP@[IoU=0.5] (bbox)FLOPst (109)
ResNet-10181.5137.70
bL-ResNet-101 (α=2, β=4)81.489.21 (1.54×)
bL-ResNet-101 (α=2, β=2)81.593.95 (1.47×)
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NetworkmAP@[IoU=0.50:0.95](bbox)|FLOPs‡ (109)
ResNet-10139.2234.85
bL-ResNet-101 (α=2, β=4)39.5151.11 (1.55×)
bL-ResNet-101 (α=2, β=2)40.4160.21 (1.47×)
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The results are shown in Fig. 5. The batch size was set to 8, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "score": 1.0, + "content": "is the largest number allowed for NASNet on a P100 GPU card. The image size for any model in this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 403, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 506, + 419 + ], + "score": 1.0, + "content": "benchmark experiment is the same as that used in the experiment reported in Fig. 3. 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NetworkmAP@[IoU=0.5] (bbox)FLOPst (109)
ResNet-10181.5137.70
bL-ResNet-101 (α=2, β=4)81.489.21 (1.54×)
bL-ResNet-101 (α=2, β=2)81.593.95 (1.47×)
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NetworkmAP@[IoU=0.50:0.95](bbox)|FLOPs‡ (109)
ResNet-10139.2234.85
bL-ResNet-101 (α=2, β=4)39.5151.11 (1.55×)
bL-ResNet-101 (α=2, β=2)40.4160.21 (1.47×)
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ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α =1, β=1)21.75%5.6534.12
bL-ResNet-50 (α =1, β=2）22.11%4.3430.14
bL-ResNet-50(α= 2,) β=2)22.72%2.9126.97
bL-ResNet-50(α =2,) β=4)22.69%2.8526.69
bL-ResNet-50(α = 4, β = 2)23.20%2.4926.31
bL-ResNet-50 (α =4, β= 4)23.15%2.4826.24
ResNet-10121.95%7.8044.54
bL-ResNet-101 (α=1,β=1)20.31%10.2963.32
bL-ResNet-101 (α = 2, β= 2)21.40%4.2743.39
bL-ResNet-101 (α = 2,β=4)21.80%3.8941.85
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All", + "type": "text" + }, + { + "bbox": [ + 441, + 688, + 453, + 698 + ], + "score": 0.43, + "content": "b L", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "-Net variants", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "end with the same projection and output layer as ResNet-22. All merges, unless otherwise specified,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "are through linear combination with unit weights per branch. 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ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α =1, β=1)21.75%5.6534.12
bL-ResNet-50 (α =1, β=2）22.11%4.3430.14
bL-ResNet-50(α= 2,) β=2)22.72%2.9126.97
bL-ResNet-50(α =2,) β=4)22.69%2.8526.69
bL-ResNet-50(α = 4, β = 2)23.20%2.4926.31
bL-ResNet-50 (α =4, β= 4)23.15%2.4826.24
ResNet-10121.95%7.8044.54
bL-ResNet-101 (α=1,β=1)20.31%10.2963.32
bL-ResNet-101 (α = 2, β= 2)21.40%4.2743.39
bL-ResNet-101 (α = 2,β=4)21.80%3.8941.85
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We report results after Cross-Entropy training, on", + "type": "text" + }, + { + "bbox": [ + 466, + 343, + 504, + 353 + ], + "score": 0.38, + "content": "{ \\mathrm { H u b } } 5 ^ { \\prime } { 0 0 }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 480, + 367 + ], + "score": 1.0, + "content": "(SWB and CH part) after decoding using the standard small 4M n-gram language model with a", + "type": "text" + }, + { + "bbox": [ + 481, + 354, + 505, + 365 + ], + "score": 0.59, + "content": "3 0 . 5 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 363, + 177, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 177, + 379 + ], + "score": 1.0, + "content": "word vocabulary.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 277, + 506, + 379 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "All models were trained in PyTorch (Paszke et al., 2017) over 16 epochs on 2 GPUs with per-GPU", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 424, + 406 + ], + "score": 1.0, + "content": "batch size 256 (total batch size of 512), gradient clipping 10.0, weight decay √", + "type": "text" + }, + { + "bbox": [ + 425, + 393, + 464, + 403 + ], + "score": 0.91, + "content": "1 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 392, + 505, + 406 + ], + "score": 1.0, + "content": ", nesterov", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 404, + 456, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 362, + 418 + ], + "score": 1.0, + "content": "accelerated momentum 0.9, and learning rate 0.03 (annealed by", + "type": "text" + }, + { + "bbox": [ + 363, + 404, + 385, + 416 + ], + "score": 0.9, + "content": "\\sqrt { 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 405, + 427, + 418 + ], + "score": 1.0, + "content": "per epoch", + "type": "text" + }, + { + "bbox": [ + 427, + 405, + 450, + 416 + ], + "score": 0.84, + "content": "\\geq 1 0 ", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 405, + 456, + 418 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 381, + 505, + 418 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 237, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 238, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 238, + 442 + ], + "score": 1.0, + "content": "B.2 NETWORK STRUCTURES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "ResNet-22 Our models follow a ResNet architecture without padding in time (Saon et al., 2017),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "which accounts for the fact that padding in time adds undesirable artifacts when processing a longer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "utterance (Sercu & Goel, 2016). 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The output then goes through", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 588, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 385, + 600 + ], + "score": 1.0, + "content": "a bottleneck projection layer to the 32k-dimensional output CD state:", + "type": "text" + }, + { + "bbox": [ + 385, + 588, + 434, + 600 + ], + "score": 0.88, + "content": "( 4 \\times 1 , 5 1 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 588, + 453, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 454, + 588, + 502, + 600 + ], + "score": 0.87, + "content": "( 1 \\times 1 , 3 2 \\mathbf { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 588, + 504, + 600 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 565, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 170, + 617 + ], + "score": 1.0, + "content": "bL-ResNet-22", + "type": "text" + }, + { + "bbox": [ + 170, + 605, + 235, + 616 + ], + "score": 0.88, + "content": "( \\alpha = 4 , \\beta = 1 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ") Table 12 displays two bL-Net architectures that we experimented", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 407, + 629 + ], + "score": 1.0, + "content": "with based on the ResNet-22 baseline. 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LayersOutput Size 一bL-ResNet-22 (α=4,β=1)bL-ResNet-22 (α= 2,β=3)
Convolution32×T T=49→455×5,64,s2
bL-module32×T T=45→35（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1(3×3,16) （3×3,16）L (3 ×3,16), × 1×2（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1（3×3,32） （3×3,32)×1
transition layer16×T T=35→33(3×3,64,s2)×1
bL-module16×T T= 33→23（3×3,128) ×2 （3×3,128） B（3×3,32) （3×3,32）×2 L（3×3,128) ×2 3×3,128 B（3×3,64) 3×3,64）×1 L
transition layer8×T T=23→21（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1
bL-module8×T T=21→11（3×3,256) ×2 3×3,256 B(3×3,64) （3×3,64） L×2（3×3,256) ×2 3×3,256 B（3×3,128) 3×3,128）×1 L
transition layer4×T(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1
bL-moduleT=11→9 4×T（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2
ConvolutionT=9→1 1×1（3×3,512） （3×3,128） （3×3,512) B L
Convolution1×14×1,512 1 × 1,32k
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To resolve this, we apply a", + "type": "text" + }, + { + "bbox": [ + 480, + 462, + 505, + 472 + ], + "score": 0.89, + "content": "1 \\times 1", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 472, + 500, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 500, + 486 + ], + "score": 1.0, + "content": "convolution to reduce the number of channels accordingly and fuse the two separate feature maps.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 171, + 502 + ], + "score": 1.0, + "content": "bL-ResNet-22", + "type": "text" + }, + { + "bbox": [ + 171, + 489, + 253, + 501 + ], + "score": 0.85, + "content": "( \\alpha = 4 , \\beta = 2 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 488, + 443, + 502 + ], + "score": 1.0, + "content": "We explored two more models where we fix", + "type": "text" + }, + { + "bbox": [ + 443, + 490, + 472, + 500 + ], + "score": 0.89, + "content": "\\alpha = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 488, + 506, + 502 + ], + "score": 1.0, + "content": ", one in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 167, + 513 + ], + "score": 1.0, + "content": "which we take", + "type": "text" + }, + { + "bbox": [ + 168, + 501, + 194, + 511 + ], + "score": 0.91, + "content": "\\beta = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 500, + 273, + 513 + ], + "score": 1.0, + "content": "and another where", + "type": "text" + }, + { + "bbox": [ + 273, + 501, + 299, + 512 + ], + "score": 0.91, + "content": "\\beta = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 500, + 506, + 513 + ], + "score": 1.0, + "content": ". 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The difference in each Little-Branch is based on the setting of", + "type": "text" + }, + { + "bbox": [ + 496, + 512, + 503, + 523 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 510, + 507, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "Since the number of convolutions we use in the bL-Net baseline is uneven in the first three stages, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 126, + 546 + ], + "score": 1.0, + "content": "take", + "type": "text" + }, + { + "bbox": [ + 127, + 533, + 149, + 545 + ], + "score": 0.92, + "content": "\\lceil L / \\beta \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 533, + 324, + 546 + ], + "score": 1.0, + "content": "to be the depth of the Little-Branch, where", + "type": "text" + }, + { + "bbox": [ + 324, + 534, + 333, + 543 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 533, + 477, + 546 + ], + "score": 1.0, + "content": "is the depth of the Big-Branch. For", + "type": "text" + }, + { + "bbox": [ + 477, + 533, + 502, + 545 + ], + "score": 0.91, + "content": "\\beta = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 533, + 507, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "the first three stages of the network have a Little-Branch consisting of one residual block with two", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 130, + 566 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 555, + 318, + 567 + ], + "score": 1.0, + "content": "convolutions and one residual block with one", + "type": "text" + }, + { + "bbox": [ + 318, + 555, + 342, + 566 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "convolution. 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For", + "type": "text" + }, + { + "bbox": [ + 460, + 566, + 486, + 577 + ], + "score": 0.93, + "content": "\\beta = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 480, + 589 + ], + "score": 1.0, + "content": "first three stages of the network have a Little-Branch consisting of one residual block with two", + "type": "text" + }, + { + "bbox": [ + 481, + 578, + 504, + 588 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "convolutions, resulting in a reduction in the number of convolutions from 5 in the Big-Branch to 2 in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "the Little-Branch. 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LayersOutput Size 一bL-ResNet-22 (α=4,β=1)bL-ResNet-22 (α= 2,β=3)
Convolution32×T T=49→455×5,64,s2
bL-module32×T T=45→35（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1(3×3,16) （3×3,16）L (3 ×3,16), × 1×2（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1（3×3,32） （3×3,32)×1
transition layer16×T T=35→33(3×3,64,s2)×1
bL-module16×T T= 33→23（3×3,128) ×2 （3×3,128） B（3×3,32) （3×3,32）×2 L（3×3,128) ×2 3×3,128 B（3×3,64) 3×3,64）×1 L
transition layer8×T T=23→21（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1
bL-module8×T T=21→11（3×3,256) ×2 3×3,256 B(3×3,64) （3×3,64） L×2（3×3,256) ×2 3×3,256 B（3×3,128) 3×3,128）×1 L
transition layer4×T(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1
bL-moduleT=11→9 4×T（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2
ConvolutionT=9→1 1×1（3×3,512） （3×3,128） （3×3,512) B L
Convolution1×14×1,512 1 × 1,32k
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All Big-Branchs in these models are the same as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 510, + 507, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 169, + 524 + ], + "score": 1.0, + "content": "bL-ResNet-22", + "type": "text" + }, + { + "bbox": [ + 170, + 512, + 230, + 523 + ], + "score": 0.89, + "content": "( \\alpha = 4 , \\beta = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 510, + 495, + 524 + ], + "score": 1.0, + "content": "). The difference in each Little-Branch is based on the setting of", + "type": "text" + }, + { + "bbox": [ + 496, + 512, + 503, + 523 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 510, + 507, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "Since the number of convolutions we use in the bL-Net baseline is uneven in the first three stages, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 126, + 546 + ], + "score": 1.0, + "content": "take", + "type": "text" + }, + { + "bbox": [ + 127, + 533, + 149, + 545 + ], + "score": 0.92, + "content": "\\lceil L / \\beta \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 533, + 324, + 546 + ], + "score": 1.0, + "content": "to be the depth of the Little-Branch, where", + "type": "text" + }, + { + "bbox": [ + 324, + 534, + 333, + 543 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 533, + 477, + 546 + ], + "score": 1.0, + "content": "is the depth of the Big-Branch. For", + "type": "text" + }, + { + "bbox": [ + 477, + 533, + 502, + 545 + ], + "score": 0.91, + "content": "\\beta = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 533, + 507, + 546 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "the first three stages of the network have a Little-Branch consisting of one residual block with two", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 130, + 566 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 555, + 318, + 567 + ], + "score": 1.0, + "content": "convolutions and one residual block with one", + "type": "text" + }, + { + "bbox": [ + 318, + 555, + 342, + 566 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "convolution. This results in a reduction", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 459, + 578 + ], + "score": 1.0, + "content": "of the number of convolutions from 5 in the Big-Branch to 3 in the Little-Branch. For", + "type": "text" + }, + { + "bbox": [ + 460, + 566, + 486, + 577 + ], + "score": 0.93, + "content": "\\beta = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 566, + 505, + 578 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 480, + 589 + ], + "score": 1.0, + "content": "first three stages of the network have a Little-Branch consisting of one residual block with two", + "type": "text" + }, + { + "bbox": [ + 481, + 578, + 504, + 588 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "convolutions, resulting in a reduction in the number of convolutions from 5 in the Big-Branch to 2 in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "the Little-Branch. For both models, the Little-Branch in the final stage consists of a single residual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 169, + 622 + ], + "score": 1.0, + "content": "block with two", + "type": "text" + }, + { + "bbox": [ + 169, + 610, + 193, + 620 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "convolutions, since there are only four convolutions in the Big-Branch of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 620, + 244, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 167, + 633 + ], + "score": 1.0, + "content": "final stage and", + "type": "text" + }, + { + "bbox": [ + 167, + 621, + 240, + 633 + ], + "score": 0.93, + "content": "\\lceil 4 / 3 \\rceil = \\lceil 4 / 2 \\rceil = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 620, + 244, + 633 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 488, + 507, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 230, + 650 + ], + "score": 1.0, + "content": "In all bL-Net variants in which", + "type": "text" + }, + { + "bbox": [ + 231, + 638, + 256, + 649 + ], + "score": 0.91, + "content": "\\beta > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 638, + 505, + 650 + ], + "score": 1.0, + "content": ", because we can’t pad in time, we see that the time dimension", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "will get out of sync between the Big-Branch and Little-Branch. Therefore, before merging we need", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "to match the output size of each branch. To do this, we crop the shallower branches in time to match", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "score": 1.0, + "content": "the deepest branch (i.e. the Big-Branch will always have a smaller time dimension due to having", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "more convolutions, so we crop to match it). This is similar to the way the shortcut in ResNet is dealt", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 706 + ], + "score": 1.0, + "content": "with in (Saon et al., 2017), and does not introduce edge artifacts when processing longer sequences.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 638, + 506, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 502, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 170, + 721 + ], + "score": 1.0, + "content": "bL-ResNet-22", + "type": "text" + }, + { + "bbox": [ + 171, + 709, + 241, + 721 + ], + "score": 0.87, + "content": "( \\alpha = 2 , \\beta = 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 708, + 427, + 721 + ], + "score": 1.0, + "content": ") The last of our two-branch models is where", + "type": "text" + }, + { + "bbox": [ + 428, + 710, + 456, + 720 + ], + "score": 0.9, + "content": "\\alpha = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 708, + 475, + 721 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 475, + 709, + 502, + 721 + ], + "score": 0.9, + "content": "\\beta = 3", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 718, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 368, + 734 + ], + "score": 1.0, + "content": "which is also presented in Table 12. This variant is well-defined in", + "type": "text" + }, + { + "bbox": [ + 369, + 722, + 376, + 730 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 718, + 393, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 721, + 401, + 732 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 718, + 506, + 734 + ], + "score": 1.0, + "content": "up to the first three stages", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of the network. In the last stage, however, we opt to not branch and instead follow identically the final", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "stage of ResNet-22 with two residual blocks operating at the full input resolution with 512 channels.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 708, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "of the network. 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ModelTop-1 ErrorFLOPs (109)Params (106)Speed (ms/batch)t
ResNet-10121.95%7.8044.54186
bL-ResNet-101 (α= 2,β= 4)21.80%3.89 (2.01×)41.85140 (1.33×)
bL-ResNet-101@256(α = 2,β= 4)21.04%5.08 (1.54×)41.85162 (1.15×)
ResNet-15221.51%11.5160.19266
bL-ResNet-152(α = 2,β =4)21.16%5.04 (2.28×)57.36178 (1.49×)
bL-ResNet-152@256 (α= 2, β=4)20.34%6.58 (1.75x)57.36205 (1.30×)
ResNeXt-50 (32×4d)22.20%4.2325.03157
bL-ResNeXt-50 (32×4d) (α =2,β= 4)21.60%3.03 (1.40×)26.19125 (1.26×)
bL-ResNeXt-50@256(α = 2, β= 4)20.96%3.95 (1.08×)26.19153 (1.03×)
ResNeXt-101 (32×4d)21.20%7.9744.17269
bL-ResNeXt-101 (32×4d) (α =2,β =4)21.08%4.08 (1.95×)41.51169 (1.59×)
bL-ResNeXt-101@256(α = 2, β=4)20.48%5.33 (1.50x)41.51203 (1.33×)
ResNeXt-101(64×4d)20.73%15.4683.46485
bL-ResNeXt-101 (64×4d) (α = 2,β =4)20.48%7.14 (2.17×)77.36263 (1.98×)
bL-ResNeXt-101@256 (64×4d) (α = 2, β = 4)19.65%9.32 (1.66x)77.36318 (1.53×)
SEResNeXt-50 (32×4d)21.78%4.2327.56216
bL-SEResNeXt-50 (32×4d) (α = 2,β=4) bL-SEResNeXt-50@256 (32×4d) (α= 2, β = 4)21.44%3.03 (1.40×)28.77163 (1.33×)
20.74%3.95 (1.08×)28.77192 (1.03×)
SEResNeXt-101 (32×4d)21.00%7.9748.96376
bL-SEResNeXt-101 (32×4d) (α = 2, β = 4)20.87%4.08 (1.95×)45.88235 (1.60×)
bL-SEResNeXt-101@256 (32×4d) (α = 2,β= 4)19.87%5.33 (1.50×)45.88270 (1.39×)
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ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α = 2, β = 2)22.72%2.91 (1.41×)26.97
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bL-ResNet-50 (α= 4, β= 4)23.15%2.48 (1.65×)26.24
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ModelFLOPs (109)Params (106)WER AvgHub5Hub5 CH
1Baseline: ResNet-221.113.0214.67%11.15%18.17%
2bL-ResNet-22 (α = 4,β = 1)0.68 (1.63×)3.1514.72%11.24%18.18%
3bL-ResNet-22 (α = 4,β = 2)0.66 (1.68×)3.1114.47%10.95%17.95%
4bL-ResNet-22(α = 4,β= 3)0.65 (1.70×)3.1014.66%11.25%18.05%
5bL-ResNet-22 (α = 2,β = 3)0.77 (1.43×)3.0714.46%11.10%17.80%
6bL-ResNet-22 (α = 4,β=1) cat0.70 (1.58×)3.1814.67%11.31%18.00%
7bL-PYR-ResNet-22 (α = 4, β =1)0.98 (1.13×)3.3214.50%11.05%17.92%
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LayersbL-ResNet-50ResNet-50
Convolution7 × 7,64, s2 (112 × 112)
bL-module3×3,323 × 3,64,s2 3×3,32,s2(56 × 56)(1×1,64MaxPooling (56 × 56)
bL-moduleResBlockg,256 ×2 ResBlockL,128 ×1ResBlock,256,s2 (28 × 28)ResBlock,256 ×3, s1(56 × 56)
bL-moduleResBlockp,512 ×3 ResBlockL,256 ×1ResBlock, 512, s2 (14 × 14)ResBlock,512 ×4, s2(28× 28)
bL-moduleResBlockb,1024 ×5 ResBlockL,512 ×1ResBlock,1024 (14 × 14)ResBlock,1024 ×6, s2(14 × 14)
ResBlockResBlock,2048 ×3, s2 (7 × 7)
Average pool7 ×7 average pooling
FC, softmax1000
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LayersOutput SizebL-ResNet-101 bL-ResNet-152
Convolution112 × 1127 × 7,64,s2
bL-module56× 56（3×3,32 3 ×3,64,s2 3×3,32,s2 （1×1,64
bL-module56×56/1×1,64 3×3,64 ×3(2) (1×1,256) B/1×1,32 3×3,32 （1×1,128） L×1/1×1,64 3×3,64 × 4(2) （1×1,256) B/1×1,32 3×3,32 （1×1,128)×1 L
transition layer28×28/1×1,64 3×3,64,s2 ×1 （1×1,256
bL-module28×28(1×1,128) /1×1,64 3×3,128 ×7(3) 3×3,64 (1×1,512) B×1 （1×1,256） L(1×1,128) 3×3,128 (1×1,512)× 11(7)/1×1,64 3×3,64 ×2 (1×1,256) L
transition layer14 × 14/1×1,128 3×3,128,s2 ×1 (1×1,512B
bL-module14 × 14/1×1,256 3×3,256 × 17(22) （1×1,1024）B(1×1,128) 3×3,128 （1×1,512 L×3/1×1,256 3×3,256 (1×1,1024) B× 29(35)(1×1,128) 3×3,128 （1×1,512）×6 L
transition layer14 × 14/1×1,256 3×3,256 ×1 （1×1,1024）
ResBlock7×7/1×1,512 3 ×3,512,s2 ×3
Average pool1×1（1×1,2048 7×7 average pooling
FC, softmax1000
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ModelTop-1 ErrorFLOPs (109)Params (106)
ResNet-5023.66%4.0925.55
bL-ResNet-50 (α =1, β=1)21.75%5.6534.12
bL-ResNet-50 (α =1, β=2）22.11%4.3430.14
bL-ResNet-50(α= 2,) β=2)22.72%2.9126.97
bL-ResNet-50(α =2,) β=4)22.69%2.8526.69
bL-ResNet-50(α = 4, β = 2)23.20%2.4926.31
bL-ResNet-50 (α =4, β= 4)23.15%2.4826.24
ResNet-10121.95%7.8044.54
bL-ResNet-101 (α=1,β=1)20.31%10.2963.32
bL-ResNet-101 (α = 2, β= 2)21.40%4.2743.39
bL-ResNet-101 (α = 2,β=4)21.80%3.8941.85
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LayersOutput Size 一bL-ResNet-22 (α=4,β=1)bL-ResNet-22 (α= 2,β=3)
Convolution32×T T=49→455×5,64,s2
bL-module32×T T=45→35（3×3,64) 3×3,64 ×2 B (3×3,64)g ×1(3×3,16) （3×3,16）L (3 ×3,16), × 1×2（3×3,64) ×2 3×3,64) B (3 ×3,64)B ×1（3×3,32） （3×3,32)×1
transition layer16×T T=35→33(3×3,64,s2)×1
bL-module16×T T= 33→23（3×3,128) ×2 （3×3,128） B（3×3,32) （3×3,32）×2 L（3×3,128) ×2 3×3,128 B（3×3,64) 3×3,64）×1 L
transition layer8×T T=23→21（3×3,128)B ×1 (3×3,32) ×1 （3×3,128)g ×1 （3×3,128,s2)×1
bL-module8×T T=21→11（3×3,256) ×2 3×3,256 B(3×3,64) （3×3,64） L×2（3×3,256) ×2 3×3,256 B（3×3,128) 3×3,128）×1 L
transition layer4×T(3×3,256)B ×1 (3×3,64) ×1 (3×3,256)g ×1 (3×3,256,s2) ×1
bL-moduleT=11→9 4×T（3×3,512) （3×3,128) （3×3,512) ×2 ×2 ×2
ConvolutionT=9→1 1×1（3×3,512） （3×3,128） （3×3,512) B L
Convolution1×14×1,512 1 × 1,32k
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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. + +
CategoryL²AEL²AE gradDSAEDSAE gradVAEVAE gradγ-VAEγ-VAE grad
TJ annescarpet0.5390.7340.5450.7740.5800.7350.6480.727
grid0.9600.9810.9600.9800.8880.9610.9500.979
leather0.7510.9210.7100.6020.8340.9250.8180.897
tile0.4760.5750.4960.6260.4650.6540.4910.581
wood0.6300.8050.6410.7380.6950.8380.6650.809
seetqtbottle0.9090.9160.9330.9510.9020.9220.9130.931
cable0.7320.8640.7900.8590.8280.9100.7770.880
capsule0.7860.9520.7690.8840.8620.9170.8140.917
hazelnut0.9760.9840.9660.9660.9770.9760.9770.988
metalnut0.8800.8990.8810.9200.8810.9070.8830.914
pill0.8850.9120.8950.9270.8880.9300.8970.935
screw0.9790.9800.9830.9250.9580.9450.9760.972
toothbrush0.9710.9830.9730.9840.9710.9850.9710.983
transistor0.9060.9210.9040.9340.8940.9190.8960.931
zipper0.6800.8890.8280.8870.8140.8690.7060.871
+ +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. 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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). + +
CategoryLr(x)VxL(x)IVxL(x)) OLr(x)IVxLKL(x)lIVxLKL(x)) Lr(x)VAE-grad
TJssnetscarpet0.5370.5800.5660.5530.5550.735
grid0.8230.6350.8120.5070.7900.961
leather0.7830.6500.7920.6270.7910.925
tile0.5470.6060.5810.6230.5880.654
wood0.6860.6910.7260.6430.7130.838
seetqtbottle0.8310.7620.8320.6290.8300.922
cable0.8310.7960.8460.6740.8410.910
capsule0.7650.7540.7720.6420.7950.917.
hazelnut0.9070.8310.9080.4680.8850.976
metalnut0.8330.8310.8700.7100.8340.907
pill0.8690.8330.8720.4800.8260.930
screw0.8510.7260.8420.4120.7950.945
toothbrush0.9420.7980.9430.6190.9390.985
transistor0.7880.8430.8340.8010.8360.919
zipper0.7250.6740.7290.5620.7270.869
+ +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 $1 5 . 8 6 \%$ . \ No newline at end of file diff --git a/parse/train/HJx81ySKwr/HJx81ySKwr_content_list.json b/parse/train/HJx81ySKwr/HJx81ySKwr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..456277c1937a5d407bcf76d4557313a11d6686f0 --- /dev/null +++ b/parse/train/HJx81ySKwr/HJx81ySKwr_content_list.json @@ -0,0 +1,1442 @@ +[ + { + "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, + 501 + ], + "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, + 825, + 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": [ + 174, + 707, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 138, + 825, + 251 + ], + "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, + 825, + 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": [ + 266, + 422, + 732, + 531 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 619, + 326, + 636 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GENERATIVE MODELS ", + "text_level": 1, + "bbox": [ + 176, + 654, + 369, + 667 + ], + "page_idx": 1 + }, + { + "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": [ + 173, + 680, + 825, + 821 + ], + "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": [ + 173, + 825, + 825, + 924 + ], + "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": [ + 173, + 102, + 825, + 202 + ], + "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": [ + 282, + 219, + 715, + 276 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 294, + 825, + 351 + ], + "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. ", + "bbox": [ + 173, + 356, + 825, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 ANOMALY DETECTION AND LOCALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 501, + 516, + 515 + ], + "page_idx": 2 + }, + { + "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": [ + 173, + 531, + 825, + 643 + ], + "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": [ + 173, + 650, + 825, + 747 + ], + "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: ", + "bbox": [ + 174, + 753, + 825, + 811 + ], + "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": [ + 308, + 830, + 689, + 848 + ], + "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": [ + 174, + 867, + 825, + 924 + ], + "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": [ + 236, + 102, + 761, + 227 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 291, + 372, + 308 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 ADVERSARIAL PROJECTIONS ", + "text_level": 1, + "bbox": [ + 176, + 324, + 415, + 338 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 349, + 825, + 392 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 398, + 823, + 428 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 434, + 825, + 477 + ], + "page_idx": 3 + }, + { + "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": [ + 401, + 483, + 584, + 518 + ], + "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": [ + 174, + 531, + 825, + 574 + ], + "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": [ + 382, + 580, + 614, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "An iteration is done by calculating $\\mathbf { x } _ { t + 1 }$ as ", + "bbox": [ + 176, + 613, + 455, + 628 + ], + "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, + 589, + 652 + ], + "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": [ + 174, + 666, + 825, + 710 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 REGULARIZATION TERM ", + "text_level": 1, + "bbox": [ + 174, + 727, + 385, + 741 + ], + "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": [ + 173, + 752, + 825, + 852 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 OPTIMIZATION IN INPUT SPACE ", + "text_level": 1, + "bbox": [ + 176, + 869, + 431, + 883 + ], + "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": [ + 174, + 895, + 823, + 924 + ], + "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": [ + 171, + 103, + 823, + 132 + ], + "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, + 137, + 601, + 155 + ], + "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": [ + 173, + 159, + 825, + 299 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 OPTIMIZING GRADIENT DESCENT ", + "text_level": 1, + "bbox": [ + 176, + 315, + 446, + 329 + ], + "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, + 825, + 452 + ], + "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": [ + 174, + 469, + 400, + 484 + ], + "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": [ + 176, + 491, + 825, + 534 + ], + "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": [ + 174, + 563, + 825, + 647 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 STOP CRITERION ", + "text_level": 1, + "bbox": [ + 174, + 664, + 333, + 678 + ], + "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": [ + 174, + 689, + 825, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 765, + 326, + 781 + ], + "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": [ + 174, + 796, + 825, + 853 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 UNSUPERVISED ANOMALY SEGMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 869, + 511, + 883 + ], + "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": [ + 174, + 895, + 821, + 924 + ], + "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. ", + "bbox": [ + 173, + 99, + 825, + 239 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/d00279df37aa0dacefee388952cb7e0aa063872f82bbe3b6dc66c907155516b5.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CategoryL²AEL²AE gradDSAEDSAE gradVAEVAE gradγ-VAEγ-VAE grad
TJ annescarpet0.5390.7340.5450.7740.5800.7350.6480.727
grid0.9600.9810.9600.9800.8880.9610.9500.979
leather0.7510.9210.7100.6020.8340.9250.8180.897
tile0.4760.5750.4960.6260.4650.6540.4910.581
wood0.6300.8050.6410.7380.6950.8380.6650.809
seetqtbottle0.9090.9160.9330.9510.9020.9220.9130.931
cable0.7320.8640.7900.8590.8280.9100.7770.880
capsule0.7860.9520.7690.8840.8620.9170.8140.917
hazelnut0.9760.9840.9660.9660.9770.9760.9770.988
metalnut0.8800.8990.8810.9200.8810.9070.8830.914
pill0.8850.9120.8950.9270.8880.9300.8970.935
screw0.9790.9800.9830.9250.9580.9450.9760.972
toothbrush0.9710.9830.9730.9840.9710.9850.9710.983
transistor0.9060.9210.9040.9340.8940.9190.8960.931
zipper0.6800.8890.8280.8870.8140.8690.7060.871
", + "bbox": [ + 192, + 257, + 805, + 497 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 176, + 534, + 821, + 561 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 176, + 568, + 821, + 597 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 603, + 825, + 799 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 805, + 825, + 888 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 173, + 222, + 825, + 305 + ], + "page_idx": 6 + }, + { + "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": [], + "bbox": [ + 184, + 330, + 805, + 637 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 INPAINTING ", + "text_level": 1, + "bbox": [ + 174, + 779, + 299, + 792 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 804, + 825, + 875 + ], + "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. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "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": [ + 241, + 112, + 756, + 342 + ], + "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). ", + "bbox": [ + 174, + 443, + 825, + 540 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 568, + 344, + 583 + ], + "page_idx": 7 + }, + { + "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": [ + 174, + 603, + 823, + 672 + ], + "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": [ + 174, + 680, + 825, + 750 + ], + "page_idx": 7 + }, + { + "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": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 117 + ], + "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. ", + "bbox": [ + 173, + 133, + 825, + 342 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 363, + 285, + 378 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 387, + 825, + 429 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Jinwon An and Sungzoon Cho. Variational autoencoder based anomaly detection using reconstruction probability. Technical report, SNU Data Mining Center, 2015. ", + "bbox": [ + 169, + 439, + 821, + 468 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "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. 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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. ", + "bbox": [ + 173, + 539, + 825, + 597 + ], + "page_idx": 9 + }, + { + "type": "text", + "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": "
CategoryLr(x)VxL(x)IVxL(x)) OLr(x)IVxLKL(x)lIVxLKL(x)) Lr(x)VAE-grad
TJssnetscarpet0.5370.5800.5660.5530.5550.735
grid0.8230.6350.8120.5070.7900.961
leather0.7830.6500.7920.6270.7910.925
tile0.5470.6060.5810.6230.5880.654
wood0.6860.6910.7260.6430.7130.838
seetqtbottle0.8310.7620.8320.6290.8300.922
cable0.8310.7960.8460.6740.8410.910
capsule0.7650.7540.7720.6420.7950.917.
hazelnut0.9070.8310.9080.4680.8850.976
metalnut0.8330.8310.8700.7100.8340.907
pill0.8690.8330.8720.4800.8260.930
screw0.8510.7260.8420.4120.7950.945
toothbrush0.9420.7980.9430.6190.9390.985
transistor0.7880.8430.8340.8010.8360.919
zipper0.7250.6740.7290.5620.7270.869
", + "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 + } +] \ No newline at end of file diff --git a/parse/train/HJx81ySKwr/HJx81ySKwr_middle.json b/parse/train/HJx81ySKwr/HJx81ySKwr_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..77f2e29a6ff8ec511b79af4c0f921fa2de1b8bd9 --- /dev/null +++ b/parse/train/HJx81ySKwr/HJx81ySKwr_middle.json @@ -0,0 +1,32699 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 99 + ], + "score": 1.0, + "content": "ITERATIVE ENERGY-BASED PROJECTION ON A NOR-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 496, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 496, + 118 + ], + "score": 1.0, + "content": "MAL DATA MANIFOLD FOR ANOMALY LOCALIZATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 134, + 405, + 168 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 402, + 147 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 402, + 147 + ], + "score": 1.0, + "content": "David Dehaene∗, Oriel Frigo∗, Sébastien Combrexelle, Pierre Eline", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 145, + 227, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 227, + 158 + ], + "score": 1.0, + "content": "AnotherBrain, Paris, France", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 157, + 406, + 169 + ], + "spans": [ + { + "bbox": [ + 113, + 157, + 406, + 169 + ], + "score": 1.0, + "content": "{david, oriel, sebastien, pierre}@anotherbrain.ai", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 209 + ], + "lines": [ + { + "bbox": [ + 276, + 197, + 335, + 210 + ], + "spans": [ + { + "bbox": [ + 276, + 197, + 335, + 210 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 223, + 468, + 397 + ], + "lines": [ + { + "bbox": [ + 142, + 222, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 142, + 222, + 469, + 235 + ], + "score": 1.0, + "content": "Autoencoder reconstructions are widely used for the task of unsupervised anomaly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 246 + ], + "score": 1.0, + "content": "localization. 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Our method achieves state-of-the-art results on various", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 470, + 389 + ], + "score": 1.0, + "content": "anomaly localization datasets. It also shows promising results at an inpainting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 388, + 254, + 398 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 254, + 398 + ], + "score": 1.0, + "content": "task on the CelebA dataset.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 13.5, + "bbox_fs": [ + 141, + 222, + 471, + 398 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 205, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 208, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 208, + 435 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 458 + ], + "score": 1.0, + "content": "Automating visual inspection on production lines with artificial intelligence has gained popularity", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 454, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 470 + ], + "score": 1.0, + "content": "and interest in recent years. Indeed, the analysis of images to segment potential manufacturing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "defects seems well suited to computer vision algorithms. However these solutions remain data", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "hungry and require knowledge transfer from human to machine via image annotations. Furthermore,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "score": 1.0, + "content": "the classification in a limited number of user-predefined categories such as non-defective, greasy,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "scratched and so on, will not generalize well if a previously unseen defect appears. This is even more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "critical on production lines where a defective product is a rare occurrence. For visual inspection, a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "better-suited task is unsupervised anomaly detection, in which the segmentation of the defect must", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "be done only via prior knowledge of non-defective samples, constraining the issue to a two-class", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 200, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 200, + 556 + ], + "score": 1.0, + "content": "segmentation problem.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 443, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "From a statistical point of view, an anomaly may be seen as a distribution outlier, or an observation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "that deviates so much from other observations as to arouse suspicion that it was generated by a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "different mechanism (Hawkins, 1980). In this setting, generative models such as Variational Au-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "toEncoders (VAE, Kingma & Welling (2014)), are especially interesting because they are capable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "to infer possible sampling mechanisms for a given dataset. The original autoencoder (AE) jointly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "learns an encoder model, that compresses input samples into a low dimensional space, and a de-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 624, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 640 + ], + "score": 1.0, + "content": "coder, that decompresses the low dimensional samples into the original input space, by minimizing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "the distance between the input of the encoder and the output of the decoder. The more recent variant,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "score": 1.0, + "content": "VAE, replaces the deterministic encoder and decoder by stochastic functions, enabling the modeling", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "of the distribution of the dataset samples as well as the generation of new, unseen samples. In both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "score": 1.0, + "content": "models, the output decompressed sample given an input is often called the reconstruction, and is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "used as some sort of projection of the input on the support of the normal data distribution, which", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "we will call the normal manifold. In most unsupervised anomaly detection methods based on VAE,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 716 + ], + "score": 1.0, + "content": "models are trained on flawless data and defect detection and localization is then performed using a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "distance metric between the input sample and its reconstruction (Bergmann et al., 2018; 2019; An", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 331, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 331, + 105 + ], + "score": 1.0, + "content": "& Cho, 2015; Baur et al., 2018; Matsubara et al., 2018).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 561, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "distance metric between the input sample and its reconstruction (Bergmann et al., 2018; 2019; An", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 331, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 331, + 105 + ], + "score": 1.0, + "content": "& Cho, 2015; Baur et al., 2018; Matsubara et al., 2018).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "One fundamental issue in this approach is that the models learn on the normal manifold, hence there", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "is no guarantee of the generalization of their behavior outside this manifold. This is problematic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "since it is precisely outside the dataset distribution that such methods intend to use the VAE for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "anomaly localization. Even in the case of a model that always generates credible samples from the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "dataset distribution, there is no way to ensure that the reconstruction will be connected to the input", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "sample in any useful way. An example illustrating this limitation is given in figure 1, where a VAE", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "trained on regular grid images provides a globally poor reconstruction despite a local perturbation,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 290, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 290, + 200 + ], + "score": 1.0, + "content": "making the anomaly localization challenging.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "In this paper, instead of using the VAE reconstruction, we propose to find a better projection of an", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "input sample on the normal manifold, by optimizing an energy function defined by an autoencoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 240 + ], + "score": 1.0, + "content": "architecture. Starting at the input sample, we iterate gradient descent steps on the input to converge", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "to an optimum, simultaneously located on the data manifold and closest to the starting input. This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "method allows us to add prior knowledge about the expected anomalies via regularization terms,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "which is not possible with the raw VAE reconstruction. We show that such an optimum is better than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "previously proposed autoencoder reconstructions to localize anomalies on a variety of unsupervised", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "anomaly localization datasets (Bergmann et al., 2019) and present its inpainting capabilities on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "CelebA dataset (Liu et al., 2015). We also propose a variant of the standard gradient descent that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 438, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 438, + 317 + ], + "score": 1.0, + "content": "uses the pixel-wise reconstruction error to speed up the convergence of the energy.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "image", + "bbox": [ + 163, + 335, + 448, + 421 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 163, + 335, + 448, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 335, + 448, + 421 + ], + "spans": [ + { + "bbox": [ + 163, + 335, + 448, + 421 + ], + "score": 0.971, + "type": "image", + "image_path": "03e889e1384f99d5b868282f7f8d3e7022b6ec4f46e54b7acbfdf86789ae3a07.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 163, + 335, + 448, + 363.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 163, + 363.6666666666667, + 448, + 392.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 163, + 392.33333333333337, + 448, + 421.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 430, + 504, + 463 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Figure 1: Even though an anomaly is a local perturbation in the image (b), the whole VAE-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 440, + 504, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 454 + ], + "score": 1.0, + "content": "reconstructed image can be disturbed (c). Our gradient descent-based method gives better quality", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 452, + 186, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 186, + 464 + ], + "score": 1.0, + "content": "reconstructions (d).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 491, + 200, + 504 + ], + "lines": [ + { + "bbox": [ + 104, + 489, + 202, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 202, + 506 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 226, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 227, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 227, + 530 + ], + "score": 1.0, + "content": "2.1 GENERATIVE MODELS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 465, + 552 + ], + "score": 1.0, + "content": "In unsupervised anomaly detection, the only data available during training are samples", + "type": "text" + }, + { + "bbox": [ + 465, + 541, + 474, + 550 + ], + "score": 0.42, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "from a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 203, + 563 + ], + "score": 1.0, + "content": "non-anomalous dataset", + "type": "text" + }, + { + "bbox": [ + 203, + 550, + 241, + 561 + ], + "score": 0.92, + "content": "\\check { \\mathbb { X } } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 549, + 505, + 563 + ], + "score": 1.0, + "content": ". In a generative setting, we suppose the existence of a proba-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 213, + 574 + ], + "score": 1.0, + "content": "bility function of density", + "type": "text" + }, + { + "bbox": [ + 213, + 563, + 219, + 573 + ], + "score": 0.73, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 560, + 330, + 574 + ], + "score": 1.0, + "content": ", having its support on all", + "type": "text" + }, + { + "bbox": [ + 330, + 561, + 343, + 572 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { \\bar { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 560, + 505, + 574 + ], + "score": 1.0, + "content": ", from which the dataset was sampled.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 380, + 585 + ], + "score": 1.0, + "content": "The generative objective is then to model an estimate of density", + "type": "text" + }, + { + "bbox": [ + 380, + 574, + 387, + 584 + ], + "score": 0.74, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 572, + 505, + 585 + ], + "score": 1.0, + "content": ", from which we can obtain", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "new samples close to the dataset. Popular generative architectures are Generative Adversarial Net-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 422, + 606 + ], + "score": 1.0, + "content": "works (GAN, Goodfellow et al. 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This model converges to the equilibrium of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "expectation over both real and generated datasets of the binary cross entropy loss of the classifier", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 375, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 162, + 650 + ], + "score": 0.31, + "content": "m i n _ { G } m a x _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 637, + 168, + 652 + ], + "score": 1.0, + "content": "[", + "type": "text" + }, + { + "bbox": [ + 168, + 637, + 369, + 651 + ], + "score": 0.71, + "content": "\\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] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 637, + 375, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Disadvantages of GANs are that they are notoriously difficult to train (Goodfellow, 2017), and they", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "suffer from mode collapse, meaning that they have the tendency to only generate a subset of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "original dataset. This can be problematic for anomaly detection, in which we do not want some", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "subset of the normal data to be considered as anomalous (Bergmann et al., 2019). Recent works", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "such as Thanh-Tung et al. 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An example illustrating this limitation is given in figure 1, where a VAE", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "trained on regular grid images provides a globally poor reconstruction despite a local perturbation,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 290, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 290, + 200 + ], + "score": 1.0, + "content": "making the anomaly localization challenging.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 110, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "In this paper, instead of using the VAE reconstruction, we propose to find a better projection of an", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "input sample on the normal manifold, by optimizing an energy function defined by an autoencoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 240 + ], + "score": 1.0, + "content": "architecture. 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This model converges to the equilibrium of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "expectation over both real and generated datasets of the binary cross entropy loss of the classifier", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 375, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 162, + 650 + ], + "score": 0.31, + "content": "m i n _ { G } m a x _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 637, + 168, + 652 + ], + "score": 1.0, + "content": "[", + "type": "text" + }, + { + "bbox": [ + 168, + 637, + 369, + 651 + ], + "score": 0.71, + "content": "\\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] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 637, + 375, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 539, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Disadvantages of GANs are that they are notoriously difficult to train (Goodfellow, 2017), and they", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "suffer from mode collapse, meaning that they have the tendency to only generate a subset of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "original dataset. 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This leads to the esti-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 128, + 128 + ], + "score": 1.0, + "content": "mate", + "type": "text" + }, + { + "bbox": [ + 128, + 115, + 225, + 128 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p ( \\mathbf { x } ) = \\int p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { z } ) d z } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ", that we would like to optimize using maximum likelihood estimation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "score": 1.0, + "content": "on the dataset. 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The original autoencoder can be roughly considered as a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 423, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 423, + 372 + ], + "score": 1.0, + "content": "VAE whose encoded and decoded distributions have infinitely small variances.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 316, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 317, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 317, + 409 + ], + "score": 1.0, + "content": "2.2 ANOMALY DETECTION AND LOCALIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We will consider that an anomaly is a sample with low probability under our estimation of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "dataset distribution. The VAE loss, being a lower bound on the density, is a good proxy to clas-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "sify samples between the anomalous and non-anomalous categories. To this effect, a threshold", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 115, + 465 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 454, + 418, + 467 + ], + "score": 1.0, + "content": "can be defined on the loss function, delimiting anomalous samples with", + "type": "text" + }, + { + "bbox": [ + 419, + 455, + 466, + 466 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } ( \\mathbf { x } ) \\geq T", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "and nor-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 160, + 478 + ], + "score": 1.0, + "content": "mal samples", + "type": "text" + }, + { + "bbox": [ + 160, + 465, + 206, + 477 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\mathbf { x } ) < T", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 466, + 505, + 478 + ], + "score": 1.0, + "content": ". However, according to Matsubara et al. (2018), the regularization term", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 231, + 489 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { K L } ( \\mathbf { x } ) = D _ { \\mathrm { K L } } ( q ( \\mathbf { z } | \\mathbf { x } ) | | p ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "has a negative influence in the computation of anomaly scores. They", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 487, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 487, + 265, + 502 + ], + "score": 1.0, + "content": "propose instead an unregularized score", + "type": "text" + }, + { + "bbox": [ + 266, + 487, + 353, + 500 + ], + "score": 0.79, + "content": "\\mathbf { \\bar { \\mathcal { L } } } _ { r } ( \\mathbf { x } ) = - \\mathbb { E } _ { \\mathbf { z } \\sim q ( \\mathbf { z } | \\mathbf { x } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 487, + 369, + 502 + ], + "score": 1.0, + "content": "log", + "type": "text" + }, + { + "bbox": [ + 369, + 487, + 397, + 500 + ], + "score": 0.67, + "content": "\\bar { p } ( { \\bf x } | { \\bf z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 487, + 506, + 502 + ], + "score": 1.0, + "content": "which is equivalent to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 444, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 444, + 511 + ], + "score": 1.0, + "content": "reconstruction term of a standard autoencoder and claim a better anomaly detection.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "Going from anomaly detection to anomaly localization, this reconstruction term becomes crucial to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 526, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 504, + 539 + ], + "score": 1.0, + "content": "most of existing solutions. Indeed, the inability of the model to reconstruct a given part of an image", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "is used as a way to segment the anomaly, using a pixel-wise threshold on the reconstruction error.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "Actually, this segmentation is very often given by a pixel-wise (An & Cho, 2015; Baur et al., 2018;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "Matsubara et al., 2018) or patch-wise comparison of the input image, and some generated image,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "as in Bergmann et al. 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The original autoencoder can be roughly considered as a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 423, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 423, + 372 + ], + "score": 1.0, + "content": "VAE whose encoded and decoded distributions have infinitely small variances.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 282, + 506, + 372 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 316, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 317, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 317, + 409 + ], + "score": 1.0, + "content": "2.2 ANOMALY DETECTION AND LOCALIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We will consider that an anomaly is a sample with low probability under our estimation of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "dataset distribution. The VAE loss, being a lower bound on the density, is a good proxy to clas-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "sify samples between the anomalous and non-anomalous categories. 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However, according to Matsubara et al. (2018), the regularization term", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 476, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 231, + 489 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { K L } ( \\mathbf { x } ) = D _ { \\mathrm { K L } } ( q ( \\mathbf { z } | \\mathbf { x } ) | | p ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 476, + 505, + 491 + ], + "score": 1.0, + "content": "has a negative influence in the computation of anomaly scores. They", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 487, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 487, + 265, + 502 + ], + "score": 1.0, + "content": "propose instead an unregularized score", + "type": "text" + }, + { + "bbox": [ + 266, + 487, + 353, + 500 + ], + "score": 0.79, + "content": "\\mathbf { \\bar { \\mathcal { L } } } _ { r } ( \\mathbf { x } ) = - \\mathbb { E } _ { \\mathbf { z } \\sim q ( \\mathbf { z } | \\mathbf { x } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 487, + 369, + 502 + ], + "score": 1.0, + "content": "log", + "type": "text" + }, + { + "bbox": [ + 369, + 487, + 397, + 500 + ], + "score": 0.67, + "content": "\\bar { p } ( { \\bf x } | { \\bf z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 487, + 506, + 502 + ], + "score": 1.0, + "content": "which is equivalent to the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 444, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 444, + 511 + ], + "score": 1.0, + "content": "reconstruction term of a standard autoencoder and claim a better anomaly detection.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 421, + 506, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "Going from anomaly detection to anomaly localization, this reconstruction term becomes crucial to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 526, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 504, + 539 + ], + "score": 1.0, + "content": "most of existing solutions. Indeed, the inability of the model to reconstruct a given part of an image", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "is used as a way to segment the anomaly, using a pixel-wise threshold on the reconstruction error.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "Actually, this segmentation is very often given by a pixel-wise (An & Cho, 2015; Baur et al., 2018;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "Matsubara et al., 2018) or patch-wise comparison of the input image, and some generated image,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "as in Bergmann et al. 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We obtain a binary classifier defined by", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 344, + 505, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 383, + 358, + 411 + ], + "lines": [ + { + "bbox": [ + 246, + 383, + 358, + 411 + ], + "spans": [ + { + "bbox": [ + 246, + 383, + 358, + 411 + ], + "score": 0.94, + "content": "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.", + "type": "interline_equation", + "image_path": "a531f29019642757f994e7aa898c3c8061e00933d98c93d5b91f6ebd5032bea7.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 383, + 358, + 397.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 246, + 397.0, + 358, + 411.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "Our method consists in computing adversarial samples of this classifier (Szegedy et al., 2014), that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 235, + 446 + ], + "score": 1.0, + "content": "is to say, starting from a sample", + "type": "text" + }, + { + "bbox": [ + 236, + 434, + 248, + 444 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 432, + 269, + 446 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 270, + 433, + 315, + 445 + ], + "score": 0.93, + "content": "A ( \\mathbf { x } _ { 0 } ) = \\bar { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 432, + 494, + 446 + ], + "score": 1.0, + "content": ", iterate gradient descent steps over the input", + "type": "text" + }, + { + "bbox": [ + 494, + 435, + 501, + 443 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 432, + 505, + 446 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 405, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 192, + 456 + ], + "score": 1.0, + "content": "constructing samples", + "type": "text" + }, + { + "bbox": [ + 193, + 446, + 237, + 455 + ], + "score": 0.78, + "content": "\\mathbf { x } _ { 1 } , \\ldots . . . \\mathbf { x } _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 444, + 335, + 456 + ], + "score": 1.0, + "content": ", to minimize the energy", + "type": "text" + }, + { + "bbox": [ + 336, + 444, + 358, + 456 + ], + "score": 0.92, + "content": "E ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 444, + 405, + 456 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 422, + 505, + 456 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 460, + 376, + 475 + ], + "lines": [ + { + "bbox": [ + 234, + 460, + 376, + 475 + ], + "spans": [ + { + "bbox": [ + 234, + 460, + 376, + 475 + ], + "score": 0.91, + "content": "\\begin{array} { r } { E ( \\mathbf { x } _ { t } ) = \\mathcal { L } _ { r } ( \\mathbf { x } _ { t } ) + \\lambda \\cdot | | \\mathbf { x } _ { t } - \\mathbf { x } _ { 0 } | | _ { 1 } } \\end{array}", + "type": "interline_equation", + "image_path": "6d6c1c1f32e9cf8aa30fffa53409b26b381a130bfc27957c14ade92eb05865fc.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 234, + 460, + 376, + 475 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 279, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 280, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 246, + 500 + ], + "score": 1.0, + "content": "An iteration is done by calculating", + "type": "text" + }, + { + "bbox": [ + 246, + 489, + 267, + 498 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 485, + 280, + 500 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 485, + 280, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 503, + 361, + 517 + ], + "lines": [ + { + "bbox": [ + 249, + 503, + 361, + 517 + ], + "spans": [ + { + "bbox": [ + 249, + 503, + 361, + 517 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - 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We chose not to iterate over", + "type": "text" + }, + { + "bbox": [ + 420, + 140, + 427, + 148 + ], + "score": 0.66, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "in the latent space", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "for two reasons. First, because as noted in Dai & Wipf (2019) and Hoffman & Johnson (2016), the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 128, + 173 + ], + "score": 1.0, + "content": "prior", + "type": "text" + }, + { + "bbox": [ + 129, + 160, + 147, + 172 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 160, + 482, + 173 + ], + "score": 1.0, + "content": "is not always a good proxy for the real image of the distribution in the latent space", + "type": "text" + }, + { + "bbox": [ + 482, + 160, + 501, + 172 + ], + "score": 0.91, + "content": "q ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 160, + 504, + 173 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "Second, because the VAE tends to ignore some details of the original image in its reconstruction,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "considering that these details are part of the independent pixel noise allowed by the modeling of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 134, + 205 + ], + "score": 0.92, + "content": "p ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "as a diagonal Gaussian, which causes its infamous blurriness. An optimization in latent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "space would have to recreate the high frequency structure of the image, whereas iterating over the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 361, + 227 + ], + "score": 1.0, + "content": "input image space, and starting the descent on the input image", + "type": "text" + }, + { + "bbox": [ + 361, + 216, + 372, + 226 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 216, + 505, + 227 + ], + "score": 1.0, + "content": ", allows us to keep that structure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 297, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 297, + 239 + ], + "score": 1.0, + "content": "and thus to obtain projections of higher quality.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 250, + 273, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 275, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 275, + 263 + ], + "score": 1.0, + "content": "3.4 OPTIMIZING GRADIENT DESCENT", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "We observed that using the Adam optimizer (Kingma & Ba, 2015) is beneficial for the quality of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "optimization. Moreover, to speed up the convergence and further preserve the aforementioned high", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "frequency structure of the input, we propose to compute our iterative samples using the pixel-wise", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "reconstruction error of the VAE. To explain the intuition behind this improvement, we will consider", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "the inpainting task. In this setting, as in anomaly localization, a local perturbation is added on top of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "a normal image. However, in the classic inpainting task, the localization of the perturbation is known", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 302, + 349 + ], + "score": 1.0, + "content": "beforehand, and we can use the localization mask", + "type": "text" + }, + { + "bbox": [ + 303, + 337, + 312, + 347 + ], + "score": 0.82, + "content": "\\pmb { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "to only change the value of the anomalous pixels", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 202, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 202, + 361 + ], + "score": 1.0, + "content": "in the gradient descent:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 357, + 378, + 371 + ], + "lines": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "spans": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\alpha \\cdot \\left( \\nabla _ { \\mathbf { x } } E ( \\mathbf { x } _ { t } ) \\odot \\Omega \\right)", + "type": "interline_equation", + "image_path": "f23f9b783749f4859ba4a788ef18043a7df6f407c9b7dc834ea915047c2fa106.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 245, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 247, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 133, + 386 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 374, + 142, + 383 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 371, + 247, + 386 + ], + "score": 1.0, + "content": "is the Hadamard product.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 389, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 403 + ], + "score": 1.0, + "content": "For anomaly localization and blind inpainting, where this information is not available, we compute", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 455, + 413 + ], + "score": 1.0, + "content": "the pixel-wise reconstruction error which gives a rough estimate of the mask. The term", + "type": "text" + }, + { + "bbox": [ + 455, + 401, + 495, + 413 + ], + "score": 0.94, + "content": "\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 397, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 201, + 425 + ], + "score": 1.0, + "content": "therefore replaced with", + "type": "text" + }, + { + "bbox": [ + 201, + 411, + 335, + 424 + ], + "score": 0.82, + "content": "\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } ) \\odot ( { \\mathbf x _ { t } } - f _ { V A E } ( \\mathbf { x } _ { t } ) ) ^ { 2 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 411, + 397, + 425 + ], + "score": 1.0, + "content": ") in equation 5:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 428, + 411, + 442 + ], + "lines": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "spans": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "score": 0.87, + "content": "\\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)", + "type": "interline_equation", + "image_path": "af7ae15f47a600f88726caf7fdc9cf674f2a0aee385c0a8de68c0eafaf942287.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 133, + 459 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 446, + 172, + 459 + ], + "score": 0.93, + "content": "f _ { V A E } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "is the standard reconstruction of the VAE. Optimizing the energy this way, a pixel", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 458, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 469 + ], + "score": 1.0, + "content": "where the reconstruction error is high will update faster, whereas a pixel with good reconstruction", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "will not change easily. This prevents the image to update its pixels where the reconstruction is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "already good, even with a high learning rate. As can be seen in appendix B, this method converges", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "score": 1.0, + "content": "to the same performance as the method of equation 5, but with fewer iterations. An illustration of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 255, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 255, + 514 + ], + "score": 1.0, + "content": "our method can be found in figure 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 107, + 526, + 204, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 205, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 205, + 538 + ], + "score": 1.0, + "content": "3.5 STOP CRITERION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "A standard stop criterion based on the convergence of the energy can efficiently be used. Using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "the adversarial setting introduced in section 3.1, we also propose to stop the gradient descent when", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 570, + 504, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 504, + 580 + ], + "score": 1.0, + "content": "a certain predefined threshold on the VAE loss is reached. For example, such a threshold can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 580, + 450, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 450, + 591 + ], + "score": 1.0, + "content": "chosen to be a quantile of the empirical loss distribution computed on the training set.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 200, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 201, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 201, + 621 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "In this section, we evaluate the proposed method for two different applications: anomaly segmen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "tation and image inpainting. Both applications are interesting use cases of our method, where we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "search to reconstruct partially corrupted images, correcting the anomalies while preserving the un-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 665, + 208, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 208, + 677 + ], + "score": 1.0, + "content": "corrupted image regions.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 107, + 689, + 313, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 315, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 315, + 701 + ], + "score": 1.0, + "content": "4.1 UNSUPERVISED ANOMALY SEGMENTATION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In order to evaluate the proposed method for the task of anomaly segmentation, we perform experi-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "ments with the recently proposed MVTec dataset (Bergmann et al., 2019). This collection of datasets", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 495, + 95 + ], + "score": 1.0, + "content": "energy defined in equation 2. Following the aforementioned analogy between a GAN generator", + "type": "text" + }, + { + "bbox": [ + 495, + 83, + 504, + 92 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 490, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 186, + 108 + ], + "score": 1.0, + "content": "and a VAE decoder", + "type": "text" + }, + { + "bbox": [ + 186, + 94, + 205, + 104 + ], + "score": 0.54, + "content": "D e c", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 92, + 490, + 108 + ], + "score": 1.0, + "content": ", a similar approach in the context of a VAE would be to use the energy", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 504, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 109, + 368, + 123 + ], + "lines": [ + { + "bbox": [ + 243, + 109, + 368, + 123 + ], + "spans": [ + { + "bbox": [ + 243, + 109, + 368, + 123 + ], + "score": 0.93, + "content": "| | \\mathbf { x } - D e c ( \\mathbf { z } ) | | _ { 1 } - \\lambda \\cdot \\log p ( \\mathbf { z } )", + "type": "interline_equation", + "image_path": "e01f7b6f97ff6b949efb4bee6be337b31b796c1b87c053c831539f5827cbfe73.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 243, + 109, + 368, + 123 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 148, + 141 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 148, + 127, + 191, + 139 + ], + "score": 0.89, + "content": "- \\log p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 125, + 345, + 141 + ], + "score": 1.0, + "content": "term has the same role as AnoGAN’s", + "type": "text" + }, + { + "bbox": [ + 346, + 127, + 441, + 139 + ], + "score": 0.92, + "content": "| | f _ { D } ( \\mathbf { x } ) - f _ { D } ( G ( \\mathbf { z } ) ) | | _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 125, + 506, + 141 + ], + "score": 1.0, + "content": "term, to ensure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 124, + 151 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 138, + 156, + 150 + ], + "score": 0.92, + "content": "D e c ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 137, + 420, + 151 + ], + "score": 1.0, + "content": "stays within the learned manifold. We chose not to iterate over", + "type": "text" + }, + { + "bbox": [ + 420, + 140, + 427, + 148 + ], + "score": 0.66, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "in the latent space", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "for two reasons. First, because as noted in Dai & Wipf (2019) and Hoffman & Johnson (2016), the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 128, + 173 + ], + "score": 1.0, + "content": "prior", + "type": "text" + }, + { + "bbox": [ + 129, + 160, + 147, + 172 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 160, + 482, + 173 + ], + "score": 1.0, + "content": "is not always a good proxy for the real image of the distribution in the latent space", + "type": "text" + }, + { + "bbox": [ + 482, + 160, + 501, + 172 + ], + "score": 0.91, + "content": "q ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 160, + 504, + 173 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "Second, because the VAE tends to ignore some details of the original image in its reconstruction,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "considering that these details are part of the independent pixel noise allowed by the modeling of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 134, + 205 + ], + "score": 0.92, + "content": "p ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "as a diagonal Gaussian, which causes its infamous blurriness. An optimization in latent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "space would have to recreate the high frequency structure of the image, whereas iterating over the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 361, + 227 + ], + "score": 1.0, + "content": "input image space, and starting the descent on the input image", + "type": "text" + }, + { + "bbox": [ + 361, + 216, + 372, + 226 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 216, + 505, + 227 + ], + "score": 1.0, + "content": ", allows us to keep that structure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 297, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 297, + 239 + ], + "score": 1.0, + "content": "and thus to obtain projections of higher quality.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 125, + 506, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 250, + 273, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 275, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 275, + 263 + ], + "score": 1.0, + "content": "3.4 OPTIMIZING GRADIENT DESCENT", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "We observed that using the Adam optimizer (Kingma & Ba, 2015) is beneficial for the quality of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "optimization. Moreover, to speed up the convergence and further preserve the aforementioned high", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "frequency structure of the input, we propose to compute our iterative samples using the pixel-wise", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "reconstruction error of the VAE. To explain the intuition behind this improvement, we will consider", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "the inpainting task. In this setting, as in anomaly localization, a local perturbation is added on top of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "a normal image. However, in the classic inpainting task, the localization of the perturbation is known", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 302, + 349 + ], + "score": 1.0, + "content": "beforehand, and we can use the localization mask", + "type": "text" + }, + { + "bbox": [ + 303, + 337, + 312, + 347 + ], + "score": 0.82, + "content": "\\pmb { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "to only change the value of the anomalous pixels", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 202, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 202, + 361 + ], + "score": 1.0, + "content": "in the gradient descent:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 270, + 506, + 361 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 357, + 378, + 371 + ], + "lines": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "spans": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "score": 0.93, + "content": "\\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\alpha \\cdot \\left( \\nabla _ { \\mathbf { x } } E ( \\mathbf { x } _ { t } ) \\odot \\Omega \\right)", + "type": "interline_equation", + "image_path": "f23f9b783749f4859ba4a788ef18043a7df6f407c9b7dc834ea915047c2fa106.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 233, + 357, + 378, + 371 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 245, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 247, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 133, + 386 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 374, + 142, + 383 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 371, + 247, + 386 + ], + "score": 1.0, + "content": "is the Hadamard product.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 371, + 247, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 389, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 403 + ], + "score": 1.0, + "content": "For anomaly localization and blind inpainting, where this information is not available, we compute", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 455, + 413 + ], + "score": 1.0, + "content": "the pixel-wise reconstruction error which gives a rough estimate of the mask. The term", + "type": "text" + }, + { + "bbox": [ + 455, + 401, + 495, + 413 + ], + "score": 0.94, + "content": "\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 397, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 201, + 425 + ], + "score": 1.0, + "content": "therefore replaced with", + "type": "text" + }, + { + "bbox": [ + 201, + 411, + 335, + 424 + ], + "score": 0.82, + "content": "\\nabla _ { \\mathbf x } E ( \\mathbf x _ { t } ) \\odot ( { \\mathbf x _ { t } } - f _ { V A E } ( \\mathbf { x } _ { t } ) ) ^ { 2 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 411, + 397, + 425 + ], + "score": 1.0, + "content": ") in equation 5:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 388, + 505, + 425 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 428, + 411, + 442 + ], + "lines": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "spans": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "score": 0.87, + "content": "\\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)", + "type": "interline_equation", + "image_path": "af7ae15f47a600f88726caf7fdc9cf674f2a0aee385c0a8de68c0eafaf942287.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 199, + 428, + 411, + 442 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 133, + 459 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 446, + 172, + 459 + ], + "score": 0.93, + "content": "f _ { V A E } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "is the standard reconstruction of the VAE. Optimizing the energy this way, a pixel", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 458, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 469 + ], + "score": 1.0, + "content": "where the reconstruction error is high will update faster, whereas a pixel with good reconstruction", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "will not change easily. This prevents the image to update its pixels where the reconstruction is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "already good, even with a high learning rate. As can be seen in appendix B, this method converges", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 502 + ], + "score": 1.0, + "content": "to the same performance as the method of equation 5, but with fewer iterations. An illustration of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 255, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 255, + 514 + ], + "score": 1.0, + "content": "our method can be found in figure 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 446, + 506, + 514 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 526, + 204, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 205, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 205, + 538 + ], + "score": 1.0, + "content": "3.5 STOP CRITERION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "A standard stop criterion based on the convergence of the energy can efficiently be used. Using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "the adversarial setting introduced in section 3.1, we also propose to stop the gradient descent when", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 570, + 504, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 504, + 580 + ], + "score": 1.0, + "content": "a certain predefined threshold on the VAE loss is reached. For example, such a threshold can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 580, + 450, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 450, + 591 + ], + "score": 1.0, + "content": "chosen to be a quantile of the empirical loss distribution computed on the training set.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 546, + 505, + 591 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 200, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 201, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 201, + 621 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "In this section, we evaluate the proposed method for two different applications: anomaly segmen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "tation and image inpainting. Both applications are interesting use cases of our method, where we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "search to reconstruct partially corrupted images, correcting the anomalies while preserving the un-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 665, + 208, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 208, + 677 + ], + "score": 1.0, + "content": "corrupted image regions.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 631, + 506, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 689, + 313, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 315, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 315, + 701 + ], + "score": 1.0, + "content": "4.1 UNSUPERVISED ANOMALY SEGMENTATION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In order to evaluate the proposed method for the task of anomaly segmentation, we perform experi-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "ments with the recently proposed MVTec dataset (Bergmann et al., 2019). This collection of datasets", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 505, + 92 + ], + "score": 1.0, + "content": "Table 1: Results for anomaly segmentation on MVTec datasets, expressed in AUROC (Area Un-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 504, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 504, + 103 + ], + "score": 1.0, + "content": "der the Receiver Operating Characteristics). 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It can be seen that anomaly segmentation is refined by our proposed method, with a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 580, + 274, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 274, + 592 + ], + "score": 1.0, + "content": "tendency of detecting less false positives.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 617, + 183, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 184, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 184, + 630 + ], + "score": 1.0, + "content": "4.2 INPAINTING", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "Image inpainting is a well known image reconstruction problem which consists of reconstructing a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "corrupted or missing part of an image, where the region to be reconstructed is usually given by a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "known mask. Many different approaches for inpainting have been proposed in the literature, such", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "as anisotropic diffusion (Bertalmio et al., 2000), patch matching (Criminisi et al., 2004), context", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 682, + 493, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 493, + 694 + ], + "score": 1.0, + "content": "autoencoders (Pathak et al., 2016) and conditional variational autoencoders (Ivanov et al., 2019).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 638, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "If we consider that the region to be reconstructed is not known beforehand, the problem is sometimes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "called blind inpainting (Altinel et al., 2018), and the corrupted part can be seen as an anomaly to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 721, + 148, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 148, + 731 + ], + "score": 1.0, + "content": "corrected.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 698, + 505, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 148, + 89, + 463, + 271 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 148, + 89, + 463, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 89, + 463, + 271 + ], + "spans": [ + { + "bbox": [ + 148, + 89, + 463, + 271 + ], + "score": 0.975, + "type": "image", + "image_path": "3c525dc352b059d364d4bd4eb4992154d606ff497f94c348f4d97f8db18a4df2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 148, + 89, + 463, + 149.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 148, + 149.66666666666666, + 463, + 210.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 148, + 210.33333333333331, + 463, + 271.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 280, + 504, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "Figure 4: Inpainting experiment performed on CelebA dataset, where the test face images are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "masked with uniform noise. The baseline VAE reconstruction is disturbed by the noise mask, pro-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "viding a poor inpainting. The proposed gradient-based VAE provides a more convincing inpainting", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 313, + 200, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 200, + 326 + ], + "score": 1.0, + "content": "by an iterative process.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "We performed experiments with image inpainting on the CelebA dataset (Liu et al., 2015), which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "consists of celebrity faces. In figure 4 we compare the inpainting results obtained with a baseline", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 220, + 385 + ], + "score": 1.0, + "content": "VAE with learned variance", + "type": "text" + }, + { + "bbox": [ + 220, + 373, + 253, + 385 + ], + "score": 0.29, + "content": "\\left( \\gamma { \\mathrm { - V A E } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "and Resnet architecture, as described by Dai & Wipf (2019),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "with the same VAE model, augmented by our proposed gradient-based iterative reconstruction. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "that for the regular inpainting task, gradients are multiplied by the inpainting mask at each iteration", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "(equation 7), while for the blind inpainting task, the mask is unknown. See appendix D for a com-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 416, + 500, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 500, + 430 + ], + "score": 1.0, + "content": "parison with a recent method based on variational autoencoders, proposed by Ivanov et al. (2019).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 450, + 211, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 213, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 213, + 465 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Baur et al. (2018) have used autoencoder reconstructions to localize anomalies in MRI scans, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "have compared several variants using diverse per-pixel distances as well as perceptual metrics de-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "rived from a GAN-like architecture. Bergmann et al. (2018) use the structural similarity metric", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "(Wang et al., 2004) to compare the original image and its reconstruction to achieve better anomaly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 523, + 488, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 488, + 535 + ], + "score": 1.0, + "content": "localization, and also presents the SSIM autoencoder, which is trained directly with this metric.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "Zimmerer et al. (2018) use the derivative of the VAE loss function with respect to the input, called", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "the score. The amplitude of the score is supposed to indicate how abnormal a pixel is. While we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "agree that the gradient of the loss is an indication of an anomaly, we think that we have to integrate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "this gradient over the path from the input to the normal manifold to obtain meaningful information.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 453, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 453, + 595 + ], + "score": 1.0, + "content": "We compare our results to score-based results for anomaly localization in appendix A.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "The work that is the most related to ours is AnoGAN (Schlegl et al., 2017). We have mentioned", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "above the differences between the two approaches, which, apart from the change in underlying ar-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "chitectures, boil down to the ability in our method to update directly the input image instead of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "searching for the optimal latent code. This enables the method to converge faster and above all to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "keep higher-frequency structures of the input, which would have been deteriorated if it were passed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "through the AE bottleneck. Bergmann et al. (2019) compare standard AE reconstructions tech-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "niques to AnoGAN, and observes that AnoGAN’s performances on anomaly localizations tasks are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "poorer than AE’s due to the mode collapse tendency of GAN architectures. Interestingly, updates on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "AnoGAN such as fast AnoGAN (Schlegl et al., 2019) or AnoVAEGAN (Baur et al., 2018) replaced", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the gradient descent search of the optimal z with a learned encoder model, yielding an approach", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "very similar to the standard VAE reconstruction-based approaches, but with a reconstruction loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 721, + 471, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 471, + 732 + ], + "score": 1.0, + "content": "learned by a discriminator, which is still prone to mode collapse (Thanh-Tung et al., 2019).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 148, + 89, + 463, + 271 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 148, + 89, + 463, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 89, + 463, + 271 + ], + "spans": [ + { + "bbox": [ + 148, + 89, + 463, + 271 + ], + "score": 0.975, + "type": "image", + "image_path": "3c525dc352b059d364d4bd4eb4992154d606ff497f94c348f4d97f8db18a4df2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 148, + 89, + 463, + 149.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 148, + 149.66666666666666, + 463, + 210.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 148, + 210.33333333333331, + 463, + 271.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 280, + 504, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "Figure 4: Inpainting experiment performed on CelebA dataset, where the test face images are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "masked with uniform noise. The baseline VAE reconstruction is disturbed by the noise mask, pro-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "viding a poor inpainting. The proposed gradient-based VAE provides a more convincing inpainting", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 313, + 200, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 200, + 326 + ], + "score": 1.0, + "content": "by an iterative process.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "We performed experiments with image inpainting on the CelebA dataset (Liu et al., 2015), which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "consists of celebrity faces. In figure 4 we compare the inpainting results obtained with a baseline", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 220, + 385 + ], + "score": 1.0, + "content": "VAE with learned variance", + "type": "text" + }, + { + "bbox": [ + 220, + 373, + 253, + 385 + ], + "score": 0.29, + "content": "\\left( \\gamma { \\mathrm { - V A E } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "and Resnet architecture, as described by Dai & Wipf (2019),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "with the same VAE model, augmented by our proposed gradient-based iterative reconstruction. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "that for the regular inpainting task, gradients are multiplied by the inpainting mask at each iteration", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "(equation 7), while for the blind inpainting task, the mask is unknown. See appendix D for a com-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 416, + 500, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 500, + 430 + ], + "score": 1.0, + "content": "parison with a recent method based on variational autoencoders, proposed by Ivanov et al. (2019).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 351, + 505, + 430 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 450, + 211, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 213, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 213, + 465 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Baur et al. (2018) have used autoencoder reconstructions to localize anomalies in MRI scans, and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "have compared several variants using diverse per-pixel distances as well as perceptual metrics de-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "rived from a GAN-like architecture. Bergmann et al. (2018) use the structural similarity metric", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "(Wang et al., 2004) to compare the original image and its reconstruction to achieve better anomaly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 523, + 488, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 488, + 535 + ], + "score": 1.0, + "content": "localization, and also presents the SSIM autoencoder, which is trained directly with this metric.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 479, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "Zimmerer et al. (2018) use the derivative of the VAE loss function with respect to the input, called", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "the score. The amplitude of the score is supposed to indicate how abnormal a pixel is. While we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "agree that the gradient of the loss is an indication of an anomaly, we think that we have to integrate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "this gradient over the path from the input to the normal manifold to obtain meaningful information.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 582, + 453, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 453, + 595 + ], + "score": 1.0, + "content": "We compare our results to score-based results for anomaly localization in appendix A.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 539, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "The work that is the most related to ours is AnoGAN (Schlegl et al., 2017). We have mentioned", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "above the differences between the two approaches, which, apart from the change in underlying ar-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "chitectures, boil down to the ability in our method to update directly the input image instead of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "searching for the optimal latent code. This enables the method to converge faster and above all to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "keep higher-frequency structures of the input, which would have been deteriorated if it were passed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "through the AE bottleneck. Bergmann et al. (2019) compare standard AE reconstructions tech-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "niques to AnoGAN, and observes that AnoGAN’s performances on anomaly localizations tasks are", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "poorer than AE’s due to the mode collapse tendency of GAN architectures. Interestingly, updates on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "AnoGAN such as fast AnoGAN (Schlegl et al., 2019) or AnoVAEGAN (Baur et al., 2018) replaced", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the gradient descent search of the optimal z with a learned encoder model, yielding an approach", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "very similar to the standard VAE reconstruction-based approaches, but with a reconstruction loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 721, + 471, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 471, + 732 + ], + "score": 1.0, + "content": "learned by a discriminator, which is still prone to mode collapse (Thanh-Tung et al., 2019).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 600, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 195, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 197, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 197, + 97 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this paper, we proposed a novel method for unsupervised anomaly localization, using gradient", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "descent of an energy defined by an autoencoder reconstruction loss. Starting from a sample under", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "test, we iteratively update this sample to reduce its autoencoder reconstruction error. This method", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 151 + ], + "score": 1.0, + "content": "offers a way to incorporate human priors into what is the optimal projection of an out-of-distribution", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "sample into the normal data manifold. 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Using gradient descent in the input data space, starting from the input sample, enables us", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "to overcome the autoencoder tendency to provide blurry reconstructions and keep normal high fre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "quency structures. This significantly reduces the number of pixels that could be wrongly classified", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "as defects when the autoencoder fails to reconstruct high frequencies. We showed that this method,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "which can easily be added to any previously trained autoencoder architecture, gives state-of-the-art", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "results on a variety of unsupervised anomaly localization datasets, as well as qualitative reconstruc-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 385, + 251 + ], + "score": 1.0, + "content": "tions on an inpainting task. Future work can focus on replacing the", + "type": "text" + }, + { + "bbox": [ + 385, + 237, + 397, + 248 + ], + "score": 0.88, + "content": "L ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "-based regularization term", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "with a Bayesian prior modeling common types of anomalies, and on further improving the speed of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 191, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 191, + 273 + ], + "score": 1.0, + "content": "the gradient descent.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 288, + 175, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 176, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 176, + 301 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "score": 1.0, + "content": "Fazil Altinel, Mete Ozay, and Takayuki Okatani. 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0000000000000000000000000000000000000000..4cd7f2f04dd1ab7da269f904ec8182f14153c611 --- /dev/null +++ b/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. + +
Task (mean ± std.) of sequence lengthListOps (888±339)Text (1296 ± 893)Retrieval (3987 ± 560)Average
ModelAcc.FLOPsAcc.FLOPsAcc.FLOPsAcc.
Full Attention [1]37.131.2165.354.5782.309.1461.59
Reformer [31] (2)36.440.2764.880.5878.641.1559.99
Linformer [17] (k=256)37.380.4156.120.8179.371.6257.62
Performer [28] (r = 256)32.780.4165.210.8281.701.6359.90
Nystromformer[18](l =128)37.340.6165.751.0281.292.0361.46
Transformer-LS (w,r = 8,32)37.500.2066.010.4081.790.8061.77
Dynamic Projection (best)37.790.1566.280.6981.862.1761.98
Transformer-LS (best)38.360.1668.400.2981.952.1762.90
+ +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. + +
Task Test PerturbTextRetrieval
NoneInsertionDeletionNoneInsertionDeletion
Linformer56.1255.9454.9179.3753.6651.75
DP66.2863.1658.9581.8670.0164.98
Linformer+ Win.59.6356.6956.2979.6852.8352.13
DP + Win. (ours)68.4066.3462.6281.9569.9364.19
+ +Table 3: Comparing the results of pretrained language models fine-tuned on IMDb. + +
ModelRoBERTa-base RoBERTa-large Longformer-base LS-base LS-large
Accuracy95.396.595.796.096.8
+ +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). + +
Method#Paramtext8
Dev Testenwik8 DevTest
T12 [49] Transformer-XL [9]44M1 1.1811.11
Reformer [31]41M -1 1 - 11 11.06 1.05
Adaptive [50]38M1.05 1.111.041.02
BP-Transformer [51] Longformer [20]38M 41M1 1.11 1.04 1.101 1.021.02 1.00
+ +
Method #Param‘ TestBPC
Transformer-XL [9]88M1.03 0.99
Transformer-XL [9] Routing [32]277M 223M0.99
Longformer [14]102M0.99
Sparse [12]95M0.99
Adaptive [50]209M0.98
Compressive [10]227M0.97
Transformer-LS110M0.97
+ +# 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. + +
Model(M)Size#Param Image FLOPs ImageNet (G)top-1 (%)Real top-1 (%) top-1 (%)V2Latency (s）
ResNet-5025224²4.176.282.563.3=
ResNet-1014522427.977.483.765.7
ResNet-1526022421178.384.167.0=
DeiT-S [36]22224²4.679.885.768.5
DeiT-B [36]86224217.681.886.770.9
PVT-Medium [5]4422426.781.211
PVT-Large [5]6122429.881.7
Swin-S [38]5022428.783.2
Swin-B[38]88224215.483.50.115
PVTv2-B4 [54]62.6224210.183.6
PVTv2-B5 [54]82.0224211.883.8
ViT-B/16 [4]86384255.577.9
ViT-L/16 [4]3073842191.176.5
DeiT-B [36]86384255.583.1
Swin-B [38]88384247.184.5110.378
CvT-13 [6]2022426.781.686.770.40.122
CvT-21 [6]32224210.182.587.271.30.165
CvT*-LS-1320.322424.981.987.070.50.083
CvT*-LS-1723.722429.882.587.271.6-
CvT*-LS-2132.1224²7.982.787.571.90.122
CvT*-LS-21S30.1224211.382.987.471.7-
CvT-13 [6]20384231.983.087.971.91
CvT-21 [6]32384245.083.387.771.9
CvT*-LS-2132.1384223.983.288.072.5
CvT*-LS-2132.1448234.283.688.272.9
ViL-Small [14]24.622424.982.4-1
ViL-Medium [14]39.722428.783.50.106
ViL-Base [14]55.7224213.483.70.164
ViL-LS-Medium39.822428.783.80.075
ViL-LS-Base55.8224²13.484.10.113
ViL-LS-Medium39.9384228.784.40.271
+ +# 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. + +
Model[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]ImageNet-9 [59]
(M)Top-1mCE(↓）Acc.Acc.|Mixed-same Mixed-rand
ResNet-50 [35]]25.676.278.96.235.387.181.6
DeiT-S [36]22.179.857.119.041.989.184.2
CvT-132081.659.625.442.990.585.7
CvT-213282.556.231.142.690.585.0
CvT*-LS-1320.381.958.727.042.690.785.6
CvT*-LS-2132.182.755.229.345.091.585.8
+ +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. 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Micro-batch training with batch-channel normalization and weight standardization. arXiv preprint arXiv:1903.10520, 2019. \ No newline at end of file diff --git a/parse/train/M_lkFOwVdYc/M_lkFOwVdYc_content_list.json b/parse/train/M_lkFOwVdYc/M_lkFOwVdYc_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..edb14edac49047740ea160517a95f689811adbf8 --- /dev/null +++ b/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, + 797, + 172 + ], + "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": [ + 220, + 222, + 777, + 252 + ], + "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, + 746, + 306 + ], + "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. ", + "bbox": [ + 232, + 378, + 766, + 667 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 693, + 310, + 710 + ], + "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": [ + 174, + 724, + 825, + 877 + ], + "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]. ", + "bbox": [ + 173, + 92, + 825, + 257 + ], + "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": [ + 174, + 263, + 825, + 333 + ], + "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": [ + 174, + 339, + 825, + 395 + ], + "page_idx": 1 + }, + { + "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": [ + 178, + 406, + 826, + 678 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 699, + 321, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Efficient Transformers ", + "text_level": 1, + "bbox": [ + 174, + 732, + 370, + 747 + ], + "page_idx": 1 + }, + { + "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": [ + 174, + 758, + 825, + 911 + ], + "page_idx": 1 + }, + { + "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": [], + "bbox": [ + 176, + 92, + 821, + 183 + ], + "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": [ + 176, + 310, + 823, + 339 + ], + "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": [ + 173, + 345, + 825, + 497 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Vision Transformers ", + "text_level": 1, + "bbox": [ + 174, + 515, + 356, + 530 + ], + "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": [ + 173, + 540, + 825, + 693 + ], + "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": [ + 174, + 698, + 825, + 837 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Long-Short Transformer ", + "text_level": 1, + "bbox": [ + 176, + 853, + 413, + 871 + ], + "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": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Preliminaries and Notations ", + "text_level": 1, + "bbox": [ + 176, + 162, + 408, + 178 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 189, + 825, + 228 + ], + "page_idx": 3 + }, + { + "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": [ + 320, + 227, + 674, + 246 + ], + "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": [ + 174, + 248, + 826, + 291 + ], + "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, + 790, + 340 + ], + "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": [ + 174, + 347, + 826, + 405 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Short-term Attention via Segment-wise Sliding Window ", + "text_level": 1, + "bbox": [ + 174, + 417, + 599, + 433 + ], + "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, + 825, + 595 + ], + "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": [ + 174, + 602, + 825, + 698 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 Long-range Attention via Dynamic Projections ", + "text_level": 1, + "bbox": [ + 174, + 712, + 539, + 727 + ], + "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": [ + 174, + 734, + 825, + 804 + ], + "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": [ + 174, + 810, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 89, + 825, + 146 + ], + "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, + 147, + 687, + 166 + ], + "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, + 169, + 825, + 227 + ], + "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": [ + 173, + 231, + 823, + 261 + ], + "page_idx": 4 + }, + { + "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": [ + 334, + 262, + 661, + 325 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 328, + 825, + 398 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 173, + 404, + 826, + 542 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 215, + 546, + 779, + 565 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 566, + 830, + 608 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 Aggregating Long-range and Short-term Attentions ", + "text_level": 1, + "bbox": [ + 173, + 623, + 573, + 638 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 173, + 648, + 826, + 762 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3fd00b25c1aa2ca8bb9d094c5b8c300f3e6c6e50b3f082fc8e95103c63b22d78.jpg", + "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", + "bbox": [ + 315, + 765, + 681, + 816 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 819, + 825, + 877 + ], + "page_idx": 4 + }, + { + "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": [ + 187, + 92, + 803, + 247 + ], + "page_idx": 5 + }, + { + "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": [ + 173, + 337, + 826, + 520 + ], + "page_idx": 5 + }, + { + "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 ", + "bbox": [ + 176, + 526, + 823, + 582 + ], + "page_idx": 5 + }, + { + "type": "equation", + "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": [ + 225, + 587, + 771, + 640 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 643, + 826, + 713 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 729, + 313, + 747 + ], + "page_idx": 5 + }, + { + "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]. ", + "bbox": [ + 174, + 758, + 823, + 787 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Bidirectional Modeling on Long Range Arena and IMDb ", + "text_level": 1, + "bbox": [ + 173, + 801, + 607, + 818 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 5 + }, + { + "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": "
Task (mean ± std.) of sequence lengthListOps (888±339)Text (1296 ± 893)Retrieval (3987 ± 560)Average
ModelAcc.FLOPsAcc.FLOPsAcc.FLOPsAcc.
Full Attention [1]37.131.2165.354.5782.309.1461.59
Reformer [31] (2)36.440.2764.880.5878.641.1559.99
Linformer [17] (k=256)37.380.4156.120.8179.371.6257.62
Performer [28] (r = 256)32.780.4165.210.8281.701.6359.90
Nystromformer[18](l =128)37.340.6165.751.0281.292.0361.46
Transformer-LS (w,r = 8,32)37.500.2066.010.4081.790.8061.77
Dynamic Projection (best)37.790.1566.280.6981.862.1761.98
Transformer-LS (best)38.360.1668.400.2981.952.1762.90
", + "bbox": [ + 176, + 117, + 823, + 290 + ], + "page_idx": 6 + }, + { + "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": "
Task Test PerturbTextRetrieval
NoneInsertionDeletionNoneInsertionDeletion
Linformer56.1255.9454.9179.3753.6651.75
DP66.2863.1658.9581.8670.0164.98
Linformer+ Win.59.6356.6956.2979.6852.8352.13
DP + Win. (ours)68.4066.3462.6281.9569.9364.19
", + "bbox": [ + 240, + 338, + 756, + 430 + ], + "page_idx": 6 + }, + { + "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": "
ModelRoBERTa-base RoBERTa-large Longformer-base LS-base LS-large
Accuracy95.396.595.796.096.8
", + "bbox": [ + 256, + 464, + 740, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 531, + 823, + 560 + ], + "page_idx": 6 + }, + { + "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": [ + 176, + 566, + 823, + 608 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 614, + 825, + 849 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/4e35688918560b95df1faea3ced37447c973045ec764520f5c395363eee27938.jpg", + "image_caption": [ + "Forward-backprop Time per Iteration (Autoregressive LM) " + ], + "image_footnote": [], + "bbox": [ + 504, + 99, + 787, + 217 + ], + "page_idx": 7 + }, + { + "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": [ + 205, + 98, + 491, + 217 + ], + "page_idx": 7 + }, + { + "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": "
Method#Paramtext8
Dev Testenwik8 DevTest
T12 [49] Transformer-XL [9]44M1 1.1811.11
Reformer [31]41M -1 1 - 11 11.06 1.05
Adaptive [50]38M1.05 1.111.041.02
BP-Transformer [51] Longformer [20]38M 41M1 1.11 1.04 1.101 1.021.02 1.00
", + "bbox": [ + 196, + 332, + 519, + 468 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a82c29f64795331a53468152cf14d4c8d87aca0d05b0b0e431c67ba64026a2f7.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Method #Param‘ TestBPC
Transformer-XL [9]88M1.03 0.99
Transformer-XL [9] Routing [32]277M 223M0.99
Longformer [14]102M0.99
Sparse [12]95M0.99
Adaptive [50]209M0.98
Compressive [10]227M0.97
Transformer-LS110M0.97
", + "bbox": [ + 562, + 332, + 812, + 469 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 503, + 825, + 532 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Autoregressive Language Modeling ", + "text_level": 1, + "bbox": [ + 174, + 558, + 459, + 573 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare our method with other efficient transformers on the character-level language modeling where each input token is a character. ", + "bbox": [ + 174, + 583, + 823, + 611 + ], + "page_idx": 7 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 618, + 825, + 811 + ], + "page_idx": 7 + }, + { + "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. ", + "bbox": [ + 174, + 818, + 823, + 915 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/c0369ed9249dfb7b22f5e42da70b51fa1c09e3e52d28a99d1162db0f458831d9.jpg", + "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": "
Model(M)Size#Param Image FLOPs ImageNet (G)top-1 (%)Real top-1 (%) top-1 (%)V2Latency (s）
ResNet-5025224²4.176.282.563.3=
ResNet-1014522427.977.483.765.7
ResNet-1526022421178.384.167.0=
DeiT-S [36]22224²4.679.885.768.5
DeiT-B [36]86224217.681.886.770.9
PVT-Medium [5]4422426.781.211
PVT-Large [5]6122429.881.7
Swin-S [38]5022428.783.2
Swin-B[38]88224215.483.50.115
PVTv2-B4 [54]62.6224210.183.6
PVTv2-B5 [54]82.0224211.883.8
ViT-B/16 [4]86384255.577.9
ViT-L/16 [4]3073842191.176.5
DeiT-B [36]86384255.583.1
Swin-B [38]88384247.184.5110.378
CvT-13 [6]2022426.781.686.770.40.122
CvT-21 [6]32224210.182.587.271.30.165
CvT*-LS-1320.322424.981.987.070.50.083
CvT*-LS-1723.722429.882.587.271.6-
CvT*-LS-2132.1224²7.982.787.571.90.122
CvT*-LS-21S30.1224211.382.987.471.7-
CvT-13 [6]20384231.983.087.971.91
CvT-21 [6]32384245.083.387.771.9
CvT*-LS-2132.1384223.983.288.072.5
CvT*-LS-2132.1448234.283.688.272.9
ViL-Small [14]24.622424.982.4-1
ViL-Medium [14]39.722428.783.50.106
ViL-Base [14]55.7224213.483.70.164
ViL-LS-Medium39.822428.783.80.075
ViL-LS-Base55.8224²13.484.10.113
ViL-LS-Medium39.9384228.784.40.271
", + "bbox": [ + 240, + 166, + 754, + 657 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 ImageNet Classification ", + "text_level": 1, + "bbox": [ + 174, + 678, + 379, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 696, + 825, + 849 + ], + "page_idx": 8 + }, + { + "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. ", + "bbox": [ + 174, + 856, + 823, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c9c2c6285db46277e4ed88f715bcfbbd4ce6709989fa3a86f3b3bd210315e90d.jpg", + "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": "
Model[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]ImageNet-9 [59]
(M)Top-1mCE(↓）Acc.Acc.|Mixed-same Mixed-rand
ResNet-50 [35]]25.676.278.96.235.387.181.6
DeiT-S [36]22.179.857.119.041.989.184.2
CvT-132081.659.625.442.990.585.7
CvT-213282.556.231.142.690.585.0
CvT*-LS-1320.381.958.727.042.690.785.6
CvT*-LS-2132.182.755.229.345.091.585.8
", + "bbox": [ + 223, + 118, + 771, + 234 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 252, + 825, + 390 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 397, + 825, + 467 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 173, + 472, + 825, + 583 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 589, + 825, + 741 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 758, + 299, + 776 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 174, + 786, + 825, + 911 + ], + "page_idx": 9 + }, + { + "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. 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For instance, Transformer-LS", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "score": 1.0, + "content": "achieves 0.97 test BPC on enwik8 using half the number of parameters than", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 464, + 470, + 476 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 378, + 476 + ], + "score": 1.0, + "content": "previous method, while being faster and is able to handle", + "type": "text" + }, + { + "bbox": [ + 378, + 464, + 392, + 474 + ], + "score": 0.88, + "content": "3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 464, + 470, + 476 + ], + "score": 1.0, + "content": "as long sequences", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 475, + 469, + 486 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 469, + 486 + ], + "score": 1.0, + "content": "compared to its full-attention version on the same hardware. On ImageNet, it can", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 486, + 470, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 384, + 498 + ], + "score": 1.0, + "content": "obtain the state-of-the-art results (e.g., a moderate size of", + "type": "text" + }, + { + "bbox": [ + 384, + 486, + 412, + 496 + ], + "score": 0.32, + "content": "5 5 . 8 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 486, + 470, + 498 + ], + "score": 1.0, + "content": "model solely", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 496, + 469, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 183, + 509 + ], + "score": 1.0, + "content": "trained on", + "type": "text" + }, + { + "bbox": [ + 184, + 497, + 226, + 507 + ], + "score": 0.88, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 496, + 387, + 509 + ], + "score": 1.0, + "content": "ImageNet-1K can obtain Top-1 accuracy", + "type": "text" + }, + { + "bbox": [ + 388, + 497, + 415, + 507 + ], + "score": 0.83, + "content": "8 4 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 496, + 469, + 509 + ], + "score": 1.0, + "content": "), while being", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 507, + 470, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 507, + 470, + 519 + ], + "score": 1.0, + "content": "more scalable on high-resolution images. The source code and models are released", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 519, + 369, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 519, + 369, + 531 + ], + "score": 1.0, + "content": "at https://github.com/NVIDIA/transformer-ls.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18, + "bbox_fs": [ + 141, + 300, + 470, + 531 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 549, + 190, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 192, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 192, + 565 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "score": 1.0, + "content": "Transformer-based models [1] have achieved great success in the domains of natural language", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "processing (NLP) [2, 3] and computer vision [4–6]. These models benefit from the self-attention", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "module, which can capture both adjacent and long-range correlations between tokens while efficiently", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "scaling on modern hardware. However, the time and memory consumed by self-attention scale", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 632 + ], + "score": 1.0, + "content": "quadratically with the input length, making it very expensive to process long sequences. Many", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "language and vision tasks benefit from modeling long sequences. In NLP, document-level tasks", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "require processing long articles [e.g., 7, 8], and the performance of language models often increases", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "with sequence length [e.g., 9, 10]. In computer vision, many tasks involve high-resolution images,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "which are converted to long sequences of image patches before being processed with Transformer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "models [4, 6, 11]. As a result, it is crucial to design an efficient attention mechanism for long sequence", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 684, + 334, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 334, + 696 + ], + "score": 1.0, + "content": "modeling that generalizes well across different domains.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 573, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Various methods have been proposed to reduce the quadratic cost of full attention. However, an", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "efficient attention mechanism that generalizes well in both language and vision domains is less", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "explored. One family of methods is to sparsify the attention matrix with predefined patterns such", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "as sliding windows [e.g., 12–15] and random sparse patterns [16]. These methods leverage strong", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "inductive biases to improve both computational and model performance, but they limit the capacity", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "of a self-attention layer because each specific token can only attend to a subset of tokens. Another", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "family of methods leverages low-rank projections to form a low resolution representation of the input", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 147, + 507, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 507, + 164 + ], + "score": 1.0, + "content": "sequence, but the successful application of these methods has been limited to certain NLP tasks [e.g.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "17–19]. Unlike sparse attention, this family of methods allows each token to attend to the entire", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "input sequence. However, due to the loss of high-fidelity token-wise information, their performance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "sometimes is not as good as full attention or sparse attention on tasks that require fine-grained local", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 193, + 418, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 418, + 205 + ], + "score": 1.0, + "content": "information, including standard benchmarks in language [20] and vision [21].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "Despite the rapid progress in efficient Transformers, some proposed architectures can only be applied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "to bidirectional models [e.g., 15, 16, 18]. Transformer-based autoregressive models have achieved", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "score": 1.0, + "content": "great successes in language modeling [22], image synthesis [23], and text-to-image synthesis [24],", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "which also involve long texts or high-resolution images. It is desirable to design an efficient trans-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 406, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 406, + 264 + ], + "score": 1.0, + "content": "former that can be applied to both autoregressive and bidirectional models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "In this work, we unify a local window attention and a novel long-range attention into a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "efficient attention mechanism. We show that these two kinds of attention have complementary effects", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "that together yield the state-of-the-art results on a range of tasks in language and vision, for both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 473, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 473, + 315 + ], + "score": 1.0, + "content": "autoregressive and bidirectional models. Specifically, we make the following contributions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 109, + 322, + 506, + 537 + ], + "lines": [ + { + "bbox": [ + 109, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 109, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "• We propose Long-Short Transformer (Transformer-LS), an efficient Transformer that integrates a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 118, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 118, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "dynamic projection based attention to model long-range correlations, and a local window attention", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 117, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 117, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "to capture fine-grained correlations. Transformer-LS can be applied to both autoregressive and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 117, + 354, + 370, + 370 + ], + "spans": [ + { + "bbox": [ + 117, + 354, + 370, + 370 + ], + "score": 1.0, + "content": "bidirectional models with linear time and memory complexity.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 110, + 372, + 507, + 385 + ], + "spans": [ + { + "bbox": [ + 110, + 372, + 507, + 385 + ], + "score": 1.0, + "content": "• We compute a dynamic low-rank projection, which depends on the content of the input sequence.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 117, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 117, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "In contrast to previous low-rank projection methods, our dynamic projection method is more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 117, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 117, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "flexible and robust to semantic-preserving positional variations (e.g., insertion, paraphrasing).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 117, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 117, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "We demonstrate that it outperforms previous low-rank methods [17, 18] on Long Range Arena", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 118, + 417, + 186, + 428 + ], + "spans": [ + { + "bbox": [ + 118, + 417, + 186, + 428 + ], + "score": 1.0, + "content": "benchmark [20].", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 111, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 111, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "• We identify a scale mismatch problem between the embeddings from the long-range and short-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 117, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 117, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "term attentions, and design a simple but effective dual normalization strategy, termed DualLN, to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 117, + 455, + 420, + 468 + ], + "spans": [ + { + "bbox": [ + 117, + 455, + 420, + 468 + ], + "score": 1.0, + "content": "account for the mismatch and enhance the effectiveness of the aggregation.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 109, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 109, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "• We demonstrate that Long-Short Transformer, despite its low memory and runtime complexity,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 118, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 118, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "outperforms the state-of-the-art models on a set of tasks from Long Range Arena, and autore-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 117, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 117, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "gressive language modeling on enwik8 and text8. In addition, the proposed efficient attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 118, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 118, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "mechanism can be easily applied to the most recent vision transformer architectures [6, 11] and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 118, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 118, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "provides state-of-the-art results, while being more scalable to high-resolution images. We also", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 118, + 527, + 478, + 538 + ], + "spans": [ + { + "bbox": [ + 118, + 527, + 478, + 538 + ], + "score": 1.0, + "content": "investigate the robustness properties of the Transformer-LS on diverse ImageNet datasets.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 554, + 197, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 198, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 198, + 570 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 580, + 227, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 228, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 228, + 593 + ], + "score": 1.0, + "content": "2.1 Efficient Transformers", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "In recent years, many methods have been introduced for dealing with the quadratic cost of full atten-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "tion. In general, they can be categorized as follows: i) Sparse attention mechanism with predefined", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "patterns (e.g., sliding window), including Sparse Transformer [12], Image Transformer [13], Axial", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "Transformer [25] for modeling images, and Longformer [14], blockwise self-attention [26], ETC [15],", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "Big Bird [16] for modeling language. ii) Low-rank projection attention, including Linformer [17],", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Nyströmformer [18], Synthesizer [19]. For example, Linformer uses linear layers to project the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 223, + 680 + ], + "score": 1.0, + "content": "original high resolution keys", + "type": "text" + }, + { + "bbox": [ + 223, + 667, + 238, + 678 + ], + "score": 0.65, + "content": "( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 667, + 284, + 680 + ], + "score": 1.0, + "content": "and values", + "type": "text" + }, + { + "bbox": [ + 285, + 668, + 299, + 678 + ], + "score": 0.68, + "content": "( V )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 667, + 348, + 680 + ], + "score": 1.0, + "content": "with length", + "type": "text" + }, + { + "bbox": [ + 348, + 669, + 355, + 677 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 667, + 462, + 680 + ], + "score": 1.0, + "content": "to low resolution with size", + "type": "text" + }, + { + "bbox": [ + 463, + 669, + 469, + 677 + ], + "score": 0.43, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 668, + 503, + 678 + ], + "score": 0.74, + "content": "( r \\ll n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 667, + 506, + 680 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 225, + 691 + ], + "score": 1.0, + "content": "and allows all query tokens", + "type": "text" + }, + { + "bbox": [ + 225, + 678, + 239, + 689 + ], + "score": 0.76, + "content": "( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "to attend these compressed representations. iii) Memory-based", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "mechanisms like Compressive Transformer [10] and Set Transformer [27], which use extra mem-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "ories for caching global long-range information for use in computing attention between distant", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "tokens. iv) Kernel-based approximation of the attention matrix, including Performer [28], Linear", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Various methods have been proposed to reduce the quadratic cost of full attention. However, an", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "efficient attention mechanism that generalizes well in both language and vision domains is less", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "explored. One family of methods is to sparsify the attention matrix with predefined patterns such", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "as sliding windows [e.g., 12–15] and random sparse patterns [16]. These methods leverage strong", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "inductive biases to improve both computational and model performance, but they limit the capacity", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "of a self-attention layer because each specific token can only attend to a subset of tokens. Another", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "family of methods leverages low-rank projections to form a low resolution representation of the input", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 147, + 507, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 507, + 164 + ], + "score": 1.0, + "content": "sequence, but the successful application of these methods has been limited to certain NLP tasks [e.g.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "17–19]. Unlike sparse attention, this family of methods allows each token to attend to the entire", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "input sequence. However, due to the loss of high-fidelity token-wise information, their performance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "sometimes is not as good as full attention or sparse attention on tasks that require fine-grained local", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 193, + 418, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 418, + 205 + ], + "score": 1.0, + "content": "information, including standard benchmarks in language [20] and vision [21].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 72, + 507, + 205 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "Despite the rapid progress in efficient Transformers, some proposed architectures can only be applied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "to bidirectional models [e.g., 15, 16, 18]. Transformer-based autoregressive models have achieved", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 244 + ], + "score": 1.0, + "content": "great successes in language modeling [22], image synthesis [23], and text-to-image synthesis [24],", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "which also involve long texts or high-resolution images. It is desirable to design an efficient trans-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 406, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 406, + 264 + ], + "score": 1.0, + "content": "former that can be applied to both autoregressive and bidirectional models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 209, + 507, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "In this work, we unify a local window attention and a novel long-range attention into a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "efficient attention mechanism. We show that these two kinds of attention have complementary effects", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "that together yield the state-of-the-art results on a range of tasks in language and vision, for both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 473, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 473, + 315 + ], + "score": 1.0, + "content": "autoregressive and bidirectional models. Specifically, we make the following contributions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 269, + 506, + 315 + ] + }, + { + "type": "list", + "bbox": [ + 109, + 322, + 506, + 537 + ], + "lines": [ + { + "bbox": [ + 109, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 109, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "• We propose Long-Short Transformer (Transformer-LS), an efficient Transformer that integrates a", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 118, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "dynamic projection based attention to model long-range correlations, and a local window attention", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 117, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 117, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "to capture fine-grained correlations. Transformer-LS can be applied to both autoregressive and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 117, + 354, + 370, + 370 + ], + "spans": [ + { + "bbox": [ + 117, + 354, + 370, + 370 + ], + "score": 1.0, + "content": "bidirectional models with linear time and memory complexity.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 372, + 507, + 385 + ], + "spans": [ + { + "bbox": [ + 110, + 372, + 507, + 385 + ], + "score": 1.0, + "content": "• We compute a dynamic low-rank projection, which depends on the content of the input sequence.", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 117, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "In contrast to previous low-rank projection methods, our dynamic projection method is more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 117, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 117, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "flexible and robust to semantic-preserving positional variations (e.g., insertion, paraphrasing).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 117, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 117, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "We demonstrate that it outperforms previous low-rank methods [17, 18] on Long Range Arena", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 118, + 417, + 186, + 428 + ], + "spans": [ + { + "bbox": [ + 118, + 417, + 186, + 428 + ], + "score": 1.0, + "content": "benchmark [20].", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 111, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "• We identify a scale mismatch problem between the embeddings from the long-range and short-", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 117, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "term attentions, and design a simple but effective dual normalization strategy, termed DualLN, to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 117, + 455, + 420, + 468 + ], + "spans": [ + { + "bbox": [ + 117, + 455, + 420, + 468 + ], + "score": 1.0, + "content": "account for the mismatch and enhance the effectiveness of the aggregation.", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 109, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 109, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "• We demonstrate that Long-Short Transformer, despite its low memory and runtime complexity,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 118, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "outperforms the state-of-the-art models on a set of tasks from Long Range Arena, and autore-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 117, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 117, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "gressive language modeling on enwik8 and text8. In addition, the proposed efficient attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 118, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 118, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "mechanism can be easily applied to the most recent vision transformer architectures [6, 11] and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 118, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 118, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "provides state-of-the-art results, while being more scalable to high-resolution images. 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See Figure 1 for", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 294, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 294, + 118 + ], + "score": 1.0, + "content": "an illustration of our long-short term attention.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "introduce the preliminaries of multi-head attention in Transformer. 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For simplicity, our discussion below is based on the case of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "1D input sequences. 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Note that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 107, + 157, + 138, + 167 + ], + "score": 0.9, + "content": "K = V", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 157, + 324, + 169 + ], + "score": 1.0, + "content": "in all the experiments we have considered, so", + "type": "text" + }, + { + "bbox": [ + 324, + 157, + 335, + 168 + ], + "score": 0.87, + "content": "P _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 157, + 473, + 169 + ], + "score": 1.0, + "content": "remains the same if it depends on", + "type": "text" + }, + { + "bbox": [ + 474, + 157, + 482, + 167 + ], + "score": 0.79, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 157, + 505, + 169 + ], + "score": 1.0, + "content": ". 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Note the effective attention", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "weights of a query on all tokens still sum to 1. Our global attention allows each query to attend", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "to all token embeddings within the same self-attention layer. In contrast, the sparse attention", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 304, + 404, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 404, + 316 + ], + "score": 1.0, + "content": "mechanisms [14, 16] need stack multiple layers to build such correlations.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 259, + 506, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "Application to Autoregressive Models: In autoregressive models, each token can only attend to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "the previous tokens, so the long-range attention should have a different range for different tokens. A", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 371, + 354 + ], + "score": 1.0, + "content": "straightforward way to implement our global attention is to update", + "type": "text" + }, + { + "bbox": [ + 372, + 342, + 398, + 353 + ], + "score": 0.72, + "content": "\\bar { K } _ { i } , \\bar { V } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "for each query recurrently,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 352, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 506, + 367 + ], + "score": 1.0, + "content": "but this requires re-computing the projection in Eq. (3) for every token due to the nonlinearity of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 209, + 378 + ], + "score": 1.0, + "content": "softmax, which results in", + "type": "text" + }, + { + "bbox": [ + 210, + 364, + 241, + 376 + ], + "score": 0.92, + "content": "O ( r n ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 363, + 506, + 378 + ], + "score": 1.0, + "content": "computational complexity. To preserve the linear complexity, for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 498, + 388 + ], + "score": 1.0, + "content": "autoregressive models, we first divide the input sequence into equal-length segments with length", + "type": "text" + }, + { + "bbox": [ + 498, + 376, + 502, + 385 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 375, + 506, + 388 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 279, + 398 + ], + "score": 1.0, + "content": "and apply our dynamic projection to extract", + "type": "text" + }, + { + "bbox": [ + 280, + 385, + 306, + 397 + ], + "score": 0.86, + "content": "\\bar { \\bar { K } } _ { i } , \\bar { \\bar { V } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 386, + 506, + 398 + ], + "score": 1.0, + "content": "from each segment. Each token can only attend to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 395, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 107, + 397, + 133, + 408 + ], + "score": 0.84, + "content": "\\bar { K } _ { i } , \\bar { V _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 395, + 387, + 410 + ], + "score": 1.0, + "content": "of segments that do not contain its future tokens. Formally, let", + "type": "text" + }, + { + "bbox": [ + 387, + 397, + 399, + 408 + ], + "score": 0.87, + "content": "Q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 395, + 498, + 410 + ], + "score": 1.0, + "content": "be the query at position", + "type": "text" + }, + { + "bbox": [ + 498, + 399, + 502, + 407 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 395, + 507, + 410 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 107, + 408, + 190, + 421 + ], + "score": 0.88, + "content": "K _ { ( l - 1 ) s : l s } , V _ { ( l - 1 ) s : l s }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 407, + 313, + 423 + ], + "score": 1.0, + "content": "be the key-value pairs from the", + "type": "text" + }, + { + "bbox": [ + 313, + 409, + 319, + 417 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 407, + 384, + 423 + ], + "score": 1.0, + "content": "-th segment, and", + "type": "text" + }, + { + "bbox": [ + 384, + 408, + 426, + 419 + ], + "score": 0.91, + "content": "s _ { t } = \\lfloor t / l \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 407, + 506, + 423 + ], + "score": 1.0, + "content": ". For autoregressive", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 460, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 300, + 432 + ], + "score": 1.0, + "content": "models, we compute the long-range attention of", + "type": "text" + }, + { + "bbox": [ + 300, + 419, + 312, + 430 + ], + "score": 0.88, + "content": "Q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 419, + 376, + 432 + ], + "score": 1.0, + "content": "by attending to", + "type": "text" + }, + { + "bbox": [ + 376, + 420, + 413, + 431 + ], + "score": 0.92, + "content": "K _ { i , t } , V _ { i , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 419, + 460, + 432 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 320, + 507, + 432 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 433, + 477, + 448 + ], + "lines": [ + { + "bbox": [ + 132, + 433, + 477, + 448 + ], + "spans": [ + { + "bbox": [ + 132, + 433, + 477, + 448 + ], + "score": 0.75, + "content": "\\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 } .", + "type": "interline_equation", + "image_path": "2b8af8a2115394307ff82943ba6e7fc875b3af1ef6ab96d766c21315e07cc9fc.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 132, + 433, + 477, + 448 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 508, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "In this way, the dynamic low-rank projection is applied to each segment only once in parallel,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "preserving the linear complexity and the high training speed. By comparison, Random Feature", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 470, + 392, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 392, + 483 + ], + "score": 1.0, + "content": "Attention [30] is slow at training due to the requirement for recurrence.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 448, + 506, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 494, + 351, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 352, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 352, + 509 + ], + "score": 1.0, + "content": "3.4 Aggregating Long-range and Short-term Attentions", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 506, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "To aggregate the local and long-range attentions, instead of adopting different attention mechanisms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 525, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 330, + 537 + ], + "score": 1.0, + "content": "for different heads [12, 14, 34], we let each query at", + "type": "text" + }, + { + "bbox": [ + 331, + 526, + 335, + 535 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 525, + 506, + 537 + ], + "score": 1.0, + "content": "-th head attend to the union of keys and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "values from the local window and global low-rank projections, thus it can learn to select important", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "information from either of them. We find this aggregation strategy works better than separating the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 464, + 570 + ], + "score": 1.0, + "content": "heads in our initial trials with the autoregressive language models. Specifically, for the", + "type": "text" + }, + { + "bbox": [ + 465, + 559, + 469, + 568 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "-th head,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 345, + 581 + ], + "score": 1.0, + "content": "we denote the global low-rank projected keys and values as", + "type": "text" + }, + { + "bbox": [ + 346, + 568, + 410, + 580 + ], + "score": 0.92, + "content": "\\bar { K } _ { i } , \\bar { V } _ { i } \\in \\dot { \\mathbb { R } ^ { r \\times d _ { k } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 567, + 506, + 581 + ], + "score": 1.0, + "content": ", and the local keys and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 146, + 594 + ], + "score": 1.0, + "content": "values as", + "type": "text" + }, + { + "bbox": [ + 147, + 579, + 215, + 592 + ], + "score": 0.94, + "content": "\\tilde { K } _ { t } , \\tilde { \\tilde { V } } _ { t } \\in \\mathbb { R } ^ { 2 w \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 578, + 367, + 594 + ], + "score": 1.0, + "content": "within the local window of position", + "type": "text" + }, + { + "bbox": [ + 367, + 582, + 372, + 591 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 578, + 430, + 594 + ], + "score": 1.0, + "content": "for the query", + "type": "text" + }, + { + "bbox": [ + 430, + 581, + 442, + 593 + ], + "score": 0.89, + "content": "Q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 578, + 488, + 594 + ], + "score": 1.0, + "content": ". 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Furthermore, we find a scale", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 258, + 674 + ], + "score": 1.0, + "content": "mismatch between the initial norms of", + "type": "text" + }, + { + "bbox": [ + 258, + 660, + 289, + 673 + ], + "score": 0.92, + "content": "\\tilde { K } _ { t } W _ { i } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 659, + 307, + 674 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 661, + 320, + 672 + ], + "score": 0.87, + "content": "{ \\bar { K } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 659, + 506, + 674 + ], + "score": 1.0, + "content": ", which biases the attention to the local window", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 673, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 506, + 684 + ], + "score": 1.0, + "content": "at initialization for both language and vision tasks. 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Without DualLN, the sparse and low-rank embeddings have a magnitude mismatch. With", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 231, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 241 + ], + "score": 1.0, + "content": "DualLN, the ratios will be 1.0 at every layer, which will facilitate optimization. Right: The validation loss of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 240, + 338, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 338, + 251 + ], + "score": 1.0, + "content": "Transformer-LS with and without DualLN on enwik8 and text8.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 506, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 477, + 281 + ], + "score": 1.0, + "content": "DualLN: For Transformers with Layer Normalization (LN) (see [44] for an illustration), the", + "type": "text" + }, + { + "bbox": [ + 477, + 268, + 504, + 280 + ], + "score": 0.87, + "content": "K _ { i } , V _ { i }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 278, + 507, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 507, + 292 + ], + "score": 1.0, + "content": "embeddings are the outputs of LN layers, so they have zero mean and unit variance at initialization.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 124, + 303 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 124, + 290, + 134, + 301 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "norm of vectors with zero-mean entries is proportional to their variance in expectation. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "note a weighted average will reduce the variance and therefore the norm of such zero-mean vectors.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 437, + 324 + ], + "score": 1.0, + "content": "As a result, the embedding vectors from low-rank attention in the weighted average", + "type": "text" + }, + { + "bbox": [ + 437, + 311, + 464, + 324 + ], + "score": 0.55, + "content": "\\bar { K } _ { i } , \\bar { V } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "of Eq. 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First, the inner product", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 343, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 107, + 345, + 150, + 359 + ], + "score": 0.93, + "content": "Q _ { t } W _ { i } ^ { Q } { \\bar { K } } _ { i } ^ { \\mathsf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 343, + 507, + 361 + ], + "score": 1.0, + "content": "from local-rank component tends to have smaller magnitude than the local window one,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "thus the attention scores on long-range attention is systematically smaller. 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In practice, to maintain the consistency between the local attention and dynamic projection,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 137, + 545 + ], + "score": 1.0, + "content": "we use", + "type": "text" + }, + { + "bbox": [ + 137, + 532, + 214, + 544 + ], + "score": 0.92, + "content": "\\mathsf { L } \\bar { \\mathsf { N } } _ { L } ( K ) , \\mathsf { L } \\mathsf { N } _ { L } ( V )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 532, + 259, + 545 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 259, + 532, + 281, + 543 + ], + "score": 0.91, + "content": "K , V", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 532, + 331, + 545 + ], + "score": 1.0, + "content": "to compute", + "type": "text" + }, + { + "bbox": [ + 332, + 531, + 358, + 543 + ], + "score": 0.92, + "content": "\\bar { K } _ { i } , \\bar { V } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "in Eq. 3. As illustrated in Figure 2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "Right, the Transformer-LS models trained with DualLN has consistently lower validation loss than", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 554, + 222, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 222, + 565 + ], + "score": 1.0, + "content": "the models without DualLN.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 578, + 192, + 592 + ], + "lines": [ + { + "bbox": [ + 104, + 576, + 193, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 193, + 595 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "In this section, we demonstrate the effectiveness and efficiency of our method in both language and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 612, + 480, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 480, + 625 + ], + "score": 1.0, + "content": "vision domains. 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Without DualLN, the sparse and low-rank embeddings have a magnitude mismatch. With", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 231, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 241 + ], + "score": 1.0, + "content": "DualLN, the ratios will be 1.0 at every layer, which will facilitate optimization. Right: The validation loss of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 240, + 338, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 338, + 251 + ], + "score": 1.0, + "content": "Transformer-LS with and without DualLN on enwik8 and text8.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 506, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 477, + 281 + ], + "score": 1.0, + "content": "DualLN: For Transformers with Layer Normalization (LN) (see [44] for an illustration), the", + "type": "text" + }, + { + "bbox": [ + 477, + 268, + 504, + 280 + ], + "score": 0.87, + "content": "K _ { i } , V _ { i }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 278, + 507, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 507, + 292 + ], + "score": 1.0, + "content": "embeddings are the outputs of LN layers, so they have zero mean and unit variance at initialization.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 124, + 303 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 124, + 290, + 134, + 301 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "norm of vectors with zero-mean entries is proportional to their variance in expectation. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "note a weighted average will reduce the variance and therefore the norm of such zero-mean vectors.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 437, + 324 + ], + "score": 1.0, + "content": "As a result, the embedding vectors from low-rank attention in the weighted average", + "type": "text" + }, + { + "bbox": [ + 437, + 311, + 464, + 324 + ], + "score": 0.55, + "content": "\\bar { K } _ { i } , \\bar { V } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "of Eq. 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The results", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "on these three tasks are given in Table 1. 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All results are averages of 4 runs with different random seeds.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 93, + 504, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 93, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 108, + 93, + 504, + 230 + ], + "score": 0.983, + "html": "
Task (mean ± std.) of sequence lengthListOps (888±339)Text (1296 ± 893)Retrieval (3987 ± 560)Average
ModelAcc.FLOPsAcc.FLOPsAcc.FLOPsAcc.
Full Attention [1]37.131.2165.354.5782.309.1461.59
Reformer [31] (2)36.440.2764.880.5878.641.1559.99
Linformer [17] (k=256)37.380.4156.120.8179.371.6257.62
Performer [28] (r = 256)32.780.4165.210.8281.701.6359.90
Nystromformer[18](l =128)37.340.6165.751.0281.292.0361.46
Transformer-LS (w,r = 8,32)37.500.2066.010.4081.790.8061.77
Dynamic Projection (best)37.790.1566.280.6981.862.1761.98
Transformer-LS (best)38.360.1668.400.2981.952.1762.90
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Task Test PerturbTextRetrieval
NoneInsertionDeletionNoneInsertionDeletion
Linformer56.1255.9454.9179.3753.6651.75
DP66.2863.1658.9581.8670.0164.98
Linformer+ Win.59.6356.6956.2979.6852.8352.13
DP + Win. (ours)68.4066.3462.6281.9569.9364.19
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ModelRoBERTa-base RoBERTa-large Longformer-base LS-base LS-large
Accuracy95.396.595.796.096.8
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Results of the other two image-based tasks of LRA, as well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 432, + 371, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 371, + 445 + ], + "score": 1.0, + "content": "as models implemented in JAX, are given in Appendix C and C.2.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 108, + 449, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "In addition, we follow the pretraining procedure of Longformer [14] to pretrain our models based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "on RoBERTa-base and RoBERTa-large [47], and fine-tune it on the IMDb sentiment classification", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 470, + 270, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 270, + 483 + ], + "score": 1.0, + "content": "dataset. 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Transformer-LS (best) with the best", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 178, + 532 + ], + "score": 1.0, + "content": "configurations of", + "type": "text" + }, + { + "bbox": [ + 178, + 522, + 196, + 531 + ], + "score": 0.87, + "content": "w , r", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "for each task are given in Table 7 in Appendix B. We also report the results", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 232, + 543 + ], + "score": 1.0, + "content": "of using fixed hyperparameter", + "type": "text" + }, + { + "bbox": [ + 232, + 531, + 295, + 542 + ], + "score": 0.92, + "content": "w = 8 , r = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "on all tasks. 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All results are averages of 4 runs with different random seeds.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 93, + 504, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 93, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 108, + 93, + 504, + 230 + ], + "score": 0.983, + "html": "
Task (mean ± std.) of sequence lengthListOps (888±339)Text (1296 ± 893)Retrieval (3987 ± 560)Average
ModelAcc.FLOPsAcc.FLOPsAcc.FLOPsAcc.
Full Attention [1]37.131.2165.354.5782.309.1461.59
Reformer [31] (2)36.440.2764.880.5878.641.1559.99
Linformer [17] (k=256)37.380.4156.120.8179.371.6257.62
Performer [28] (r = 256)32.780.4165.210.8281.701.6359.90
Nystromformer[18](l =128)37.340.6165.751.0281.292.0361.46
Transformer-LS (w,r = 8,32)37.500.2066.010.4081.790.8061.77
Dynamic Projection (best)37.790.1566.280.6981.862.1761.98
Transformer-LS (best)38.360.1668.400.2981.952.1762.90
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Task Test PerturbTextRetrieval
NoneInsertionDeletionNoneInsertionDeletion
Linformer56.1255.9454.9179.3753.6651.75
DP66.2863.1658.9581.8670.0164.98
Linformer+ Win.59.6356.6956.2979.6852.8352.13
DP + Win. (ours)68.4066.3462.6281.9569.9364.19
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ModelRoBERTa-base RoBERTa-large Longformer-base LS-base LS-large
Accuracy95.396.595.796.096.8
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The results are given in Table 3.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 449, + 505, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "Results. From Table 3, our base model outperforms Longformer-base, and our large model achieves", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "score": 1.0, + "content": "improvements over RoBERTa-large, demonstrating the benefits of learning to model long sequences.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "Comparisons with models on LRA are given in Table 1. Transformer-LS (best) with the best", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 178, + 532 + ], + "score": 1.0, + "content": "configurations of", + "type": "text" + }, + { + "bbox": [ + 178, + 522, + 196, + 531 + ], + "score": 0.87, + "content": "w , r", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "for each task are given in Table 7 in Appendix B. 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Method#Paramtext8
Dev Testenwik8 DevTest
T12 [49] Transformer-XL [9]44M1 1.1811.11
Reformer [31]41M -1 1 - 11 11.06 1.05
Adaptive [50]38M1.05 1.111.041.02
BP-Transformer [51] Longformer [20]38M 41M1 1.11 1.04 1.101 1.021.02 1.00
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Method #Param‘ TestBPC
Transformer-XL [9]88M1.03 0.99
Transformer-XL [9] Routing [32]277M 223M0.99
Longformer [14]102M0.99
Sparse [12]95M0.99
Adaptive [50]209M0.98
Compressive [10]227M0.97
Transformer-LS110M0.97
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Method#Paramtext8
Dev Testenwik8 DevTest
T12 [49] Transformer-XL [9]44M1 1.1811.11
Reformer [31]41M -1 1 - 11 11.06 1.05
Adaptive [50]38M1.05 1.111.041.02
BP-Transformer [51] Longformer [20]38M 41M1 1.11 1.04 1.101 1.021.02 1.00
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Method #Param‘ TestBPC
Transformer-XL [9]88M1.03 0.99
Transformer-XL [9] Routing [32]277M 223M0.99
Longformer [14]102M0.99
Sparse [12]95M0.99
Adaptive [50]209M0.98
Compressive [10]227M0.97
Transformer-LS110M0.97
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Model(M)Size#Param Image FLOPs ImageNet (G)top-1 (%)Real top-1 (%) top-1 (%)V2Latency (s）
ResNet-5025224²4.176.282.563.3=
ResNet-1014522427.977.483.765.7
ResNet-1526022421178.384.167.0=
DeiT-S [36]22224²4.679.885.768.5
DeiT-B [36]86224217.681.886.770.9
PVT-Medium [5]4422426.781.211
PVT-Large [5]6122429.881.7
Swin-S [38]5022428.783.2
Swin-B[38]88224215.483.50.115
PVTv2-B4 [54]62.6224210.183.6
PVTv2-B5 [54]82.0224211.883.8
ViT-B/16 [4]86384255.577.9
ViT-L/16 [4]3073842191.176.5
DeiT-B [36]86384255.583.1
Swin-B [38]88384247.184.5110.378
CvT-13 [6]2022426.781.686.770.40.122
CvT-21 [6]32224210.182.587.271.30.165
CvT*-LS-1320.322424.981.987.070.50.083
CvT*-LS-1723.722429.882.587.271.6-
CvT*-LS-2132.1224²7.982.787.571.90.122
CvT*-LS-21S30.1224211.382.987.471.7-
CvT-13 [6]20384231.983.087.971.91
CvT-21 [6]32384245.083.387.771.9
CvT*-LS-2132.1384223.983.288.072.5
CvT*-LS-2132.1448234.283.688.272.9
ViL-Small [14]24.622424.982.4-1
ViL-Medium [14]39.722428.783.50.106
ViL-Base [14]55.7224213.483.70.164
ViL-LS-Medium39.822428.783.80.075
ViL-LS-Base55.8224²13.484.10.113
ViL-LS-Medium39.9384228.784.40.271
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We", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 104, + 639, + 120, + 653 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 641, + 146, + 650 + ], + "score": 0.88, + "content": "r = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "for the dynamic projections for both ViL-LS-Medium and ViL-LS-Base. 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Model(M)Size#Param Image FLOPs ImageNet (G)top-1 (%)Real top-1 (%) top-1 (%)V2Latency (s）
ResNet-5025224²4.176.282.563.3=
ResNet-1014522427.977.483.765.7
ResNet-1526022421178.384.167.0=
DeiT-S [36]22224²4.679.885.768.5
DeiT-B [36]86224217.681.886.770.9
PVT-Medium [5]4422426.781.211
PVT-Large [5]6122429.881.7
Swin-S [38]5022428.783.2
Swin-B[38]88224215.483.50.115
PVTv2-B4 [54]62.6224210.183.6
PVTv2-B5 [54]82.0224211.883.8
ViT-B/16 [4]86384255.577.9
ViT-L/16 [4]3073842191.176.5
DeiT-B [36]86384255.583.1
Swin-B [38]88384247.184.5110.378
CvT-13 [6]2022426.781.686.770.40.122
CvT-21 [6]32224210.182.587.271.30.165
CvT*-LS-1320.322424.981.987.070.50.083
CvT*-LS-1723.722429.882.587.271.6-
CvT*-LS-2132.1224²7.982.787.571.90.122
CvT*-LS-21S30.1224211.382.987.471.7-
CvT-13 [6]20384231.983.087.971.91
CvT-21 [6]32384245.083.387.771.9
CvT*-LS-2132.1384223.983.288.072.5
CvT*-LS-2132.1448234.283.688.272.9
ViL-Small [14]24.622424.982.4-1
ViL-Medium [14]39.722428.783.50.106
ViL-Base [14]55.7224213.483.70.164
ViL-LS-Medium39.822428.783.80.075
ViL-LS-Base55.8224²13.484.10.113
ViL-LS-Medium39.9384228.784.40.271
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Model[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]ImageNet-9 [59]
(M)Top-1mCE(↓）Acc.Acc.|Mixed-same Mixed-rand
ResNet-50 [35]]25.676.278.96.235.387.181.6
DeiT-S [36]22.179.857.119.041.989.184.2
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CvT-213282.556.231.142.690.585.0
CvT*-LS-1320.381.958.727.042.690.785.6
CvT*-LS-2132.182.755.229.345.091.585.8
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Model[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]ImageNet-9 [59]
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CvT*-LS-1320.381.958.727.042.690.785.6
CvT*-LS-2132.182.755.229.345.091.585.8
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We compare", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "our methods with standard classification methods, including CNN-based model (ResNet [35]) and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Transformer-based models (DeiT [36]) with similar numbers of parameters. As shown in Table 6, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 544, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 555 + ], + "score": 1.0, + "content": "observe that our method significantly outperforms the CNN-based method (ResNet-50). Compared", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "to DeiT, our models also achieve favorable improvements. These results indicate that the design of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "score": 1.0, + "content": "different attention mechanisms plays an important role for model robustness, which sheds new light", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 577, + 497, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 497, + 589 + ], + "score": 1.0, + "content": "on the design of robust vision transformers. More details and results can be found in Appendix E.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 468, + 506, + 589 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 601, + 183, + 615 + ], + "lines": [ + { + "bbox": [ + 104, + 600, + 185, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 185, + 617 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "In this paper, we introduced Long-Short Transformer, an efficient transformer for long sequence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "modeling for both language and vision domain, including both bidirectional and autoregressive", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "models. We design a novel global attention mechanism with linear computational and memory", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "complexity in sequence length based on a dynamic projection. We identify the scale mismatch issue", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "and propose the DualLN technique to eliminate the mismatch at initialization and more effectively", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "aggregate the local and global attentions. We demonstrate that our method obtains the state-of-the-art", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "results on the Long Range Arena, char-level language modeling and ImageNet classification. We", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 504, + 712 + ], + "score": 1.0, + "content": "look forward to extending our methods to more domains, including document QA, object detection", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 325, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 325, + 724 + ], + "score": 1.0, + "content": "and semantic segmentation on high-resolution images.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 623, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 70, + 507, + 724 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 109, + 88, + 506, + 102 + ], + "spans": [ + { + "bbox": [ + 109, + 88, + 506, + 102 + ], + "score": 1.0, + "content": "[1] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 126, + 100, + 424, + 111 + ], + "spans": [ + { + "bbox": [ + 126, + 100, + 424, + 111 + ], + "score": 1.0, + "content": "Kaiser, and Illia Polosukhin. 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Task (mean ± std.) of sequence lengthListOps (888±339)Text (1296 ± 893)Retrieval (3987 ± 560)Average
ModelAcc.FLOPsAcc.FLOPsAcc.FLOPsAcc.
Full Attention [1]37.131.2165.354.5782.309.1461.59
Reformer [31] (2)36.440.2764.880.5878.641.1559.99
Linformer [17] (k=256)37.380.4156.120.8179.371.6257.62
Performer [28] (r = 256)32.780.4165.210.8281.701.6359.90
Nystromformer[18](l =128)37.340.6165.751.0281.292.0361.46
Transformer-LS (w,r = 8,32)37.500.2066.010.4081.790.8061.77
Dynamic Projection (best)37.790.1566.280.6981.862.1761.98
Transformer-LS (best)38.360.1668.400.2981.952.1762.90
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Task Test PerturbTextRetrieval
NoneInsertionDeletionNoneInsertionDeletion
Linformer56.1255.9454.9179.3753.6651.75
DP66.2863.1658.9581.8670.0164.98
Linformer+ Win.59.6356.6956.2979.6852.8352.13
DP + Win. (ours)68.4066.3462.6281.9569.9364.19
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ModelRoBERTa-base RoBERTa-large Longformer-base LS-base LS-large
Accuracy95.396.595.796.096.8
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Method#Paramtext8
Dev Testenwik8 DevTest
T12 [49] Transformer-XL [9]44M1 1.1811.11
Reformer [31]41M -1 1 - 11 11.06 1.05
Adaptive [50]38M1.05 1.111.041.02
BP-Transformer [51] Longformer [20]38M 41M1 1.11 1.04 1.101 1.021.02 1.00
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Method #Param‘ TestBPC
Transformer-XL [9]88M1.03 0.99
Transformer-XL [9] Routing [32]277M 223M0.99
Longformer [14]102M0.99
Sparse [12]95M0.99
Adaptive [50]209M0.98
Compressive [10]227M0.97
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Model(M)Size#Param Image FLOPs ImageNet (G)top-1 (%)Real top-1 (%) top-1 (%)V2Latency (s）
ResNet-5025224²4.176.282.563.3=
ResNet-1014522427.977.483.765.7
ResNet-1526022421178.384.167.0=
DeiT-S [36]22224²4.679.885.768.5
DeiT-B [36]86224217.681.886.770.9
PVT-Medium [5]4422426.781.211
PVT-Large [5]6122429.881.7
Swin-S [38]5022428.783.2
Swin-B[38]88224215.483.50.115
PVTv2-B4 [54]62.6224210.183.6
PVTv2-B5 [54]82.0224211.883.8
ViT-B/16 [4]86384255.577.9
ViT-L/16 [4]3073842191.176.5
DeiT-B [36]86384255.583.1
Swin-B [38]88384247.184.5110.378
CvT-13 [6]2022426.781.686.770.40.122
CvT-21 [6]32224210.182.587.271.30.165
CvT*-LS-1320.322424.981.987.070.50.083
CvT*-LS-1723.722429.882.587.271.6-
CvT*-LS-2132.1224²7.982.787.571.90.122
CvT*-LS-21S30.1224211.382.987.471.7-
CvT-13 [6]20384231.983.087.971.91
CvT-21 [6]32384245.083.387.771.9
CvT*-LS-2132.1384223.983.288.072.5
CvT*-LS-2132.1448234.283.688.272.9
ViL-Small [14]24.622424.982.4-1
ViL-Medium [14]39.722428.783.50.106
ViL-Base [14]55.7224213.483.70.164
ViL-LS-Medium39.822428.783.80.075
ViL-LS-Base55.8224²13.484.10.113
ViL-LS-Medium39.9384228.784.40.271
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Model[Params|ImageNet|IN-C [56]|IN-A [57]|IN-R [58]]ImageNet-9 [59]
(M)Top-1mCE(↓）Acc.Acc.|Mixed-same Mixed-rand
ResNet-50 [35]]25.676.278.96.235.387.181.6
DeiT-S [36]22.179.857.119.041.989.184.2
CvT-132081.659.625.442.990.585.7
CvT-213282.556.231.142.690.585.0
CvT*-LS-1320.381.958.727.042.690.785.6
CvT*-LS-2132.182.755.229.345.091.585.8
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+ ], + "page_info": { + "page_no": 13, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4.md b/parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4.md new file mode 100644 index 0000000000000000000000000000000000000000..d7e66f3ded7c8e8578f951ef21ed32d86b9e6d16 --- /dev/null +++ b/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. + +
LayerBlock 1 layer 2Block1 layer 3Block 2 layer 2
AugMax-DuBN0.03690.04500.0301
品0.04690.05850.0382
AugMax-DuBIN0.03060.04030.0264
0.03480.04660.0292
R
+ +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. + +
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)95.5695.7995.76 (-0.03)
RA(%)74.7589.4990.36 (+0.87)
WRN40-2SA(%)94.7895.6795.68 (+0.01)
RA(%)73.7189.0190.67 (+1.66)
ResNeXt29SA(%)95.6096.2596.39 (+0.14)
RA(%)71.7089.0892.11 (+3.03)
+ +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. + +
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)77.9978.2378.69 (+0.46)
RA(%)48.4662.6765.75 (+3.08)
WRN40-2SA(%)76.1977.0376.80 (-0.23)
RA(%)46.8064.5666.35 (+1.79)
ResNeXt29SA(%)79.9578.5880.70 (+2.12)
RA(%)47.7665.3768.86 (+3.49)
+ +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. + +
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal69.8330.9187.47
AugMix68.06 67.6234.58 35.0183.08 82.56
AugMax-DuBIN(-0.44)(+0.43)(-0.52)
DeepAugment + AugMix65.3245.8469.29
DeepAugment +64.4346.5568.47
AugMax-DuBIN(-0.89)(+0.71)(-0.82)
+ +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. + +
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal61.6423.91100.00
AugMix61.79 62.2136.85 38.6783.04 80.72
AugMax-DuBIN(+0.42)(+1.82)(-2.32)
DeepAugment + AugMix59.5940.6778.28
DeepAugment +59.7240.9977.83
AugMax-DuBIN(+0.13)(+0.32)(-0.45)
+ +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. + +
MethodSA(%,↑)RA(%,↑)mCE(%,）
Normal61.6424.38100.00
AugMix61.7937.6382.54
ANT61.2635.3085.70
AugMax-DuBIN62.2138.6680.29
+ +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. + +
MethodTime (sec/epoch)
Normal2669
AugMix3622
AT[6]37162
AugMax5264
+ +# 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. + +
MethodCIFAR10-CCIFAR100-C
AugMix-BN89.01 (± 0.03)64.56 (± 0.04)
AugMix-IBN89.17 (± 0.24)63.94 (± 0.28)
AugMix-DuBN89.11 (± 0.21)64.07 (± 0.38)
AugMix-DuBIN89.74(± 0.35)65.02 (± 0.58)
AugMax-DuBN89.60 (± 0.62)65.06 (± 0.28)
AugMax-DuBIN90.67 (± 0.16)66.35 (± 0.21)
+ +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. + +
n
5101k231入 1015
SA(%)95.7895.6895.6895.6095.5195.9095.6895.88
RA (%)90.4990.6790.6790.3890.2188.8990.6790.30
+ +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. + +
MethodStrategySA(%)RA(%)
Normal-77.9948.46
AugMix [5](diversity)78.2362.67
PGDAT [6](hardness)60.9447.71
FAT[28]61.5148.70
AdvMax56.6138.65
AugMix+PGDAT61.6851.39
AdvMix(diversity &adversity)72.3656.89
AugMax-DuBIN78.5264.02
+ +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. + +
MethodCIFAR10.1CIFAR10-STA
Normal88.9030.30
AugMix88.9054.30
AugMax-DuBIN90.6463.20
+ +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. 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In International Conference on Learning Representations (ICLR), 2018. 9, 15 \ No newline at end of file diff --git a/parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4_content_list.json b/parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e975e86df4fd1fd66cd2ef213891b92f396c020f --- /dev/null +++ b/parse/train/P5MtdcVdFZ4/P5MtdcVdFZ4_content_list.json @@ -0,0 +1,1315 @@ +[ + { + "type": "text", + "text": "AugMax: Adversarial Composition of Random Augmentations for Robust Training ", + "text_level": 1, + "bbox": [ + 215, + 122, + 781, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "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 ", + "bbox": [ + 281, + 222, + 717, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "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 ", + "bbox": [ + 225, + 263, + 772, + 319 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 361, + 535, + 377 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 232, + 392, + 766, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 655, + 310, + 672 + ], + "page_idx": 0 + }, + { + "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. ", + "bbox": [ + 174, + 686, + 825, + 797 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 804, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "image", + "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. " + ], + "image_footnote": [], + "bbox": [ + 178, + 88, + 823, + 213 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There are mainly two categories of data augmentation approaches: ", + "bbox": [ + 174, + 400, + 607, + 415 + ], + "page_idx": 1 + }, + { + "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). ", + "bbox": [ + 174, + 420, + 825, + 518 + ], + "page_idx": 1 + }, + { + "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). ", + "bbox": [ + 174, + 523, + 825, + 635 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 641, + 825, + 696 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Summary of contributions: ", + "text_level": 1, + "bbox": [ + 174, + 704, + 362, + 718 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 720, + 825, + 844 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 174, + 856, + 823, + 911 + ], + "page_idx": 1 + }, + { + "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 $\\textbf { \\em w }$ ). " + ], + "image_footnote": [], + "bbox": [ + 173, + 87, + 816, + 223 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 320, + 825, + 472 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 479, + 825, + 590 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 595, + 825, + 666 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 686, + 321, + 704 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Robustness to Distributional Shifts ", + "text_level": 1, + "bbox": [ + 176, + 718, + 455, + 733 + ], + "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]. ", + "bbox": [ + 174, + 744, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Data Augmentation: Random and Adversarial ", + "text_level": 1, + "bbox": [ + 174, + 200, + 535, + 215 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 236, + 825, + 388 + ], + "page_idx": 3 + }, + { + "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]. ", + "bbox": [ + 173, + 393, + 826, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 Normalization ", + "text_level": 1, + "bbox": [ + 174, + 683, + 313, + 696 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 717, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Method ", + "text_level": 1, + "bbox": [ + 174, + 89, + 271, + 106 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 Preliminary: The AugMix Framework ", + "text_level": 1, + "bbox": [ + 174, + 119, + 480, + 136 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 146, + 826, + 339 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 AugMax: Augmented Training with Unified Diversity and Hardness ", + "text_level": 1, + "bbox": [ + 176, + 354, + 686, + 371 + ], + "page_idx": 4 + }, + { + "type": "text", + "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 $\\pmb { w }$ . ", + "bbox": [ + 173, + 381, + 825, + 424 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 174, + 428, + 823, + 472 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 398, + 478, + 599, + 501 + ], + "page_idx": 4 + }, + { + "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: ", + "bbox": [ + 174, + 508, + 826, + 537 + ], + "page_idx": 4 + }, + { + "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": [ + 416, + 544, + 578, + 561 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $g ( \\cdot )$ denotes the AugMax augmentation function, $\\pmb { x } _ { o r i g }$ is the original image, and ", + "bbox": [ + 173, + 568, + 750, + 583 + ], + "page_idx": 4 + }, + { + "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": [ + 197, + 590, + 781, + 618 + ], + "page_idx": 4 + }, + { + "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: ", + "bbox": [ + 174, + 625, + 831, + 654 + ], + "page_idx": 4 + }, + { + "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": [ + 269, + 660, + 728, + 689 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training with AugMax augmentation can then be written altogether as minimax optimization: ", + "bbox": [ + 173, + 694, + 787, + 710 + ], + "page_idx": 4 + }, + { + "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": [ + 186, + 715, + 787, + 776 + ], + "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": [ + 173, + 782, + 825, + 811 + ], + "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": [ + 354, + 818, + 643, + 835 + ], + "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": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 4 + }, + { + "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": [ + 173, + 92, + 826, + 215 + ], + "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": [ + 173, + 222, + 826, + 333 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 DuBIN: Disentangled Normalization for Heterogeneous Features ", + "text_level": 1, + "bbox": [ + 174, + 349, + 663, + 364 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 377, + 549, + 541 + ], + "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": [ + 174, + 547, + 549, + 672 + ], + "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": [ + 562, + 381, + 823, + 530 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 672, + 823, + 700 + ], + "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": [ + 174, + 707, + 485, + 844 + ], + "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": "
LayerBlock 1 layer 2Block1 layer 3Block 2 layer 2
AugMax-DuBN0.03690.04500.0301
品0.04690.05850.0382
AugMax-DuBIN0.03060.04030.0264
0.03480.04660.0292
R
", + "bbox": [ + 509, + 755, + 808, + 842 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "encoding instance-level diversities. With lower feature variation, BNs can converge better with improved performance [66]. ", + "bbox": [ + 174, + 844, + 823, + 872 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 89, + 312, + 107 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 122, + 825, + 165 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 183, + 351, + 198 + ], + "page_idx": 6 + }, + { + "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": [ + 173, + 210, + 826, + 294 + ], + "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": [ + 174, + 311, + 485, + 435 + ], + "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": "
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)95.5695.7995.76 (-0.03)
RA(%)74.7589.4990.36 (+0.87)
WRN40-2SA(%)94.7895.6795.68 (+0.01)
RA(%)73.7189.0190.67 (+1.66)
ResNeXt29SA(%)95.6096.2596.39 (+0.14)
RA(%)71.7089.0892.11 (+3.03)
", + "bbox": [ + 498, + 345, + 821, + 435 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 435, + 825, + 505 + ], + "page_idx": 6 + }, + { + "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": "
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)77.9978.2378.69 (+0.46)
RA(%)48.4662.6765.75 (+3.08)
WRN40-2SA(%)76.1977.0376.80 (-0.23)
RA(%)46.8064.5666.35 (+1.79)
ResNeXt29SA(%)79.9578.5880.70 (+2.12)
RA(%)47.7665.3768.86 (+3.49)
", + "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": "
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal69.8330.9187.47
AugMix68.06 67.6234.58 35.0183.08 82.56
AugMax-DuBIN(-0.44)(+0.43)(-0.52)
DeepAugment + AugMix65.3245.8469.29
DeepAugment +64.4346.5568.47
AugMax-DuBIN(-0.89)(+0.71)(-0.82)
", + "bbox": [ + 500, + 734, + 820, + 854 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 854, + 599, + 868 + ], + "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": "
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal61.6423.91100.00
AugMix61.79 62.2136.85 38.6783.04 80.72
AugMax-DuBIN(+0.42)(+1.82)(-2.32)
DeepAugment + AugMix59.5940.6778.28
DeepAugment +59.7240.9977.83
AugMax-DuBIN(+0.13)(+0.32)(-0.45)
", + "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": [ + 174, + 353, + 826, + 409 + ], + "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": "
MethodSA(%,↑)RA(%,↑)mCE(%,）
Normal61.6424.38100.00
AugMix61.7937.6382.54
ANT61.2635.3085.70
AugMax-DuBIN62.2138.6680.29
", + "bbox": [ + 501, + 599, + 821, + 666 + ], + "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": "
MethodTime (sec/epoch)
Normal2669
AugMix3622
AT[6]37162
AugMax5264
", + "bbox": [ + 594, + 790, + 820, + 868 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 Ablation Study ", + "text_level": 1, + "bbox": [ + 174, + 856, + 316, + 871 + ], + "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": [ + 174, + 883, + 820, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 826, + 160 + ], + "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, + 485, + 291 + ], + "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": "
MethodCIFAR10-CCIFAR100-C
AugMix-BN89.01 (± 0.03)64.56 (± 0.04)
AugMix-IBN89.17 (± 0.24)63.94 (± 0.28)
AugMix-DuBN89.11 (± 0.21)64.07 (± 0.38)
AugMix-DuBIN89.74(± 0.35)65.02 (± 0.58)
AugMax-DuBN89.60 (± 0.62)65.06 (± 0.28)
AugMax-DuBIN90.67 (± 0.16)66.35 (± 0.21)
", + "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": [ + 173, + 340, + 825, + 395 + ], + "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, + 527 + ], + "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": "
n
5101k231入 1015
SA(%)95.7895.6895.6895.6095.5195.9095.6895.88
RA (%)90.4990.6790.6790.3890.2188.8990.6790.30
", + "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": "
MethodStrategySA(%)RA(%)
Normal-77.9948.46
AugMix [5](diversity)78.2362.67
PGDAT [6](hardness)60.9447.71
FAT[28]61.5148.70
AdvMax56.6138.65
AugMix+PGDAT61.6851.39
AdvMix(diversity &adversity)72.3656.89
AugMax-DuBIN78.5264.02
", + "bbox": [ + 500, + 686, + 821, + 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": [ + 173, + 806, + 826, + 861 + ], + "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": [ + 174, + 92, + 825, + 161 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.4 Robustness against Other Distribution Shifts ", + "text_level": 1, + "bbox": [ + 174, + 178, + 524, + 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": "
MethodCIFAR10.1CIFAR10-STA
Normal88.9030.30
AugMix88.9054.30
AugMax-DuBIN90.6463.20
", + "bbox": [ + 552, + 262, + 816, + 321 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 376, + 299, + 393 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 401, + 825, + 470 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 478, + 825, + 534 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 554, + 330, + 570 + ], + "page_idx": 9 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 579, + 825, + 608 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 627, + 266, + 642 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. 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Thus it is critical to develop", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 622, + 447, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 447, + 633 + ], + "score": 1.0, + "content": "techniques that improve robustness even when training with relatively clean datasets.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 545, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Several techniques have been proposed to consolidate the robustness against unforeseen corruptions,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 507, + 661 + ], + "score": 1.0, + "content": "including robust data augmentation [5, 3, 8, 6], Lipschitz continuity [9–11], stability training [12],", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 658, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 507, + 673 + ], + "score": 1.0, + "content": "pre-training [13–16], and robust network structures [17–19], to name a few. Among these techniques,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 669, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 507, + 684 + ], + "score": 1.0, + "content": "data augmentation is of particular interest due to its empirical effectiveness, ease of implementation,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 329, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 329, + 694 + ], + "score": 1.0, + "content": "low computational overhead, and plug-and-play nature.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 637, + 507, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 70, + 504, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 70, + 504, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 70, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 109, + 70, + 504, + 169 + ], + "score": 0.903, + "type": "image", + "image_path": "96da970a0935f651f016706faf1673e344b89a5c13bdb2ecde02923b0a6543ea.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 70, + 504, + 103.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 103.0, + 504, + 136.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 136.0, + 504, + 169.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 175, + 505, + 296 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "Figure 1: The effects of diversity and hardness in data augmentations. We visualize features of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "augmented images fed to the network during training. The features are from the penultimate layer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "score": 1.0, + "content": "of a ResNeXt29 trained on CIFAR10 training data using only standard data augmentation (random", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "score": 1.0, + "content": "flipping and translation) following [4]. For visualization, we randomly selected 300 fixed images", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "from 3 fixed classes (denoted by different colors) to which we apply different augmentation methods:", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "(a) standard augmentation (random flipping and translation); (b) AugMix [5]; (c) PGD Attack [6, 7];", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "(d) AugMax (ours). In order to achieve good model robustness, the augmented training data should", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "both be diverse and also contain enough hard cases. As can be seen, PGD Attack generates hard cases", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "(which the network cannot separate) but are not diverse enough (they are all clustered together) while", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "AugMix creates diverse but not hard samples (they are well separated). By contrast, our approach", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 285, + 379, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 379, + 297 + ], + "score": 1.0, + "content": "(AugMax) achieves a unification between hard and diverse samples.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 372, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 375, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 375, + 331 + ], + "score": 1.0, + "content": "There are mainly two categories of data augmentation approaches:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 346 + ], + "score": 1.0, + "content": "The first category aims to increase diversity of the training data by composing multiple random", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "transformations [20–22, 5, 3]. While standard data augmentation methods (e.g., random flipping and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "translation) lead to poor robustness [5], more aggressive combinations of multiple augmentations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "have shown promise. One such successful example is AugMix [5], which stochastically samples from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "diverse augmentation operations and randomly mixing them to produce highly diverse augmented", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "images. AugMix can increase the sample diversity coverage compared to standard augmentations, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 398, + 228, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 228, + 412 + ], + "score": 1.0, + "content": "shown in Figure 1 (a) and (b).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "The second category aims to boost the hardness of the training data by sampling from the worst-case", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "augmentations that tries to fool the model into misclassifying the samples. A common technique to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "achieve this goal is adversarial perturbation [6, 23, 8]. Training over such worst-case samples allows", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "a model to actively fix its generalization weaknesses [24] and empirically improves its robustness", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "against corruptions [25, 26, 7]. Due to the extra complexity to generate adversarial perturbations,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "the improved robustness usually comes at the cost of largely increased training time compared with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "non-adversarial methods [6, 27]. An example falling into this category is the PGD attack [6], as is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 197, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 197, + 505 + ], + "score": 1.0, + "content": "shown in Figure 1 (c).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 507, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 507, + 522 + ], + "score": 1.0, + "content": "Previous work has focused on leveraging one of these category to improve robustness. In this paper,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "we unify both approaches in a single framework. In particular, we show that diversity and hardness", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "are complementary and that a unification between the two is necessary to achieve robustness. We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 455, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 455, + 554 + ], + "score": 1.0, + "content": "propose a new strategy to achieve this unification and successfully increase robustness.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 222, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 224, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 224, + 570 + ], + "score": 1.0, + "content": "Summary of contributions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "• We propose AugMax, a novel augmentation framework which achieves robustness through a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 113, + 581, + 483, + 596 + ], + "spans": [ + { + "bbox": [ + 113, + 581, + 483, + 596 + ], + "score": 1.0, + "content": "unification between diversity and hardness, by searching for the worst-case mixing strategy.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "• Being a stronger form of data augmentation, AugMax leads to a significantly augmented and more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 113, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "heterogeneous input distribution, which also makes model training more challenging. To solve this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 618, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 114, + 618, + 505, + 634 + ], + "score": 1.0, + "content": "problem, we design a new normalization strategy, termed DuBIN, to disentangle the instance-wise", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 631, + 287, + 643 + ], + "spans": [ + { + "bbox": [ + 114, + 631, + 287, + 643 + ], + "score": 1.0, + "content": "feature heterogeneity of AugMax samples.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "• We show that combination of AugMax and DuBIN (AugMax-DuBIN) achieves state-of-the-art", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 113, + 658, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 113, + 658, + 504, + 669 + ], + "score": 1.0, + "content": "robustness against corruptions and improves robustness against other common distribution shifts.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "AugMax achieves a good unification between sample diversity and hard corner-case generation during", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "data augmentation. AugMax is built on top of the AugMix framework [5] which mixes multiple data", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "augmentation operators in a multi-branch and layered pipeline. However, different from AugMix", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "where augmentation operators and mixing weights are both randomly sampled, operators are first", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 70, + 504, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 70, + 504, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 70, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 109, + 70, + 504, + 169 + ], + "score": 0.903, + "type": "image", + "image_path": "96da970a0935f651f016706faf1673e344b89a5c13bdb2ecde02923b0a6543ea.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 70, + 504, + 103.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 103.0, + 504, + 136.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 136.0, + 504, + 169.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 175, + 505, + 296 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "Figure 1: The effects of diversity and hardness in data augmentations. We visualize features of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "augmented images fed to the network during training. The features are from the penultimate layer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "score": 1.0, + "content": "of a ResNeXt29 trained on CIFAR10 training data using only standard data augmentation (random", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "score": 1.0, + "content": "flipping and translation) following [4]. For visualization, we randomly selected 300 fixed images", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "from 3 fixed classes (denoted by different colors) to which we apply different augmentation methods:", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "(a) standard augmentation (random flipping and translation); (b) AugMix [5]; (c) PGD Attack [6, 7];", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "(d) AugMax (ours). In order to achieve good model robustness, the augmented training data should", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "both be diverse and also contain enough hard cases. As can be seen, PGD Attack generates hard cases", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "(which the network cannot separate) but are not diverse enough (they are all clustered together) while", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "AugMix creates diverse but not hard samples (they are well separated). By contrast, our approach", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 285, + 379, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 379, + 297 + ], + "score": 1.0, + "content": "(AugMax) achieves a unification between hard and diverse samples.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 372, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 375, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 375, + 331 + ], + "score": 1.0, + "content": "There are mainly two categories of data augmentation approaches:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 315, + 375, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 504, + 346 + ], + "score": 1.0, + "content": "The first category aims to increase diversity of the training data by composing multiple random", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "transformations [20–22, 5, 3]. While standard data augmentation methods (e.g., random flipping and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "translation) lead to poor robustness [5], more aggressive combinations of multiple augmentations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "have shown promise. One such successful example is AugMix [5], which stochastically samples from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "diverse augmentation operations and randomly mixing them to produce highly diverse augmented", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "images. AugMix can increase the sample diversity coverage compared to standard augmentations, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 398, + 228, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 228, + 412 + ], + "score": 1.0, + "content": "shown in Figure 1 (a) and (b).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 334, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "The second category aims to boost the hardness of the training data by sampling from the worst-case", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "augmentations that tries to fool the model into misclassifying the samples. A common technique to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "achieve this goal is adversarial perturbation [6, 23, 8]. Training over such worst-case samples allows", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "a model to actively fix its generalization weaknesses [24] and empirically improves its robustness", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "against corruptions [25, 26, 7]. Due to the extra complexity to generate adversarial perturbations,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "the improved robustness usually comes at the cost of largely increased training time compared with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "non-adversarial methods [6, 27]. An example falling into this category is the PGD attack [6], as is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 197, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 197, + 505 + ], + "score": 1.0, + "content": "shown in Figure 1 (c).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 415, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 507, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 507, + 522 + ], + "score": 1.0, + "content": "Previous work has focused on leveraging one of these category to improve robustness. In this paper,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "we unify both approaches in a single framework. In particular, we show that diversity and hardness", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "are complementary and that a unification between the two is necessary to achieve robustness. We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 455, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 455, + 554 + ], + "score": 1.0, + "content": "propose a new strategy to achieve this unification and successfully increase robustness.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 507, + 507, + 554 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 222, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 224, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 224, + 570 + ], + "score": 1.0, + "content": "Summary of contributions:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "• We propose AugMax, a novel augmentation framework which achieves robustness through a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 113, + 581, + 483, + 596 + ], + "spans": [ + { + "bbox": [ + 113, + 581, + 483, + 596 + ], + "score": 1.0, + "content": "unification between diversity and hardness, by searching for the worst-case mixing strategy.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "• Being a stronger form of data augmentation, AugMax leads to a significantly augmented and more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 113, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "heterogeneous input distribution, which also makes model training more challenging. To solve this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 114, + 618, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 114, + 618, + 505, + 634 + ], + "score": 1.0, + "content": "problem, we design a new normalization strategy, termed DuBIN, to disentangle the instance-wise", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 631, + 287, + 643 + ], + "spans": [ + { + "bbox": [ + 114, + 631, + 287, + 643 + ], + "score": 1.0, + "content": "feature heterogeneity of AugMax samples.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "• We show that combination of AugMax and DuBIN (AugMax-DuBIN) achieves state-of-the-art", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 113, + 658, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 113, + 658, + 504, + 669 + ], + "score": 1.0, + "content": "robustness against corruptions and improves robustness against other common distribution shifts.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 570, + 505, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "AugMax achieves a good unification between sample diversity and hard corner-case generation during", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "data augmentation. AugMax is built on top of the AugMix framework [5] which mixes multiple data", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "augmentation operators in a multi-branch and layered pipeline. However, different from AugMix", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "where augmentation operators and mixing weights are both randomly sampled, operators are first", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "score": 1.0, + "content": "randomly sampled, followed by adversarially trained mixture of the selected operators in Augmax.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "This simple change from AugMix to AugMax results in considerable difference in their feature", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "distributions. From the visualizations (in Figure 1), AugMax generates more adversarially “hard", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "samples\", while still keeping a good amount of diversity compared to AugMix and PGD Attack.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "Searching for adversarial mixing strategies in AugMax is slightly more expensive since it involves", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "adversarial mixing. To make this efficient, we adopt an efficient adversarial training strategy [28]. As", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "a result, AugMax adds only a reasonable amount of extra complexity compared to non-adversarial", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "training methods, while improving robustness significantly. For example, AugMax training time is", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 127, + 354 + ], + "score": 1.0, + "content": "only", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 127, + 342, + 153, + 352 + ], + "score": 0.85, + "content": "\\sim 1 . 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 153, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "times that of AugMix on ImageNet (see Table 7). Moreover, Augmax is significantly", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 359, + 365 + ], + "score": 1.0, + "content": "more efficient than traditional adversarial training which has", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 359, + 353, + 383, + 363 + ], + "score": 0.83, + "content": "\\sim 1 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 383, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "times training time overhead", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 364, + 207, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 207, + 375 + ], + "score": 1.0, + "content": "compared with AugMix.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 677, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 69, + 500, + 177 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 69, + 500, + 177 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 69, + 500, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 69, + 500, + 177 + ], + "score": 0.964, + "type": "image", + "image_path": "5c2b2929baf2ee2ecdb10d564875ad242f4ebe5bd3f8d0fcce5c19740424a479.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 69, + 500, + 105.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 105.0, + 500, + 141.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 141.0, + 500, + 177.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 188, + 505, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "Figure 2: AugMax overview. Black and red arrows represent forward paths and back-propagation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "paths to generate AugMax images, respectively. In contrast to AugMax, where the mixing parameters", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 107, + 212, + 117, + 220 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 210, + 133, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 133, + 212, + 143, + 220 + ], + "score": 0.64, + "content": "\\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 210, + 352, + 222 + ], + "score": 1.0, + "content": "are adversarially learned, AugMix randomly samples", + "type": "text" + }, + { + "bbox": [ + 352, + 212, + 362, + 220 + ], + "score": 0.69, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 210, + 379, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 212, + 389, + 220 + ], + "score": 0.62, + "content": "\\pmb { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "from predefined distributions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 222, + 287, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 242, + 234 + ], + "score": 1.0, + "content": "(and thus no backpropergation on", + "type": "text" + }, + { + "bbox": [ + 243, + 223, + 253, + 231 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 222, + 270, + 234 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 223, + 280, + 231 + ], + "score": 0.69, + "content": "\\textbf { \\em w }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 222, + 287, + 234 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "score": 1.0, + "content": "randomly sampled, followed by adversarially trained mixture of the selected operators in Augmax.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "This simple change from AugMix to AugMax results in considerable difference in their feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "distributions. From the visualizations (in Figure 1), AugMax generates more adversarially “hard", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "samples\", while still keeping a good amount of diversity compared to AugMix and PGD Attack.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "Searching for adversarial mixing strategies in AugMax is slightly more expensive since it involves", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "adversarial mixing. To make this efficient, we adopt an efficient adversarial training strategy [28]. 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[1] pioneer the study of prediction errors exhibited by machine learning models on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "unseen natural corruptions and perturbations. They introduce two variants of the original ImageNet", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "score": 1.0, + "content": "validation set to evaluate the model’s robustness: the ImageNet-C for input corruption robustness,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "and the ImageNet-P for input perturbation robustness. The former applies 15 diverse corruptions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "drawn from four main categories (noise, blur, weather, and digital) on ImageNet validation images.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "The latter contains perturbation sequences generated from each image. Recht et al. [30, 2] collect", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "CIFAR10.1 and ImageNet-V2 as a reproduction of the original CIFAR10 [31] and ImageNet [32]", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "test sets, and find that a minute natural distribution shift caused by the minutiae in dataset collection", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "process leads to a large drop in accuracy for a broad range of image classifiers. Geirhos et al. [33]", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "observe that deep models trained on ImageNet are biased towards textures, and the model robustness", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "can be improved by emphasizing more on global shape features [33–36]. Wang et al. 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AugMax-DuBIN trains on clean images", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "and achieves state-of-the-art robustness on natural corruption benchmarks [1], and also improves", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "model robustness against other common distribution shifts [30, 2]. In particular, our method surpasses", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 507, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 475, + 518 + ], + "score": 1.0, + "content": "state-of-the-art method on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C by", + "type": "text" + }, + { + "bbox": [ + 476, + 505, + 502, + 516 + ], + "score": 0.85, + "content": "3 . 0 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 505, + 507, + 518 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 262, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 133, + 527 + ], + "score": 0.88, + "content": "3 . 4 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 514, + 137, + 530 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 137, + 516, + 164, + 527 + ], + "score": 0.83, + "content": "1 . 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 514, + 182, + 530 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 516, + 209, + 527 + ], + "score": 0.87, + "content": "0 . 7 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 514, + 262, + 530 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 471, + 507, + 530 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 544, + 197, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 198, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 198, + 559 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 569, + 279, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 279, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 279, + 582 + ], + "score": 1.0, + "content": "2.1 Robustness to Distributional Shifts", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "Hendrycks et al. [1] pioneer the study of prediction errors exhibited by machine learning models on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "unseen natural corruptions and perturbations. They introduce two variants of the original ImageNet", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "score": 1.0, + "content": "validation set to evaluate the model’s robustness: the ImageNet-C for input corruption robustness,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "and the ImageNet-P for input perturbation robustness. The former applies 15 diverse corruptions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "drawn from four main categories (noise, blur, weather, and digital) on ImageNet validation images.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "The latter contains perturbation sequences generated from each image. Recht et al. [30, 2] collect", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "CIFAR10.1 and ImageNet-V2 as a reproduction of the original CIFAR10 [31] and ImageNet [32]", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "test sets, and find that a minute natural distribution shift caused by the minutiae in dataset collection", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "process leads to a large drop in accuracy for a broad range of image classifiers. Geirhos et al. [33]", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "observe that deep models trained on ImageNet are biased towards textures, and the model robustness", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "can be improved by emphasizing more on global shape features [33–36]. Wang et al. [37] find", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "that the benchmark test performance does not always translate to the real-world generalizability,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 73, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 107, + 73, + 506, + 83 + ], + "score": 1.0, + "content": "and propose an efficient method to troubleshoot trained models using real-world unlabeled images.", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "Hendrycks et al. [38] collect ImageNet-A with the new concept of natural adversarial examples:", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "unmodified real-world images that falsify common machine learning models. Other representative", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "benchmark efforts include [39–42, 3], all demonstrating the brittleness of deep models under various", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 402, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 402, + 129 + ], + "score": 1.0, + "content": "distribution shifts. Readers of interest are referred to a recent survey [43].", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 590, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 107, + 73, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 107, + 73, + 506, + 83 + ], + "score": 1.0, + "content": "and propose an efficient method to troubleshoot trained models using real-world unlabeled images.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "Hendrycks et al. [38] collect ImageNet-A with the new concept of natural adversarial examples:", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "unmodified real-world images that falsify common machine learning models. Other representative", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "benchmark efforts include [39–42, 3], all demonstrating the brittleness of deep models under various", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 402, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 402, + 129 + ], + "score": 1.0, + "content": "distribution shifts. Readers of interest are referred to a recent survey [43].", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 159, + 328, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 330, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 330, + 172 + ], + "score": 1.0, + "content": "2.2 Data Augmentation: Random and Adversarial", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "Data augmentations have been widely used to increase training set diversity and improve model", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "generalization ability [44, 21, 33, 22]. Among those methods, AugMix [5] achieves the outstanding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "robustness against natural corruptions by randomly mixing multiple diverse augmentations. It", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "establishes high performance bars on many natural corruption benchmarks including CIFAR10-C,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "CIFAR100-C and ImageNet-C [1]. DeepAugment [3] feeds clean images to image-to-image models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "with randomly perturbed model weights to generate visually diverse augmented images. Combined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "with AugMix, it achieves state-of-the-art robustness on ImageNet-C [1]. Lately, a concurrent work", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "score": 1.0, + "content": "MaxUp [45] generates hard training samples by selecting the worst-case weights in the MixUp [20]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "pipeline, to improve both model generalization ability and adversarial robustness. Our work shared a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "similar mindset to MaxUp, but we build on the more sophisticated AugMix pipeline and handle the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 401, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 401, + 309 + ], + "score": 1.0, + "content": "resulting higher feature heterogeneity using a new normalization module.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "Besides, Adversarial training (AT) [6, 46–52] utilizes adversarial samples as a special data augmenta-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "tion method, and has also shown promise in improving model robustness against natural corruptions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "[23] and domain gaps [25], usually at the cost of increasing training time [27] and degrading the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 343, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 506, + 359 + ], + "score": 1.0, + "content": "standard accuracy on clean images [53]. Volpi et al. [25] and Zhao et al. [26] augment the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "score": 1.0, + "content": "set with samples from a fictitious adversarial domain to improve domain adaptation performance.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "Xie et al. [7] show that adversarial training can improve both standard accuracy and robustness", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 376, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 506, + 392 + ], + "score": 1.0, + "content": "against distribution shifts, with the help of auxiliary batch normalization. Adversarial Noise Training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "(ANT) [8] uses random Gaussian noise with learned adversarial hyperparameters (mean and variance)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "score": 1.0, + "content": "as the augmentation. Gowal et al. [54] propose to generate new augmented images by adversarially", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "composing the representations of different original images. Wang et al. [55] improve the robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "of pose estimation models by adversarially mixing different corrupted images as data augmentation.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "Based on DeepAugment [3], Calian et al. [56] propose AdversarialAugment, which optimizes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "the parameters of image-to-image models to generate adversarially augmented images, achieving", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "score": 1.0, + "content": "promising results against natural corruptions. Robey et al. [57] proposed model-based robust learning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "which generates augmented images with adversarial natural variations to improve model robustness", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "against naturally-occurring conditions (e.g., snow, decolorization, shadow, and many others). The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "authors further generalized such idea to the domain generalization problem and achieved impressive", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 156, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 156, + 510 + ], + "score": 1.0, + "content": "results [58].", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 541, + 192, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 193, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 193, + 555 + ], + "score": 1.0, + "content": "2.3 Normalization", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 504, + 581 + ], + "score": 1.0, + "content": "Batch Normalization (BN) [59] has been a cornerstone for training deep networks. Inspired by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "BN, more task-specific modifications are proposed by exploiting different normalization axes, such", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "as Instance Normalization (IN) [60], Layer Normalization (LN) [61] and Group Normalization", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "(GN) [62]. A few latest works investigate to use multiple normalization layers instead of one BN.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "Zajkac et al. [63] use two separate BNs to handle the domain shift between labeled and unlabeled", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "data in semi-supervised learning. Xie et al. [64, 7] observe the difference between standard and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "adversarial feature statistics during AT, and craft dual batch normalization (DuBN) to disentangle the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "standard and adversarial feature statistic to improve both the standard accuracy and robustness. While", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "most deep models just use one normalization type throughout the network, IBN-Net [29] unifies IN", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "and BN in one model. It finds that when applied together, IN tends to learn features invariant to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "appearance (colors, styles, etc.) changes, while BN is essential for preserving high-level semantic", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "contents. IBN-Net shows that combining IN and BN in an appropriate manner can improve model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "generalization. A similar combination was also explored in the style transfer field [65], learning to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 405, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 405, + 723 + ], + "score": 1.0, + "content": "selectively normalize only disturbing styles while preserving useful styles.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 127 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 73, + 506, + 129 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 159, + 328, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 330, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 330, + 172 + ], + "score": 1.0, + "content": "2.2 Data Augmentation: Random and Adversarial", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "Data augmentations have been widely used to increase training set diversity and improve model", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "generalization ability [44, 21, 33, 22]. Among those methods, AugMix [5] achieves the outstanding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "robustness against natural corruptions by randomly mixing multiple diverse augmentations. It", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "establishes high performance bars on many natural corruption benchmarks including CIFAR10-C,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "CIFAR100-C and ImageNet-C [1]. DeepAugment [3] feeds clean images to image-to-image models", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "with randomly perturbed model weights to generate visually diverse augmented images. Combined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "with AugMix, it achieves state-of-the-art robustness on ImageNet-C [1]. Lately, a concurrent work", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "score": 1.0, + "content": "MaxUp [45] generates hard training samples by selecting the worst-case weights in the MixUp [20]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "pipeline, to improve both model generalization ability and adversarial robustness. Our work shared a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "similar mindset to MaxUp, but we build on the more sophisticated AugMix pipeline and handle the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 401, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 401, + 309 + ], + "score": 1.0, + "content": "resulting higher feature heterogeneity using a new normalization module.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 187, + 506, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 506, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "Besides, Adversarial training (AT) [6, 46–52] utilizes adversarial samples as a special data augmenta-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "tion method, and has also shown promise in improving model robustness against natural corruptions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "[23] and domain gaps [25], usually at the cost of increasing training time [27] and degrading the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 343, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 506, + 359 + ], + "score": 1.0, + "content": "standard accuracy on clean images [53]. Volpi et al. [25] and Zhao et al. [26] augment the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "score": 1.0, + "content": "set with samples from a fictitious adversarial domain to improve domain adaptation performance.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "Xie et al. [7] show that adversarial training can improve both standard accuracy and robustness", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 376, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 506, + 392 + ], + "score": 1.0, + "content": "against distribution shifts, with the help of auxiliary batch normalization. Adversarial Noise Training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "(ANT) [8] uses random Gaussian noise with learned adversarial hyperparameters (mean and variance)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "score": 1.0, + "content": "as the augmentation. Gowal et al. [54] propose to generate new augmented images by adversarially", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 423 + ], + "score": 1.0, + "content": "composing the representations of different original images. Wang et al. [55] improve the robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "of pose estimation models by adversarially mixing different corrupted images as data augmentation.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "Based on DeepAugment [3], Calian et al. [56] propose AdversarialAugment, which optimizes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "the parameters of image-to-image models to generate adversarially augmented images, achieving", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "score": 1.0, + "content": "promising results against natural corruptions. Robey et al. [57] proposed model-based robust learning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "which generates augmented images with adversarial natural variations to improve model robustness", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "against naturally-occurring conditions (e.g., snow, decolorization, shadow, and many others). The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "authors further generalized such idea to the domain generalization problem and achieved impressive", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 156, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 156, + 510 + ], + "score": 1.0, + "content": "results [58].", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 312, + 507, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 541, + 192, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 193, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 193, + 555 + ], + "score": 1.0, + "content": "2.3 Normalization", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 504, + 581 + ], + "score": 1.0, + "content": "Batch Normalization (BN) [59] has been a cornerstone for training deep networks. Inspired by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "BN, more task-specific modifications are proposed by exploiting different normalization axes, such", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "as Instance Normalization (IN) [60], Layer Normalization (LN) [61] and Group Normalization", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "(GN) [62]. A few latest works investigate to use multiple normalization layers instead of one BN.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "Zajkac et al. [63] use two separate BNs to handle the domain shift between labeled and unlabeled", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "data in semi-supervised learning. Xie et al. [64, 7] observe the difference between standard and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "adversarial feature statistics during AT, and craft dual batch normalization (DuBN) to disentangle the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "standard and adversarial feature statistic to improve both the standard accuracy and robustness. While", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "most deep models just use one normalization type throughout the network, IBN-Net [29] unifies IN", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "and BN in one model. It finds that when applied together, IN tends to learn features invariant to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "appearance (colors, styles, etc.) changes, while BN is essential for preserving high-level semantic", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "contents. IBN-Net shows that combining IN and BN in an appropriate manner can improve model", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "generalization. A similar combination was also explored in the style transfer field [65], learning to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 711, + 405, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 405, + 723 + ], + "score": 1.0, + "content": "selectively normalize only disturbing styles while preserving useful styles.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 569, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 166, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 168, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 168, + 86 + ], + "score": 1.0, + "content": "3 Method", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 95, + 294, + 108 + ], + "lines": [ + { + "bbox": [ + 105, + 93, + 295, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 295, + 110 + ], + "score": 1.0, + "content": "3.1 Preliminary: The AugMix Framework", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 116, + 506, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "score": 1.0, + "content": "At a high level, AugMix creates a new composite image from a clean sample by first applying", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "several simple augmentation operations to it before combining the results through random linear", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "combinations. This augmentation scheme is coupled with a Jensen-Shannon consistency loss that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "enforces feature similarity among clean and augmented images. Specifically, AugMix consists of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "multiple augmentation chains (3 by default), each applied in parallel to the same input image. Each", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "chain composes one to three randomly selected augmentation operations. These augmentations are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "the same as AutoAugment [21], but excluding those that overlap with ImageNet-C corruptions. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "augmented images from each augmentation chains are then combined together with the original,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "clean sample using a linear combinations with randomly samples coefficients. The final image thus", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "incorporates several sources of randomness, coming respectively from the choice of operations, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "severity level of these operations, the lengths of the augmentation chains and the mixing weights.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "Despite its simplicity, AugMix achieves state-of-the-art corruption robustness. In the remaining of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 261 + ], + "score": 1.0, + "content": "this section, we build on the previous AugMix work and propose AugMax, an augmentation strategy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 240, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 240, + 270 + ], + "score": 1.0, + "content": "that further improves robustness.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 420, + 294 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 421, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 421, + 298 + ], + "score": 1.0, + "content": "3.2 AugMax: Augmented Training with Unified Diversity and Hardness", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "In this section, we rigorously introduce our new data augmentation method, AugMax, illustrated in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "Figure 2. 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The final image thus", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "incorporates several sources of randomness, coming respectively from the choice of operations, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "severity level of these operations, the lengths of the augmentation chains and the mixing weights.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "Despite its simplicity, AugMix achieves state-of-the-art corruption robustness. In the remaining of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 245, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 261 + ], + "score": 1.0, + "content": "this section, we build on the previous AugMix work and propose AugMax, an augmentation strategy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 240, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 240, + 270 + ], + "score": 1.0, + "content": "that further improves robustness.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 115, + 506, + 270 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 281, + 420, + 294 + ], + "lines": [ + { + "bbox": [ + 104, + 280, + 421, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 421, + 298 + ], + "score": 1.0, + "content": "3.2 AugMax: Augmented Training with Unified Diversity and Hardness", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "In this section, we rigorously introduce our new data augmentation method, AugMax, illustrated in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "Figure 2. 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Specifically, we visualize the impact of various data augmen-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "tation methods on the feature representations obtained with a ResNeXt29 that is normally trained on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "CIFAR-10. Note that the model is trained using only default standard augmentations (i.e., random", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "flipping and translation), and therefore has never seen augmentation from AugMix, AugMax, or PGD", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "attack. Following [4], we visualize the penultimate layer feature distributions of training samples from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "three different classes. Three hundred training images are selected from each class for visualization.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 492, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 175, + 173 + ], + "score": 1.0, + "content": "For AugMix and", + "type": "text" + }, + { + "bbox": [ + 176, + 160, + 190, + 171 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 159, + 492, + 173 + ], + "score": 1.0, + "content": "PGD attack, we use the same hyperparameters in the original papers [5, 6].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 506, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 507, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 507, + 190 + ], + "score": 1.0, + "content": "We observe in Figure 1 that the clean images form well-separated clusters in the feature space,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "leaving large regions between those clusters empty. This potentially causes uncertain and unreliable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "generalization if new test samples (from a shifted distribution) fall into those regions. AugMix", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "enlarges each cluster to cover a broader region, effectively increasing sample diversity. However,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "after augmentation, very few samples are close to the decision boundaries, revealing an inability to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "generate hard samples. Samples from the PGD attack, on the other hand, collapse into a small region", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "around the classification borders, losing feature diversity. We further investigate the method through", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 231, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 231, + 265 + ], + "score": 1.0, + "content": "ablation studies in Section 4.3.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 277, + 406, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 407, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 407, + 292 + ], + "score": 1.0, + "content": "3.3 DuBIN: Disentangled Normalization for Heterogeneous Features", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 336, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 336, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 336, + 312 + ], + "score": 1.0, + "content": "AugMax leads to better coverage of the input space by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 336, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 336, + 321 + ], + "score": 1.0, + "content": "unifying sample diversity and hardness, as demonstrated", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 337, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 337, + 332 + ], + "score": 1.0, + "content": "in Figure 1. While such more versatile distribution in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 338, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 338, + 344 + ], + "score": 1.0, + "content": "data augmentation has the potential to increase robust-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 337, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 337, + 354 + ], + "score": 1.0, + "content": "ness, we do observe that naive incorporation of AugMax", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 337, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 337, + 365 + ], + "score": 1.0, + "content": "leads to relatively marginal improvement over AugMix", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 364, + 337, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 337, + 376 + ], + "score": 1.0, + "content": "(see Section 4.3). This is due to the comprehensiveness of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 338, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 338, + 387 + ], + "score": 1.0, + "content": "AugMax can also lead to a higher feature heterogeneity,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 338, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 338, + 397 + ], + "score": 1.0, + "content": "which may require larger model capacity to encode. To ad-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 338, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 338, + 409 + ], + "score": 1.0, + "content": "dress this problem, we design a novel normalization layer,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 338, + 420 + ], + "score": 1.0, + "content": "termed Dual Batch-and-Instance Normalization (DuBIN),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 416, + 295, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 295, + 433 + ], + "score": 1.0, + "content": "to disentangle the instance-level heterogeneity.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 336, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "score": 1.0, + "content": "As illustrated in Figure 3, DuBIN consists of two parallel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 336, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 336, + 457 + ], + "score": 1.0, + "content": "parts: (1) a dual batch normalization (DuBN) [7], which is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 336, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 336, + 468 + ], + "score": 1.0, + "content": "the by-default normalization layer to use for adversarial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "training [7, 64], to disentangle the group-level statistics of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 337, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 337, + 491 + ], + "score": 1.0, + "content": "the clean and corner-case augmented samples2; and (2) an", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 489, + 337, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 337, + 501 + ], + "score": 1.0, + "content": "additional instance normalization, to account for instance-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 500, + 337, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 337, + 511 + ], + "score": 1.0, + "content": "level feature statistics due to augmentation diversity. 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The input feature", + "type": "text" + }, + { + "bbox": [ + 464, + 456, + 471, + 464 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 454, + 495, + 465 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 495, + 454, + 504, + 464 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 343, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 343, + 465, + 473, + 477 + ], + "score": 1.0, + "content": "channels is split into halves (i.e.,", + "type": "text" + }, + { + "bbox": [ + 474, + 467, + 487, + 476 + ], + "score": 0.86, + "content": "x _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 344, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 344, + 478, + 357, + 487 + ], + "score": 0.83, + "content": "x _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 476, + 402, + 487 + ], + "score": 1.0, + "content": ", each with", + "type": "text" + }, + { + "bbox": [ + 402, + 476, + 421, + 488 + ], + "score": 0.88, + "content": "C / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 476, + 506, + 487 + ], + "score": 1.0, + "content": "channels). Each half", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 343, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 343, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "goes into different normalization layers", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 343, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 343, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "(either IN or DuBN) and the outputs are", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 343, + 509, + 507, + 520 + ], + "spans": [ + { + "bbox": [ + 343, + 509, + 507, + 520 + ], + "score": 1.0, + "content": "concatenated back along the channel di-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 343, + 520, + 382, + 530 + ], + "spans": [ + { + "bbox": [ + 343, + 520, + 382, + 530 + ], + "score": 1.0, + "content": "mension.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 52.5 + } + ], + "index": 47.75 + }, + { + "type": "text", + "bbox": [ + 109, + 533, + 504, + 555 + ], + "lines": [ + { + "bbox": [ + 107, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "other fed into DuBN. Finally, the outputs of IN and DuBN are concatenated back along the channel", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 543, + 223, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 223, + 556 + ], + "score": 1.0, + "content": "dimension for the next layer.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 58.5 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 297, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 297, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 297, + 572 + ], + "score": 1.0, + "content": "To provide additional insights of DuBIN, we", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 571, + 297, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 297, + 582 + ], + "score": 1.0, + "content": "compare the BN statistics of AugMax-DuBN", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 102, + 577, + 295, + 600 + ], + "spans": [ + { + "bbox": [ + 102, + 577, + 250, + 600 + ], + "score": 1.0, + "content": "and AugMax-DuBIN in Table 1.", + "type": "text" + }, + { + "bbox": [ + 250, + 581, + 262, + 594 + ], + "score": 0.88, + "content": "\\overline { { \\sigma } } _ { c } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 577, + 282, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 283, + 581, + 295, + 594 + ], + "score": 0.88, + "content": "\\overline { { \\sigma } } _ { a } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 593, + 298, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 298, + 604 + ], + "score": 1.0, + "content": "represent the average batch normalization vari-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 603, + 298, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 213, + 615 + ], + "score": 1.0, + "content": "ance over all channels of", + "type": "text" + }, + { + "bbox": [ + 213, + 604, + 234, + 615 + ], + "score": 0.91, + "content": "B N _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 604, + 254, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 603, + 280, + 614 + ], + "score": 0.91, + "content": "B N _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 604, + 298, + 615 + ], + "score": 1.0, + "content": ", re-", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 104, + 614, + 295, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 250, + 627 + ], + "score": 1.0, + "content": "spectively. We observe that both", + "type": "text" + }, + { + "bbox": [ + 250, + 614, + 262, + 627 + ], + "score": 0.88, + "content": "\\overline { { \\sigma } } _ { c } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 614, + 282, + 627 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 283, + 614, + 295, + 627 + ], + "score": 0.87, + "content": "\\overline { { \\sigma } } _ { a } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 626, + 297, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 297, + 637 + ], + "score": 1.0, + "content": "are smaller in the DuBIN network than those", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 636, + 297, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 297, + 648 + ], + "score": 1.0, + "content": "in the DuBN counterpart. This shows that the", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 106, + 648, + 298, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 298, + 658 + ], + "score": 1.0, + "content": "IN branch in DuBIN can reduce the feature di-", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 106, + 659, + 297, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 297, + 671 + ], + "score": 1.0, + "content": "versity that BN otherwise needs to model, by", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 64.5 + }, + { + "type": "table", + "bbox": [ + 312, + 598, + 495, + 667 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 303, + 562, + 505, + 595 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 303, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Table 1: BN statistics of different layers in", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 303, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 303, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "WRN40-2 with DuBN or DuBIN trained on CI-", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 303, + 582, + 438, + 597 + ], + "spans": [ + { + "bbox": [ + 303, + 582, + 438, + 597 + ], + "score": 1.0, + "content": "FAR100 using AugMax training.", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 71 + }, + { + "type": "table_body", + "bbox": [ + 312, + 598, + 495, + 667 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 312, + 598, + 495, + 667 + ], + "spans": [ + { + "bbox": [ + 312, + 598, + 495, + 667 + ], + "score": 0.978, + "html": "
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Specifically, we visualize the impact of various data augmen-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "tation methods on the feature representations obtained with a ResNeXt29 that is normally trained on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "CIFAR-10. Note that the model is trained using only default standard augmentations (i.e., random", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "flipping and translation), and therefore has never seen augmentation from AugMix, AugMax, or PGD", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "attack. Following [4], we visualize the penultimate layer feature distributions of training samples from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "three different classes. Three hundred training images are selected from each class for visualization.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 492, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 175, + 173 + ], + "score": 1.0, + "content": "For AugMix and", + "type": "text" + }, + { + "bbox": [ + 176, + 160, + 190, + 171 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 159, + 492, + 173 + ], + "score": 1.0, + "content": "PGD attack, we use the same hyperparameters in the original papers [5, 6].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 72, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 506, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 507, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 507, + 190 + ], + "score": 1.0, + "content": "We observe in Figure 1 that the clean images form well-separated clusters in the feature space,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "leaving large regions between those clusters empty. This potentially causes uncertain and unreliable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "generalization if new test samples (from a shifted distribution) fall into those regions. AugMix", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "enlarges each cluster to cover a broader region, effectively increasing sample diversity. However,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "after augmentation, very few samples are close to the decision boundaries, revealing an inability to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "generate hard samples. Samples from the PGD attack, on the other hand, collapse into a small region", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "around the classification borders, losing feature diversity. We further investigate the method through", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 231, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 231, + 265 + ], + "score": 1.0, + "content": "ablation studies in Section 4.3.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 176, + 507, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 277, + 406, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 407, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 407, + 292 + ], + "score": 1.0, + "content": "3.3 DuBIN: Disentangled Normalization for Heterogeneous Features", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 336, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 336, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 336, + 312 + ], + "score": 1.0, + "content": "AugMax leads to better coverage of the input space by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 336, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 336, + 321 + ], + "score": 1.0, + "content": "unifying sample diversity and hardness, as demonstrated", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 337, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 337, + 332 + ], + "score": 1.0, + "content": "in Figure 1. While such more versatile distribution in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 338, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 338, + 344 + ], + "score": 1.0, + "content": "data augmentation has the potential to increase robust-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 337, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 337, + 354 + ], + "score": 1.0, + "content": "ness, we do observe that naive incorporation of AugMax", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 337, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 337, + 365 + ], + "score": 1.0, + "content": "leads to relatively marginal improvement over AugMix", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 364, + 337, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 337, + 376 + ], + "score": 1.0, + "content": "(see Section 4.3). This is due to the comprehensiveness of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 338, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 338, + 387 + ], + "score": 1.0, + "content": "AugMax can also lead to a higher feature heterogeneity,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 338, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 338, + 397 + ], + "score": 1.0, + "content": "which may require larger model capacity to encode. To ad-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 338, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 338, + 409 + ], + "score": 1.0, + "content": "dress this problem, we design a novel normalization layer,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 338, + 420 + ], + "score": 1.0, + "content": "termed Dual Batch-and-Instance Normalization (DuBIN),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 416, + 295, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 295, + 433 + ], + "score": 1.0, + "content": "to disentangle the instance-level heterogeneity.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 298, + 338, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 336, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 336, + 446 + ], + "score": 1.0, + "content": "As illustrated in Figure 3, DuBIN consists of two parallel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 336, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 336, + 457 + ], + "score": 1.0, + "content": "parts: (1) a dual batch normalization (DuBN) [7], which is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 336, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 336, + 468 + ], + "score": 1.0, + "content": "the by-default normalization layer to use for adversarial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "training [7, 64], to disentangle the group-level statistics of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 337, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 337, + 491 + ], + "score": 1.0, + "content": "the clean and corner-case augmented samples2; and (2) an", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 489, + 337, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 337, + 501 + ], + "score": 1.0, + "content": "additional instance normalization, to account for instance-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 500, + 337, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 337, + 511 + ], + "score": 1.0, + "content": "level feature statistics due to augmentation diversity. At", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 511, + 336, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 336, + 523 + ], + "score": 1.0, + "content": "each layer, the input feature is split into halves along the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 522, + 336, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 522, + 336, + 533 + ], + "score": 1.0, + "content": "channel dimension: one half is then fed into IN, and the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "other fed into DuBN. 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With lower feature variation, BNs can converge better with", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 106, + 679, + 221, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 221, + 692 + ], + "score": 1.0, + "content": "improved performance [66].", + "type": "text" + } + ], + "index": 79 + } + ], + "index": 78.5, + "bbox_fs": [ + 106, + 669, + 505, + 692 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 191, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 193, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 193, + 88 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 505, + 110 + ], + "score": 1.0, + "content": "Here, we introduce in detail the experimental setting, and report quantitative results for robustness", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 121 + ], + "score": 1.0, + "content": "against natural corruptions and distribution shifts. 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For neural architectures, we use a ResNet18 [67], WRN40-2 [68]", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "and ResNeXt29 [69] on the CIFAR datasets, and ResNet18 on ImageNet and Tiny ImageNet4. To", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "evaluate the model’s robustness against common natural corruptions, we use CIFAR10-C, CIFAR100-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "C, ImageNet-C and Tiny ImageNet-C (TIN-C) [1], which are generated by adding natural corruptions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 220, + 228, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 228, + 235 + ], + "score": 1.0, + "content": "to the original test set images.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 297, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 297, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 297, + 259 + ], + "score": 1.0, + "content": "Evaluation Metrics Following established", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 298, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 298, + 270 + ], + "score": 1.0, + "content": "conventions [1, 5], we define robustness accu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 297, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 297, + 281 + ], + "score": 1.0, + "content": "racy (RA) as the average classification accuracy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 297, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 297, + 291 + ], + "score": 1.0, + "content": "over all 15 corruptions, to measure the models’", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 297, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 297, + 302 + ], + "score": 1.0, + "content": "robustness on CIFAR10-C and CIFAR100-C.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 297, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 297, + 313 + ], + "score": 1.0, + "content": "On ImageNet-C and Tiny ImageNet-C, we use", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 297, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 297, + 325 + ], + "score": 1.0, + "content": "both RA and mean corruption error (mCE) to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 297, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 297, + 335 + ], + "score": 1.0, + "content": "evaluate robustness. As defined by [1], mCE is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 296, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 296, + 346 + ], + "score": 1.0, + "content": "the weighted average of target model corruption", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "table", + "bbox": [ + 305, + 274, + 503, + 345 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 249, + 505, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 303, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "Table 2: Evaluation results on CIFAR10 and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 304, + 259, + 492, + 272 + ], + "spans": [ + { + "bbox": [ + 304, + 259, + 492, + 272 + ], + "score": 1.0, + "content": "CIFAR10-C. The best metric is shown in bold.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "table_body", + "bbox": [ + 305, + 274, + 503, + 345 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 305, + 274, + 503, + 345 + ], + "spans": [ + { + "bbox": [ + 305, + 274, + 503, + 345 + ], + "score": 0.977, + "html": "
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)95.5695.7995.76 (-0.03)
RA(%)74.7589.4990.36 (+0.87)
WRN40-2SA(%)94.7895.6795.68 (+0.01)
RA(%)73.7189.0190.67 (+1.66)
ResNeXt29SA(%)95.6096.2596.39 (+0.14)
RA(%)71.7089.0892.11 (+3.03)
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Since [1] does", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "not provide an off-the-shelf baseline model for Tiny ImageNet, we use the conventionally trained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 376, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 391 + ], + "score": 1.0, + "content": "ResNet18 on Tiny ImageNet as the baseline model for experiments on Tiny ImageNet. We use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 388, + 495, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 495, + 402 + ], + "score": 1.0, + "content": "standard accuracy (SA) to denote the classification accuracy on the original clean testing images.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 297, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 297, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 297, + 420 + ], + "score": 1.0, + "content": "Baseline Methods On CIFAR10-C and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 419, + 297, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 297, + 432 + ], + "score": 1.0, + "content": "CIFAR100-C, we compare our method with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 430, + 298, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 298, + 443 + ], + "score": 1.0, + "content": "the state-of-the-art method, AugMix [5].", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 442, + 297, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 297, + 454 + ], + "score": 1.0, + "content": "In addition, AugMix has previously been", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 453, + 298, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 298, + 464 + ], + "score": 1.0, + "content": "combined with other orthogonal methods,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 464, + 297, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 297, + 475 + ], + "score": 1.0, + "content": "such as DeepAugment [3], to further boost", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 475, + 297, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 297, + 487 + ], + "score": 1.0, + "content": "performance on ImageNet-C. Specifically, [3]", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 486, + 298, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 298, + 497 + ], + "score": 1.0, + "content": "combine DeepAugment as an orthogonal com-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 496, + 298, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 298, + 508 + ], + "score": 1.0, + "content": "ponent to AugMix, and achieve state-of-the-art", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "table", + "bbox": [ + 305, + 433, + 503, + 505 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 409, + 504, + 431 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 303, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "Table 3: Evaluation results on CIFAR100 and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 304, + 419, + 497, + 432 + ], + "spans": [ + { + "bbox": [ + 304, + 419, + 497, + 432 + ], + "score": 1.0, + "content": "CIFAR100-C. The best metric is shown in bold.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "table_body", + "bbox": [ + 305, + 433, + 503, + 505 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 305, + 433, + 503, + 505 + ], + "spans": [ + { + "bbox": [ + 305, + 433, + 503, + 505 + ], + "score": 0.977, + "html": "
ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)77.9978.2378.69 (+0.46)
RA(%)48.4662.6765.75 (+3.08)
WRN40-2SA(%)76.1977.0376.80 (-0.23)
RA(%)46.8064.5666.35 (+1.79)
ResNeXt29SA(%)79.9578.5880.70 (+2.12)
RA(%)47.7665.3768.86 (+3.49)
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DeepAugment", + "type": "text" + }, + { + "bbox": [ + 418, + 531, + 426, + 540 + ], + "score": 0.75, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "AugMax. 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Since models tend to generalize bet-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 599, + 297, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 297, + 613 + ], + "score": 1.0, + "content": "ter on test images similar to those seen during", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 611, + 297, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 297, + 623 + ], + "score": 1.0, + "content": "training [70], ANT mainly improves robustness", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 622, + 298, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 298, + 634 + ], + "score": 1.0, + "content": "against high frequency corruptions (e.g., addi-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "score": 1.0, + "content": "tive noises), while achieving comparable robust-", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 644, + 297, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 297, + 657 + ], + "score": 1.0, + "content": "ness on other unseen corruptions (e.g., bluring)", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 655, + 297, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 297, + 667 + ], + "score": 1.0, + "content": "with AugMix. Following [8], we perform the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 666, + 297, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 297, + 678 + ], + "score": 1.0, + "content": "comparison on the subset of Tiny ImageNet-C", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 57 + }, + { + "type": "table", + "bbox": [ + 306, + 582, + 502, + 677 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 556, + 505, + 578 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 303, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 303, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "Table 4: Evaluation results on ImageNet and", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 303, + 567, + 493, + 579 + ], + "spans": [ + { + "bbox": [ + 303, + 567, + 493, + 579 + ], + "score": 1.0, + "content": "ImageNet-C. 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MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal69.8330.9187.47
AugMix68.06 67.6234.58 35.0183.08 82.56
AugMax-DuBIN(-0.44)(+0.43)(-0.52)
DeepAugment + AugMix65.3245.8469.29
DeepAugment +64.4346.5568.47
AugMax-DuBIN(-0.89)(+0.71)(-0.82)
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We show that our method induces better robustness", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 119, + 461, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 461, + 132 + ], + "score": 1.0, + "content": "than existing ones. We study the properties of our approach in thorough ablation studies.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 97, + 505, + 132 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 145, + 215, + 157 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 217, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 217, + 161 + ], + "score": 1.0, + "content": "4.1 Experimental Setup", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 506, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "score": 1.0, + "content": "Datasets and Models We evaluate our proposed method on CIFAR10, CIFAR100 [31], ImageNet", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 206, + 191 + ], + "score": 1.0, + "content": "[32] and Tiny ImageNet", + "type": "text" + }, + { + "bbox": [ + 207, + 177, + 235, + 189 + ], + "score": 0.4, + "content": "\\left( \\mathrm { T I N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 177, + 505, + 191 + ], + "score": 1.0, + "content": ". For neural architectures, we use a ResNet18 [67], WRN40-2 [68]", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "and ResNeXt29 [69] on the CIFAR datasets, and ResNet18 on ImageNet and Tiny ImageNet4. To", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "evaluate the model’s robustness against common natural corruptions, we use CIFAR10-C, CIFAR100-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "C, ImageNet-C and Tiny ImageNet-C (TIN-C) [1], which are generated by adding natural corruptions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 220, + 228, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 228, + 235 + ], + "score": 1.0, + "content": "to the original test set images.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 166, + 506, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 297, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 297, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 297, + 259 + ], + "score": 1.0, + "content": "Evaluation Metrics Following established", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 298, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 298, + 270 + ], + "score": 1.0, + "content": "conventions [1, 5], we define robustness accu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 297, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 297, + 281 + ], + "score": 1.0, + "content": "racy (RA) as the average classification accuracy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 297, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 297, + 291 + ], + "score": 1.0, + "content": "over all 15 corruptions, to measure the models’", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 297, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 297, + 302 + ], + "score": 1.0, + "content": "robustness on CIFAR10-C and CIFAR100-C.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 297, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 297, + 313 + ], + "score": 1.0, + "content": "On ImageNet-C and Tiny ImageNet-C, we use", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 297, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 297, + 325 + ], + "score": 1.0, + "content": "both RA and mean corruption error (mCE) to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 297, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 297, + 335 + ], + "score": 1.0, + "content": "evaluate robustness. As defined by [1], mCE is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 296, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 296, + 346 + ], + "score": 1.0, + "content": "the weighted average of target model corruption", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 246, + 298, + 346 + ] + }, + { + "type": "table", + "bbox": [ + 305, + 274, + 503, + 345 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 249, + 505, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 303, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "Table 2: Evaluation results on CIFAR10 and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 304, + 259, + 492, + 272 + ], + "spans": [ + { + "bbox": [ + 304, + 259, + 492, + 272 + ], + "score": 1.0, + "content": "CIFAR10-C. 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ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)95.5695.7995.76 (-0.03)
RA(%)74.7589.4990.36 (+0.87)
WRN40-2SA(%)94.7895.6795.68 (+0.01)
RA(%)73.7189.0190.67 (+1.66)
ResNeXt29SA(%)95.6096.2596.39 (+0.14)
RA(%)71.7089.0892.11 (+3.03)
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ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)77.9978.2378.69 (+0.46)
RA(%)48.4662.6765.75 (+3.08)
WRN40-2SA(%)76.1977.0376.80 (-0.23)
RA(%)46.8064.5666.35 (+1.79)
ResNeXt29SA(%)79.9578.5880.70 (+2.12)
RA(%)47.7665.3768.86 (+3.49)
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Since models tend to generalize bet-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 599, + 297, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 297, + 613 + ], + "score": 1.0, + "content": "ter on test images similar to those seen during", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 611, + 297, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 297, + 623 + ], + "score": 1.0, + "content": "training [70], ANT mainly improves robustness", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 622, + 298, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 298, + 634 + ], + "score": 1.0, + "content": "against high frequency corruptions (e.g., addi-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 298, + 645 + ], + "score": 1.0, + "content": "tive noises), while achieving comparable robust-", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 644, + 297, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 297, + 657 + ], + "score": 1.0, + "content": "ness on other unseen corruptions (e.g., bluring)", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 655, + 297, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 297, + 667 + ], + "score": 1.0, + "content": "with AugMix. Following [8], we perform the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 666, + 297, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 297, + 678 + ], + "score": 1.0, + "content": "comparison on the subset of Tiny ImageNet-C", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 676, + 367, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 367, + 690 + ], + "score": 1.0, + "content": "with 12 remaining corruptions after removing all additive noises.", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 57, + "bbox_fs": [ + 105, + 556, + 298, + 678 + ] + }, + { + "type": "table", + "bbox": [ + 306, + 582, + 502, + 677 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 556, + 505, + 578 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 303, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 303, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "Table 4: Evaluation results on ImageNet and", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 303, + 567, + 493, + 579 + ], + "spans": [ + { + "bbox": [ + 303, + 567, + 493, + 579 + ], + "score": 1.0, + "content": "ImageNet-C. The best metric is shown in bold.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 63.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 582, + 502, + 677 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 306, + 582, + 502, + 677 + ], + "spans": [ + { + "bbox": [ + 306, + 582, + 502, + 677 + ], + "score": 0.976, + "html": "
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal69.8330.9187.47
AugMix68.06 67.6234.58 35.0183.08 82.56
AugMax-DuBIN(-0.44)(+0.43)(-0.52)
DeepAugment + AugMix65.3245.8469.29
DeepAugment +64.4346.5568.47
AugMax-DuBIN(-0.89)(+0.71)(-0.82)
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Specifically, for all experi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "ments on CIAFR10 and CIFAR100, we use SGD optimizer with initial learning rate 0.1 and cosine", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "annealing learning rate scheduler, and train all models for 200 epochs. For all experiments on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "ImageNet, we use SGD optimizer with initial learning rate 0.1 and batch size 256 to train the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 328, + 129 + ], + "score": 1.0, + "content": "model for 90 epochs. We reduce the learning rate by", + "type": "text" + }, + { + "bbox": [ + 329, + 117, + 350, + 128 + ], + "score": 0.63, + "content": "1 / 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "at the 30-th and 60-th epoch. For all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "experiments on Tiny ImageNet, we train for 200 epochs and decay the learning rate at the 100-th", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "and 150-th epoch. Other hyperparameter settings are identical as those in ImageNet experiments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 296, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 298, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 298, + 161 + ], + "score": 1.0, + "content": "We set batch size to 256 for all experiments.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 159, + 297, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 154, + 172 + ], + "score": 1.0, + "content": "In Eq. 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AugMax follows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 215, + 297, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 297, + 226 + ], + "score": 1.0, + "content": "the same rules as AugMix [5] to randomly select", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 226, + 297, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 297, + 237 + ], + "score": 1.0, + "content": "augmentation operation type, severity level and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 236, + 296, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 296, + 248 + ], + "score": 1.0, + "content": "chain length. 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The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 304, + 164, + 420, + 175 + ], + "spans": [ + { + "bbox": [ + 304, + 164, + 420, + 175 + ], + "score": 1.0, + "content": "best metric is shown in bold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 178, + 503, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 178, + 503, + 272 + ], + "spans": [ + { + "bbox": [ + 306, + 178, + 503, + 272 + ], + "score": 0.98, + "html": "
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal61.6423.91100.00
AugMix61.79 62.2136.85 38.6783.04 80.72
AugMax-DuBIN(+0.42)(+1.82)(-2.32)
DeepAugment + AugMix59.5940.6778.28
DeepAugment +59.7240.9977.83
AugMax-DuBIN(+0.13)(+0.32)(-0.45)
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From the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "results, we see that AugMax-DuBIN achieves new state-of-the-art performance on both datasets with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "score": 1.0, + "content": "different model structures. For example, compared with AugMix, our method improves accuracy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 165, + 416 + ], + "score": 1.0, + "content": "by as high as", + "type": "text" + }, + { + "bbox": [ + 165, + 403, + 193, + 414 + ], + "score": 0.87, + "content": "3 . 0 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 402, + 213, + 416 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 404, + 241, + 414 + ], + "score": 0.87, + "content": "3 . 4 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "on CIFAR10-C and CIFAR100-C, respectively. Moreover, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "observe that our method benefits larger models more. Specifically, in both CIFAR10 and CIFAR100", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "experiments, the largest robustness gain is achieved on ResNeXt29, which has the largest capacity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 436, + 207, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 207, + 448 + ], + "score": 1.0, + "content": "among the three models.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 297, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 298, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 298, + 464 + ], + "score": 1.0, + "content": "Results on ImageNet are shown in Ta-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 462, + 297, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 297, + 475 + ], + "score": 1.0, + "content": "ble 4. AugMax-DuBIN outperforms AugMix", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 474, + 298, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 118, + 486 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 474, + 146, + 486 + ], + "score": 0.91, + "content": "0 . 5 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 474, + 191, + 486 + ], + "score": 1.0, + "content": "in terms of", + "type": "text" + }, + { + "bbox": [ + 191, + 475, + 213, + 485 + ], + "score": 0.28, + "content": "\\mathrm { m C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 474, + 298, + 486 + ], + "score": 1.0, + "content": "on ImageNet-C. Fur-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 483, + 298, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 298, + 499 + ], + "score": 1.0, + "content": "ther combining AugMax-DuBIN with DeepAug-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 495, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 215, + 509 + ], + "score": 1.0, + "content": "ment outperforms AugMix", + "type": "text" + }, + { + "bbox": [ + 215, + 497, + 224, + 506 + ], + "score": 0.69, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 495, + 297, + 509 + ], + "score": 1.0, + "content": "DeepAugment by", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 506, + 298, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 134, + 518 + ], + "score": 0.88, + "content": "0 . 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 506, + 298, + 519 + ], + "score": 1.0, + "content": "in terms of mCE, achieving a new state-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "score": 1.0, + "content": "of-the-art performance on ImageNet-C. This", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + }, + { + "type": "table", + "bbox": [ + 307, + 475, + 503, + 528 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 452, + 504, + 474 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 452, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 303, + 452, + 505, + 463 + ], + "score": 1.0, + "content": "Table 6: Evaluation results on TIN and TIN-C (w/o", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 303, + 463, + 469, + 474 + ], + "spans": [ + { + "bbox": [ + 303, + 463, + 469, + 474 + ], + "score": 1.0, + "content": "noise). The best metric is shown in bold.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + }, + { + "type": "table_body", + "bbox": [ + 307, + 475, + 503, + 528 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 307, + 475, + 503, + 528 + ], + "spans": [ + { + "bbox": [ + 307, + 475, + 503, + 528 + ], + "score": 0.97, + "html": "
MethodSA(%,↑)RA(%,↑)mCE(%,）
Normal61.6424.38100.00
AugMix61.7937.6382.54
ANT61.2635.3085.70
AugMax-DuBIN62.2138.6680.29
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From Table 5, we see that our method", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 567, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 193, + 579 + ], + "score": 1.0, + "content": "improves the mCE by", + "type": "text" + }, + { + "bbox": [ + 194, + 567, + 220, + 578 + ], + "score": 0.86, + "content": "2 . 3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 567, + 333, + 579 + ], + "score": 1.0, + "content": "compared with AugMix and", + "type": "text" + }, + { + "bbox": [ + 333, + 567, + 360, + 578 + ], + "score": 0.88, + "content": "0 . 4 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 567, + 504, + 579 + ], + "score": 1.0, + "content": "when combined with DeepAugment.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 577, + 440, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 440, + 590 + ], + "score": 1.0, + "content": "From Table 6, we can see our method outperforms ANT by a considerable margin.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 354, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 354, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 354, + 610 + ], + "score": 1.0, + "content": "Training Time Since we use an accelerated adversarial train-", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 355, + 621 + ], + "score": 1.0, + "content": "ing method [28] to solve Eq. (4), the worst-case mixing strat-", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 620, + 354, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 354, + 631 + ], + "score": 1.0, + "content": "egy can be searched efficiently. As a result, AugMax adds", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 631, + 354, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 354, + 642 + ], + "score": 1.0, + "content": "only a modest training overhead over AugMix, and is signif-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 642, + 355, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 355, + 654 + ], + "score": 1.0, + "content": "icantly more efficient than traditional adversarial training (AT).", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 652, + 278, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 278, + 664 + ], + "score": 1.0, + "content": "Detailed training time is shown in Table 7.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 61.5 + }, + { + "type": "table", + "bbox": [ + 364, + 626, + 502, + 688 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 360, + 593, + 504, + 625 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 360, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 360, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "Table 7: Training time on ImageNet", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 360, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 360, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "with ResNet18, reported on a single", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 360, + 614, + 431, + 626 + ], + "spans": [ + { + "bbox": [ + 360, + 614, + 431, + 626 + ], + "score": 1.0, + "content": "NVIDIA A6000.", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 66 + }, + { + "type": "table_body", + "bbox": [ + 364, + 626, + 502, + 688 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 364, + 626, + 502, + 688 + ], + "spans": [ + { + "bbox": [ + 364, + 626, + 502, + 688 + ], + "score": 0.968, + "html": "
MethodTime (sec/epoch)
Normal2669
AugMix3622
AT[6]37162
AugMax5264
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When using DuBN or DuBIN, we route", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 71.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 506, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "score": 1.0, + "content": "Implementation Details We follow [5] for hyperparameter settings. Specifically, for all experi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "ments on CIAFR10 and CIFAR100, we use SGD optimizer with initial learning rate 0.1 and cosine", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "annealing learning rate scheduler, and train all models for 200 epochs. For all experiments on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "ImageNet, we use SGD optimizer with initial learning rate 0.1 and batch size 256 to train the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 328, + 129 + ], + "score": 1.0, + "content": "model for 90 epochs. We reduce the learning rate by", + "type": "text" + }, + { + "bbox": [ + 329, + 117, + 350, + 128 + ], + "score": 0.63, + "content": "1 / 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "at the 30-th and 60-th epoch. For all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "experiments on Tiny ImageNet, we train for 200 epochs and decay the learning rate at the 100-th", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "and 150-th epoch. Other hyperparameter settings are identical as those in ImageNet experiments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 72, + 507, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 296, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 298, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 298, + 161 + ], + "score": 1.0, + "content": "We set batch size to 256 for all experiments.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 159, + 297, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 154, + 172 + ], + "score": 1.0, + "content": "In Eq. (4),", + "type": "text" + }, + { + "bbox": [ + 154, + 162, + 165, + 170 + ], + "score": 0.5, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 159, + 185, + 172 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 185, + 162, + 193, + 171 + ], + "score": 0.78, + "content": "\\pmb { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 159, + 297, + 172 + ], + "score": 1.0, + "content": "are uniformly randomly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "score": 1.0, + "content": "initialized between 0 and 1. We use PGD to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 182, + 297, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 136, + 194 + ], + "score": 1.0, + "content": "update", + "type": "text" + }, + { + "bbox": [ + 137, + 183, + 147, + 192 + ], + "score": 0.69, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 182, + 267, + 194 + ], + "score": 1.0, + "content": ": first do gradient ascend on", + "type": "text" + }, + { + "bbox": [ + 267, + 184, + 278, + 192 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 182, + 297, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 192, + 297, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 220, + 205 + ], + "score": 1.0, + "content": "then project it back into the", + "type": "text" + }, + { + "bbox": [ + 221, + 192, + 241, + 204 + ], + "score": 0.53, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 192, + 278, + 205 + ], + "score": 1.0, + "content": "interval.", + "type": "text" + }, + { + "bbox": [ + 279, + 194, + 286, + 204 + ], + "score": 0.78, + "content": "\\pmb { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 192, + 297, + 205 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 204, + 297, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 297, + 215 + ], + "score": 1.0, + "content": "updated using gradient ascend. AugMax follows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 215, + 297, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 297, + 226 + ], + "score": 1.0, + "content": "the same rules as AugMix [5] to randomly select", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 226, + 297, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 297, + 237 + ], + "score": 1.0, + "content": "augmentation operation type, severity level and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 236, + 296, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 296, + 248 + ], + "score": 1.0, + "content": "chain length. Following [6], we iteratively solve", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 296, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 296, + 258 + ], + "score": 1.0, + "content": "the inner maximization and outer minimization", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 259, + 297, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 297, + 270 + ], + "score": 1.0, + "content": "problems in Eq. (5): the inner maximization is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 269, + 297, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 155, + 282 + ], + "score": 1.0, + "content": "updated for", + "type": "text" + }, + { + "bbox": [ + 155, + 271, + 163, + 279 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 269, + 297, + 282 + ], + "score": 1.0, + "content": "steps, for every 1 step update on", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 149, + 298, + 282 + ] + }, + { + "type": "table", + "bbox": [ + 306, + 178, + 503, + 272 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 303, + 154, + 504, + 175 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 304, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "Table 5: Evaluation results on TIN and TIN-C. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 304, + 164, + 420, + 175 + ], + "spans": [ + { + "bbox": [ + 304, + 164, + 420, + 175 + ], + "score": 1.0, + "content": "best metric is shown in bold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 178, + 503, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 178, + 503, + 272 + ], + "spans": [ + { + "bbox": [ + 306, + 178, + 503, + 272 + ], + "score": 0.98, + "html": "
MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal61.6423.91100.00
AugMix61.79 62.2136.85 38.6783.04 80.72
AugMax-DuBIN(+0.42)(+1.82)(-2.32)
DeepAugment + AugMix59.5940.6778.28
DeepAugment +59.7240.9977.83
AugMax-DuBIN(+0.13)(+0.32)(-0.45)
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(5) to be 12 on ImageNet and 10 on all", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 300, + 507, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 366, + 315 + ], + "score": 1.0, + "content": "other datasets, except for ResNet18 on CIFAR100 where we find", + "type": "text" + }, + { + "bbox": [ + 367, + 302, + 392, + 312 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 300, + 507, + 315 + ], + "score": 1.0, + "content": "leads to better performance.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 313, + 434, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 434, + 325 + ], + "score": 1.0, + "content": "All experiments are conducted on a server with four NVIDIA RTX A6000 GPUs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 280, + 507, + 325 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 302, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 303, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 303, + 353 + ], + "score": 1.0, + "content": "4.2 Robustness against Natural Corruptions", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 506, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 507, + 372 + ], + "score": 1.0, + "content": "Results on CIFAR10-C and CIFAR100-C are shown in Table 2 and 3 respectively, where “normal”", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "means training with default standard augmentations (e.g., random flipping and translation). From the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "results, we see that AugMax-DuBIN achieves new state-of-the-art performance on both datasets with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "score": 1.0, + "content": "different model structures. For example, compared with AugMix, our method improves accuracy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 165, + 416 + ], + "score": 1.0, + "content": "by as high as", + "type": "text" + }, + { + "bbox": [ + 165, + 403, + 193, + 414 + ], + "score": 0.87, + "content": "3 . 0 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 402, + 213, + 416 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 404, + 241, + 414 + ], + "score": 0.87, + "content": "3 . 4 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "on CIFAR10-C and CIFAR100-C, respectively. 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This", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 452, + 298, + 530 + ] + }, + { + "type": "table", + "bbox": [ + 307, + 475, + 503, + 528 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 452, + 504, + 474 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 452, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 303, + 452, + 505, + 463 + ], + "score": 1.0, + "content": "Table 6: Evaluation results on TIN and TIN-C (w/o", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 303, + 463, + 469, + 474 + ], + "spans": [ + { + "bbox": [ + 303, + 463, + 469, + 474 + ], + "score": 1.0, + "content": "noise). The best metric is shown in bold.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + }, + { + "type": "table_body", + "bbox": [ + 307, + 475, + 503, + 528 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 307, + 475, + 503, + 528 + ], + "spans": [ + { + "bbox": [ + 307, + 475, + 503, + 528 + ], + "score": 0.97, + "html": "
MethodSA(%,↑)RA(%,↑)mCE(%,）
Normal61.6424.38100.00
AugMix61.7937.6382.54
ANT61.2635.3085.70
AugMax-DuBIN62.2138.6680.29
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(4), the worst-case mixing strat-", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 620, + 354, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 354, + 631 + ], + "score": 1.0, + "content": "egy can be searched efficiently. As a result, AugMax adds", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 631, + 354, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 354, + 642 + ], + "score": 1.0, + "content": "only a modest training overhead over AugMix, and is signif-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 642, + 355, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 355, + 654 + ], + "score": 1.0, + "content": "icantly more efficient than traditional adversarial training (AT).", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 652, + 278, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 278, + 664 + ], + "score": 1.0, + "content": "Detailed training time is shown in Table 7.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 597, + 355, + 664 + ] + }, + { + "type": "table", + "bbox": [ + 364, + 626, + 502, + 688 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 360, + 593, + 504, + 625 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 360, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 360, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "Table 7: Training time on ImageNet", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 360, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 360, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "with ResNet18, reported on a single", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 360, + 614, + 431, + 626 + ], + "spans": [ + { + "bbox": [ + 360, + 614, + 431, + 626 + ], + "score": 1.0, + "content": "NVIDIA A6000.", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 66 + }, + { + "type": "table_body", + "bbox": [ + 364, + 626, + 502, + 688 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 364, + 626, + 502, + 688 + ], + "spans": [ + { + "bbox": [ + 364, + 626, + 502, + 688 + ], + "score": 0.968, + "html": "
MethodTime (sec/epoch)
Normal2669
AugMix3622
AT[6]37162
AugMax5264
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When using DuBN or DuBIN, we route", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 105, + 70, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 195, + 87 + ], + "score": 1.0, + "content": "original images to the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 195, + 73, + 216, + 84 + ], + "score": 0.88, + "content": "B N _ { c }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 216, + 70, + 351, + 87 + ], + "score": 1.0, + "content": ", and AugMax/AugMix images to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 351, + 73, + 372, + 84 + ], + "score": 0.89, + "content": "B N _ { a }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 373, + 70, + 506, + 87 + ], + "score": 1.0, + "content": ". 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This shows the poten-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 298, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 298, + 211 + ], + "score": 1.0, + "content": "tial of applying DuBIN to other diversity aug-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 298, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 298, + 221 + ], + "score": 1.0, + "content": "mentation methods for general performance im-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 154, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 154, + 231 + ], + "score": 1.0, + "content": "provement.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 297, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 297, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 297, + 248 + ], + "score": 1.0, + "content": "We also conduct stability analyses on these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "score": 1.0, + "content": "experiments, by showing the statistical sig-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 297, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 297, + 271 + ], + "score": 1.0, + "content": "nificance of the improvements achieved by", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "table", + "bbox": [ + 306, + 181, + 503, + 265 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 136, + 505, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 303, + 136, + 474, + 149 + ], + "score": 1.0, + "content": "Table 8: Ablation results on DuBIN. RA", + "type": "text" + }, + { + "bbox": [ + 474, + 138, + 491, + 148 + ], + "score": 0.75, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 303, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 303, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "CIFAR10/100-C with WRN40-2 backbone are re-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 302, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 302, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "ported. Mean and standard derivation over three", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 302, + 169, + 488, + 182 + ], + "spans": [ + { + "bbox": [ + 302, + 169, + 488, + 182 + ], + "score": 1.0, + "content": "random seeds are shown for each experiment.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 181, + 503, + 265 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 181, + 503, + 265 + ], + "spans": [ + { + "bbox": [ + 306, + 181, + 503, + 265 + ], + "score": 0.978, + "html": "
MethodCIFAR10-CCIFAR100-C
AugMix-BN89.01 (± 0.03)64.56 (± 0.04)
AugMix-IBN89.17 (± 0.24)63.94 (± 0.28)
AugMix-DuBN89.11 (± 0.21)64.07 (± 0.38)
AugMix-DuBIN89.74(± 0.35)65.02 (± 0.58)
AugMax-DuBN89.60 (± 0.62)65.06 (± 0.28)
AugMax-DuBIN90.67 (± 0.16)66.35 (± 0.21)
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As we can observe, the improvements", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 301, + 395, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 395, + 315 + ], + "score": 1.0, + "content": "achieved by AugMax-DuBIN are statistically significant and consistent.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 268, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 269, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 269, + 343 + ], + "score": 1.0, + "content": "AugMax Hyperparameters In this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 341, + 269, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 269, + 354 + ], + "score": 1.0, + "content": "paragraph, we check the sensitivity of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "score": 1.0, + "content": "AugMax with respect to its hyperparam-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 363, + 266, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 210, + 376 + ], + "score": 1.0, + "content": "eters: maximization steps", + "type": "text" + }, + { + "bbox": [ + 211, + 366, + 218, + 374 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 363, + 258, + 376 + ], + "score": 1.0, + "content": ", step size", + "type": "text" + }, + { + "bbox": [ + 258, + 366, + 266, + 374 + ], + "score": 0.65, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 374, + 269, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 269, + 386 + ], + "score": 1.0, + "content": "and consistency loss tradeoff parameter", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 385, + 269, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 114, + 396 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 385, + 269, + 397 + ], + "score": 1.0, + "content": ". 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5101k231入 1015
SA(%)95.7895.6895.6895.6095.5195.9095.6895.88
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We then fix", + "type": "text" + }, + { + "bbox": [ + 286, + 440, + 345, + 451 + ], + "score": 0.92, + "content": "n = 1 0 , k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 439, + 390, + 452 + ], + "score": 1.0, + "content": "and adjust", + "type": "text" + }, + { + "bbox": [ + 390, + 441, + 397, + 450 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". We find robustness peaks", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 451, + 317, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 146, + 463 + ], + "score": 1.0, + "content": "at around", + "type": "text" + }, + { + "bbox": [ + 146, + 451, + 176, + 461 + ], + "score": 0.9, + "content": "\\lambda = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 451, + 317, + 463 + ], + "score": 1.0, + "content": ", which we set as the default value.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "Comparison with Different Diversity and Hardness Strategies In this paragraph, we study", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "different strategies to combine diversity and hardness. The results confirm that diversity and hardness", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 502, + 465, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 465, + 515 + ], + "score": 1.0, + "content": "are two complementary dimensions and their proper unification boosts model robustness.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 297, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "score": 1.0, + "content": "Specifically, we compare AugMax with baseline", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 529, + 298, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 298, + 541 + ], + "score": 1.0, + "content": "methods from each strategy group (i.e., diver-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 540, + 297, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 297, + 552 + ], + "score": 1.0, + "content": "sity or hardness), and other possible strategies to", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 551, + 298, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 298, + 562 + ], + "score": 1.0, + "content": "combine both. Besides AugMix and Adversar-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 562, + 297, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 297, + 574 + ], + "score": 1.0, + "content": "ial Training (e.g., PDGAT [6], FAT [28]), which", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 573, + 298, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 298, + 584 + ], + "score": 1.0, + "content": "are the two representative examples for diver-", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 583, + 297, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 297, + 595 + ], + "score": 1.0, + "content": "sity and hardness as shown in Figure 1, we also", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 593, + 298, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 298, + 608 + ], + "score": 1.0, + "content": "design the following baseline methods. (1) Aug-", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 605, + 298, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 298, + 617 + ], + "score": 1.0, + "content": "Mix+PGDAT: using both AugMix and adver-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 617, + 297, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 297, + 628 + ], + "score": 1.0, + "content": "sarial images for augmentation. This is a naive", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 627, + 297, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 297, + 640 + ], + "score": 1.0, + "content": "baseline to combine diversity with hardness. (2)", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 58 + }, + { + "type": "table", + "bbox": [ + 306, + 544, + 503, + 636 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 518, + 505, + 540 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 303, + 518, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 303, + 518, + 506, + 529 + ], + "score": 1.0, + "content": "Table 10: Results of different augmentation strate-", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 303, + 528, + 462, + 540 + ], + "spans": [ + { + "bbox": [ + 303, + 528, + 462, + 540 + ], + "score": 1.0, + "content": "gies on CIFAR100(-C) with ResNet18.", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 64.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 544, + 503, + 636 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 306, + 544, + 503, + 636 + ], + "spans": [ + { + "bbox": [ + 306, + 544, + 503, + 636 + ], + "score": 0.975, + "html": "
MethodStrategySA(%)RA(%)
Normal-77.9948.46
AugMix [5](diversity)78.2362.67
PGDAT [6](hardness)60.9447.71
FAT[28]61.5148.70
AdvMax56.6138.65
AugMix+PGDAT61.6851.39
AdvMix(diversity &adversity)72.3656.89
AugMax-DuBIN78.5264.02
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AugMax-DuBN", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 298, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 298, + 155 + ], + "score": 1.0, + "content": "outperforms both AugMix-BN and AugMix-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 297, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 297, + 166 + ], + "score": 1.0, + "content": "DuBN, and AugMax-DuBIN outperforms both", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "score": 1.0, + "content": "AugMix-IBN and AugMix-DuBIN. Noticeably,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 297, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 297, + 188 + ], + "score": 1.0, + "content": "when combined with AugMix, DuBIN also", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 298, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 298, + 199 + ], + "score": 1.0, + "content": "helps improve robustness. This shows the poten-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 298, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 298, + 211 + ], + "score": 1.0, + "content": "tial of applying DuBIN to other diversity aug-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 298, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 298, + 221 + ], + "score": 1.0, + "content": "mentation methods for general performance im-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 154, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 154, + 231 + ], + "score": 1.0, + "content": "provement.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 133, + 298, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 297, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 297, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 297, + 248 + ], + "score": 1.0, + "content": "We also conduct stability analyses on these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "score": 1.0, + "content": "experiments, by showing the statistical sig-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 297, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 297, + 271 + ], + "score": 1.0, + "content": "nificance of the improvements achieved by", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 235, + 298, + 271 + ] + }, + { + "type": "table", + "bbox": [ + 306, + 181, + 503, + 265 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 136, + 505, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 303, + 136, + 474, + 149 + ], + "score": 1.0, + "content": "Table 8: Ablation results on DuBIN. RA", + "type": "text" + }, + { + "bbox": [ + 474, + 138, + 491, + 148 + ], + "score": 0.75, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 303, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 303, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "CIFAR10/100-C with WRN40-2 backbone are re-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 302, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 302, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "ported. Mean and standard derivation over three", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 302, + 169, + 488, + 182 + ], + "spans": [ + { + "bbox": [ + 302, + 169, + 488, + 182 + ], + "score": 1.0, + "content": "random seeds are shown for each experiment.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "table_body", + "bbox": [ + 306, + 181, + 503, + 265 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 181, + 503, + 265 + ], + "spans": [ + { + "bbox": [ + 306, + 181, + 503, + 265 + ], + "score": 0.978, + "html": "
MethodCIFAR10-CCIFAR100-C
AugMix-BN89.01 (± 0.03)64.56 (± 0.04)
AugMix-IBN89.17 (± 0.24)63.94 (± 0.28)
AugMix-DuBN89.11 (± 0.21)64.07 (± 0.38)
AugMix-DuBIN89.74(± 0.35)65.02 (± 0.58)
AugMax-DuBN89.60 (± 0.62)65.06 (± 0.28)
AugMax-DuBIN90.67 (± 0.16)66.35 (± 0.21)
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As we can observe, the improvements", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 301, + 395, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 395, + 315 + ], + "score": 1.0, + "content": "achieved by AugMax-DuBIN are statistically significant and consistent.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 269, + 505, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 268, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 269, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 269, + 343 + ], + "score": 1.0, + "content": "AugMax Hyperparameters In this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 341, + 269, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 269, + 354 + ], + "score": 1.0, + "content": "paragraph, we check the sensitivity of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "score": 1.0, + "content": "AugMax with respect to its hyperparam-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 363, + 266, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 210, + 376 + ], + "score": 1.0, + "content": "eters: maximization steps", + "type": "text" + }, + { + "bbox": [ + 211, + 366, + 218, + 374 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 363, + 258, + 376 + ], + "score": 1.0, + "content": ", step size", + "type": "text" + }, + { + "bbox": [ + 258, + 366, + 266, + 374 + ], + "score": 0.65, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 374, + 269, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 269, + 386 + ], + "score": 1.0, + "content": "and consistency loss tradeoff parameter", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 385, + 269, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 114, + 396 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 385, + 269, + 397 + ], + "score": 1.0, + "content": ". The results are shown in Table 9. We", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 396, + 268, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 138, + 408 + ], + "score": 1.0, + "content": "first fix", + "type": "text" + }, + { + "bbox": [ + 138, + 397, + 170, + 407 + ], + "score": 0.9, + "content": "\\lambda = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 396, + 268, + 408 + ], + "score": 1.0, + "content": "and try different values", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 406, + 270, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 120, + 419 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 409, + 128, + 417 + ], + "score": 0.8, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 406, + 147, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 147, + 408, + 154, + 417 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 406, + 187, + 419 + ], + "score": 1.0, + "content": ". 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5101k231入 1015
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We then fix", + "type": "text" + }, + { + "bbox": [ + 286, + 440, + 345, + 451 + ], + "score": 0.92, + "content": "n = 1 0 , k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 439, + 390, + 452 + ], + "score": 1.0, + "content": "and adjust", + "type": "text" + }, + { + "bbox": [ + 390, + 441, + 397, + 450 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". We find robustness peaks", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 451, + 317, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 146, + 463 + ], + "score": 1.0, + "content": "at around", + "type": "text" + }, + { + "bbox": [ + 146, + 451, + 176, + 461 + ], + "score": 0.9, + "content": "\\lambda = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 451, + 317, + 463 + ], + "score": 1.0, + "content": ", which we set as the default value.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 418, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "Comparison with Different Diversity and Hardness Strategies In this paragraph, we study", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "different strategies to combine diversity and hardness. The results confirm that diversity and hardness", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 502, + 465, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 465, + 515 + ], + "score": 1.0, + "content": "are two complementary dimensions and their proper unification boosts model robustness.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 479, + 505, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 297, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 297, + 530 + ], + "score": 1.0, + "content": "Specifically, we compare AugMax with baseline", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 529, + 298, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 298, + 541 + ], + "score": 1.0, + "content": "methods from each strategy group (i.e., diver-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 540, + 297, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 297, + 552 + ], + "score": 1.0, + "content": "sity or hardness), and other possible strategies to", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 551, + 298, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 298, + 562 + ], + "score": 1.0, + "content": "combine both. 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MethodStrategySA(%)RA(%)
Normal-77.9948.46
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PGDAT [6](hardness)60.9447.71
FAT[28]61.5148.70
AdvMax56.6138.65
AugMix+PGDAT61.6851.39
AdvMix(diversity &adversity)72.3656.89
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AugMax-DuBIN achieves the best performance, outperforming", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "all methods from either diversity or hardness group. This shows diversity and hardness to be indeed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "complementary. 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MethodCIFAR10.1CIFAR10-STA
Normal88.9030.30
AugMix88.9054.30
AugMax-DuBIN90.6463.20
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AugMax-DuBIN steadily outperforms AugMix against both distributional shifts.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 298, + 183, + 312 + ], + "lines": [ + { + "bbox": [ + 104, + 296, + 185, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 185, + 314 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In this paper, we propose AugMax, a novel data augmentation strategy, that significantly improves", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "score": 1.0, + "content": "robustness by strategically unifying diversity and hardness. To enable efficient training while facing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 338, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 355 + ], + "score": 1.0, + "content": "the resulting heterogeneous features, we design a novel normalization scheme termed DuBIN. Using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "the combination of AugMax and DuBIN, we consistently demonstrate state-of-the-art robustness on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 363, + 264, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 264, + 375 + ], + "score": 1.0, + "content": "several natural corruption benchmarks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "AugMax unifies diversity and hardness in a heuristic manner. While we showed in ablation studies", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "that AugMax outperforms other heuristic combinations, there may be better trade-offs available. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "leave such study for future work and focus in this paper on showing the benefit of unifying diversity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 412, + 384, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 384, + 423 + ], + "score": 1.0, + "content": "and hardness, which were separately considered in previous research.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 439, + 202, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 204, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 204, + 455 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "Z.W. is supported by the U.S. Army Research Laboratory Cooperative Research Agreement W911NF-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 470, + 446, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 446, + 483 + ], + "score": 1.0, + "content": "17-2-0196 (IOBT REIGN), and an NVIDIA Applied Research Accelerator Program.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 497, + 163, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 165, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 165, + 511 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 515, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 110, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "[1] Dan Hendrycks and Thomas Dietterich. 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Specifically, we evaluate on the distribution", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 326, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 326, + 217 + ], + "score": 1.0, + "content": "shifts caused by the differences in data collection pro-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 327, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 327, + 228 + ], + "score": 1.0, + "content": "cess using CIFAR10.1 [30], and against Spatial Trans-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 327, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 327, + 239 + ], + "score": 1.0, + "content": "form adversarial Attacks (STA) [71] on CIFAR10-STA.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 324, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 317, + 250 + ], + "score": 1.0, + "content": "We generate CIFAR10-STA by applying the worst-of-", + "type": "text" + }, + { + "bbox": [ + 317, + 239, + 324, + 248 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 327, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 152, + 262 + ], + "score": 1.0, + "content": "attack with", + "type": "text" + }, + { + "bbox": [ + 152, + 249, + 182, + 259 + ], + "score": 0.89, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 248, + 327, + 262 + ], + "score": 1.0, + "content": "on CIFAR10 test set following [71].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 162, + 327, + 262 + ] + }, + { + "type": "table", + "bbox": [ + 338, + 208, + 500, + 255 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 332, + 161, + 505, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 331, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 331, + 161, + 506, + 173 + ], + "score": 1.0, + "content": "Table 11: Robustness against other dis-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 331, + 171, + 507, + 184 + ], + "spans": [ + { + "bbox": [ + 331, + 171, + 453, + 184 + ], + "score": 1.0, + "content": "tribution shifts. 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MethodCIFAR10.1CIFAR10-STA
Normal88.9030.30
AugMix88.9054.30
AugMax-DuBIN90.6463.20
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AugMax-DuBIN steadily outperforms AugMix against both distributional shifts.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 298, + 183, + 312 + ], + "lines": [ + { + "bbox": [ + 104, + 296, + 185, + 314 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 185, + 314 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In this paper, we propose AugMax, a novel data augmentation strategy, that significantly improves", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "score": 1.0, + "content": "robustness by strategically unifying diversity and hardness. To enable efficient training while facing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 338, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 355 + ], + "score": 1.0, + "content": "the resulting heterogeneous features, we design a novel normalization scheme termed DuBIN. Using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "the combination of AugMax and DuBIN, we consistently demonstrate state-of-the-art robustness on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 363, + 264, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 264, + 375 + ], + "score": 1.0, + "content": "several natural corruption benchmarks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 318, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "AugMax unifies diversity and hardness in a heuristic manner. While we showed in ablation studies", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "that AugMax outperforms other heuristic combinations, there may be better trade-offs available. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "leave such study for future work and focus in this paper on showing the benefit of unifying diversity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 412, + 384, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 384, + 423 + ], + "score": 1.0, + "content": "and hardness, which were separately considered in previous research.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 379, + 505, + 423 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 439, + 202, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 204, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 204, + 455 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "Z.W. is supported by the U.S. Army Research Laboratory Cooperative Research Agreement W911NF-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 470, + 446, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 446, + 483 + ], + "score": 1.0, + "content": "17-2-0196 (IOBT REIGN), and an NVIDIA Applied Research Accelerator Program.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 458, + 507, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 497, + 163, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 165, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 165, + 511 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "list", + "bbox": [ + 108, + 515, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 110, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "[1] Dan Hendrycks and Thomas Dietterich. 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ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)77.9978.2378.69 (+0.46)
RA(%)48.4662.6765.75 (+3.08)
WRN40-2SA(%)76.1977.0376.80 (-0.23)
RA(%)46.8064.5666.35 (+1.79)
ResNeXt29SA(%)79.9578.5880.70 (+2.12)
RA(%)47.7665.3768.86 (+3.49)
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ModelMetricNormalAugMixAugMax-DuBIN
ResNet18SA(%)95.5695.7995.76 (-0.03)
RA(%)74.7589.4990.36 (+0.87)
WRN40-2SA(%)94.7895.6795.68 (+0.01)
RA(%)73.7189.0190.67 (+1.66)
ResNeXt29SA(%)95.6096.2596.39 (+0.14)
RA(%)71.7089.0892.11 (+3.03)
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MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal69.8330.9187.47
AugMix68.06 67.6234.58 35.0183.08 82.56
AugMax-DuBIN(-0.44)(+0.43)(-0.52)
DeepAugment + AugMix65.3245.8469.29
DeepAugment +64.4346.5568.47
AugMax-DuBIN(-0.89)(+0.71)(-0.82)
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MethodSA (%,↑)RA (%,↑)mCE(%,↓)
Normal61.6423.91100.00
AugMix61.79 62.2136.85 38.6783.04 80.72
AugMax-DuBIN(+0.42)(+1.82)(-2.32)
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MethodSA(%,↑)RA(%,↑)mCE(%,）
Normal61.6424.38100.00
AugMix61.7937.6382.54
ANT61.2635.3085.70
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MethodTime (sec/epoch)
Normal2669
AugMix3622
AT[6]37162
AugMax5264
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MethodCIFAR10-CCIFAR100-C
AugMix-BN89.01 (± 0.03)64.56 (± 0.04)
AugMix-IBN89.17 (± 0.24)63.94 (± 0.28)
AugMix-DuBN89.11 (± 0.21)64.07 (± 0.38)
AugMix-DuBIN89.74(± 0.35)65.02 (± 0.58)
AugMax-DuBN89.60 (± 0.62)65.06 (± 0.28)
AugMax-DuBIN90.67 (± 0.16)66.35 (± 0.21)
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MethodStrategySA(%)RA(%)
Normal-77.9948.46
AugMix [5](diversity)78.2362.67
PGDAT [6](hardness)60.9447.71
FAT[28]61.5148.70
AdvMax56.6138.65
AugMix+PGDAT61.6851.39
AdvMix(diversity &adversity)72.3656.89
AugMax-DuBIN78.5264.02
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n
5101k231入 1015
SA(%)95.7895.6895.6895.6095.5195.9095.6895.88
RA (%)90.4990.6790.6790.3890.2188.8990.6790.30
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b/parse/train/PbEHqvFtcS/PbEHqvFtcS.md new file mode 100644 index 0000000000000000000000000000000000000000..279342d92530524f65256a8dfeb3f487abf5a2e2 --- /dev/null +++ b/parse/train/PbEHqvFtcS/PbEHqvFtcS.md @@ -0,0 +1,790 @@ +# BYZANTINE-RESILIENT NON-CONVEX STOCHASTIC GRADIENT DESCENT∗ + +Zeyuan Allen-Zhu†, Faeze Ebrahimian‡, Jerry Li§, Dan Alistarh¶ + +# ABSTRACT + +We study adversary-resilient stochastic distributed optimization, in which $m$ machines can independently compute stochastic gradients, and cooperate to jointly optimize over their local objective functions. However, an $\alpha$ -fraction of the machines are Byzantine, in that they may behave in arbitrary, adversarial ways. We consider a variant of this procedure in the challenging non-convex case. Our main result is a new algorithm SafeguardSGD which can provably escape saddle points and find approximate local minima of the non-convex objective. The algorithm is based on a new concentration filtering technique, and its sample and time complexity bounds match the best known theoretical bounds in the stochastic, distributed setting when no Byzantine machines are present. + +Our algorithm is very practical: it improves upon the performance of all prior methods when training deep neural networks, it is relatively lightweight, and it is the first method to withstand two recently-proposed Byzantine attacks. + +# 1 INTRODUCTION + +Motivated by the pervasiveness of large-scale distributed machine learning, there has recently been significant interest in providing distributed optimization algorithms with strong fault-tolerance guarantees. In this context, the strongest, most stringent fault model is that of Byzantine faults (Lamport et al., 1982): given $m$ machines, each having access to private data, at most an $\alpha$ fraction of the machines can behave in arbitrary, possibly adversarial ways, with the goal of breaking or slowing down the algorithm. Although extremely harsh, this fault model is the “gold standard” in distributed computing (Lynch, 1996; Lamport et al., 1982; Castro et al., 1999), as algorithms proven to be correct in this setting are guaranteed to converge under arbitrary system behaviour. + +A setting of particular interest in this context has been that of distributed stochastic optimization. Here, the task is to minimize some stochastic function $f ( x ) = \mathbb { E } _ { s \sim \mathcal { D } } [ f _ { s } ( x ) ]$ over a distribution $\mathcal { D }$ , where $f _ { s } ( \cdot )$ can be viewed as the loss function for sample $s \sim \mathcal { D }$ . We assume there are $m$ machines (workers) and an honest master, and $\alpha < 1 / 2$ fraction of the workers may be Byzantine. In each iteration $t$ , each worker has access to a version of the global iterate $x _ { t }$ , which is maintained by the master. The worker can independently sample $s \sim \mathcal { D }$ , compute $\nabla f _ { s } ( x _ { t } )$ , and then synchronously send this stochastic gradient to the master. The master aggregates the workers’ messages, and sends an updated iterate $x _ { t + 1 }$ to all the workers. Eventually, the master has to output an approximate minimizer of $f$ . Clearly, the above description only applies to honest workers; Byzantine workers may deviate arbitrarily and return adversarial “gradient” vectors to the master in every iteration. + +This distributed framework is quite general and well studied. One of the first references in this setting studied distributed PCA and regression (Feng et al., 2014). Other early approaches (Blanchard et al., 2017; Chen et al., 2017; Su & Vaidya, 2016a;b; Xie et al., 2018a) relied on defining generalizations of the geometric median. These approaches can withstand up to half of the nodes being malicious, but can have relatively high local computational cost $\Omega ( m ^ { 2 } { \dot { d } } )$ (Blanchard et al., 2017; Chen et al., 2017), where $m$ is the number of nodes and $d$ is the problem dimension, and usually have suboptimal sample and iteration complexities. + +Follow-up work resolved this last issue when the objective $f ( \cdot )$ is convex, leading to tight sample complexity bounds. Specifically, Yin et al. (2018) provided bounds for gradient descent-type algorithms, and showed that the bounds are tight when the dimension is constant. Alistarh et al. (2018) provided a stochastic gradient descent (SGD) type algorithm and showed that its sample and time complexities are asymptotically optimal even when the dimension is large. + +Non-convex Byzantine-resilient stochastic optimization. In this paper, we focus on the more challenging non-convex setting, and shoot for the strong goal of finding approximate local minima (a.k.a. second-order critical points). In a nutshell, our main result is the following. Fix $d$ to denote the dimension, and let the objective $f : \mathbb { R } ^ { d } \mathbb { R }$ be Lipschitz smooth and second-order smooth. We have $m$ worker machines, each having access to unbiased, bounded estimators of the gradient of $f$ . Given an initial point $x _ { 0 }$ , the SafeguardSGD algorithm ensures that, even if at most $\bar { \alpha } < 1 / 2$ fraction of the machines are Byzantine, after + +$$ +\begin{array} { r } { T = \widetilde { O } \left( \left( \alpha ^ { 2 } + \frac { 1 } { m } \right) \frac { d \left( f \left( x _ { 0 } \right) - \operatorname* { m i n } f \left( x \right) \right) } { \varepsilon ^ { 4 } } \right) \qquad \mathrm { p a r a l l e l i t e r a t i o n s } . } \end{array} +$$ + +for at least a constant fraction of the indices $t \in [ T ]$ , the following hold: + +$$ +\| \nabla f ( x _ { t } ) \| \leq \varepsilon \quad { \mathrm { a n d } } \quad \nabla ^ { 2 } f ( x _ { t } ) \succeq - { \sqrt { \varepsilon } } \mathbf { I } . +$$ + +If the goal is simply $\| \nabla f ( x _ { t } ) \| \leq \varepsilon$ , then $\begin{array} { r } { T = \widetilde { O } \big ( \left( \alpha ^ { 2 } + \frac { 1 } { m } \right) \frac { \left( f ( x _ { 0 } ) - \operatorname* { m i n } f ( x ) \right) } { \varepsilon ^ { 4 } } \big ) } \end{array}$ iterations suffice. Here, the $\widetilde O$ notation serves to hide logarithmic factors for readability. We spell out these factors in the detailed analysis. + +• When $\alpha < 1 / \sqrt { m }$ , our sample complexity $( = m T ,$ ) matches the best known result in the nonByzantine case (Jin et al., 2019) without additional assumptions, and enjoys linear parallel speedup: with $m$ workers of which $< \sqrt { m }$ are Byzantine, the parallel speedup is $\widetilde \Omega ( m )$ . 1 • For $\alpha \in [ 1 / \sqrt { m } , 1 / 2 )$ , our parallel time complexity is ${ \widetilde { \cal O } } ( \alpha ^ { 2 } )$ times that needed when no parallelism is used. This still gives parallel speedup. This $\alpha ^ { 2 }$ factor appears in convex Byzantine distributed optimization, where it is tight (Yin et al., 2018; Alistarh et al., 2018). • The Lipschitz and second-order smoothness assumptions are the minimal assumptions needed to derive convergence rates for finding second-order critical points (Jin et al., 2019). + +Comparison with prior bounds. The closest known bounds are by Yin et al. (2019), who derived three gradient descent-type of algorithms (based on median, mean, and iterative filtering) to find a weaker type of approximate local minima. Since it relies on full gradients, their algorithm is arguably less practical, and their time complexities are generally higher than ours (see Section 2.1). + +Other prior works consider a weaker goal: to find approximate stationary points $\| \nabla f ( x ) \| \leq \varepsilon$ only: Bulusu et al. (2020) additionally assumed there is a guaranteed good (i.e. non-Byzantine) worker known by the master, Xie et al. (2018b) gave a practical algorithm when the Byzantine attackers have no information about the loss function or its gradient, Yang et al. (2019); Xie et al. (2018a); Blanchard et al. (2017) derived eventual convergence without an explicit complexity bound, and the non-convex result obtained in Yin et al. (2018) is subsumed by Yin et al. (2019), discussed above. + +Our algorithm and techniques. The structure of our algorithm is deceptively simple. The master node keeps track of the sum of gradients produced by each worker across time. It labels (allegedly) good workers as those whose sum of gradients “concentrate” well with respect to a surrogate of the median vector, and labels bad workers otherwise. Once a worker is labelled bad, it is removed from consideration forever. The master then performs the vanilla $S G D$ , by moving in the negative direction of the average gradients produced by those workers currently labelled as good. + +We call our algorithm SafeguardSGD, since it behaves like having a safe guard to filter away bad workers. Its processing overhead at the master is $O ( m d )$ , negligible compared to standard SGD. + +As the astute reader may have guessed, the key non-trivial technical ingredient is to identify the right quantity to check for concentration, and make it compatible with the task of non-convex optimization. In particular, we manage to construct such quantities so that (1) good non-Byzantine workers never get mislabelled as bad ones; (2) Byzantine workers may be labelled as good ones (which is inevitable) but when they do, the convergence rates are not impacted significantly; and (3) the notion does not require additional assumptions or running time overhead. + +The idea of using concentration (for each worker across time) to filter out Byzantine machines traces back to the convex setting (Alistarh et al., 2018). However, the quantities used in (Alistarh et al., 2018) to check for concentration are necessarily different from this paper, and our analysis is completely new, as deriving non-convex rates is known to be much more delicate and challenging. Recently, Bulusu et al. (2020) used similar concentration filters to Alistarh et al. (2018) in the nonconvex setting, but under stronger assumptions, and for the simpler task of finding stationary points. + +Many other algorithms do not rely on concentration filters. In each iteration, they ask each worker to compute a batch of stochastic gradients, and then use coordinate-wise median or mean over the batch average (e.g. Yin et al. (2018; 2019); Yang et al. (2019)) or iterative filtering (e.g. Su & Xu (2018); Yin et al. (2019)) by the master to derive a “robust mean.” These works fundamentally rely on each iteration to calculate an almost precise full gradient, so that they can apply a surrogate of full gradient descent. Such algorithms can introduce higher sample and time complexities (see Section 2), are less practical than stochastic gradient schemes, require additional restrictions on the resilience factor $\alpha$ , e.g. $\alpha < 1 / 4$ (Su & Xu, 2018), and, critically, have been shown to be vulnerable to recent attacks (Baruch et al., 2019; Xie et al., 2020). + +Attack resilience and experimental validation. There is a growing literature on customized attacks against Byzantine-resilient algorithms, showing that many defenses can be entirely circumvented in real-world scenarios (Baruch et al., 2019; Xie et al., 2020). Our algorithm is provably correct against these attacks, a fact we also validate experimentally. We implemented SafeguardSGD to examine its practical performance against a range of prior works (Xie et al., 2018b; Blanchard et al., 2017; Chen et al., 2017; Yin et al., 2018; 2019), and against recent attacks on the distributed task of training deep neural networks. Our experiments show that SafeguardSGD generally outperforms previous methods in convergence speed and final accuracy, sometimes by a wide accuracy margin. This is true not only against known Byzantine attacks, but also against attack variants we fine-crafted to specifically slow down our algorithm, and against transient node failures. + +# 2 STATEMENT OF OUR THEORETICAL RESULT + +We denote by $\| \cdot \|$ the Euclidean norm and $[ n ] : = \{ 1 , 2 , \dots , n \}$ . Given symmetric matrices $\mathbf { A } , \mathbf { B }$ , we let $\| \mathbf { A } \| _ { 2 }$ denote the spectral norm of A. We use $\succeq$ to denote Loewner ordering, i.e. $\mathbf { A } \succeq \mathbf { B }$ if $\mathbf { A } - \mathbf { B }$ is positive semi-definite. We denote by $\lambda _ { \mathrm { m i n } } ( \mathbf { A } )$ the minimum eigenvalue of matrix $\mathbf { A }$ . + +We consider arbitrary $d$ -dimensional non-convex functions $f \colon { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ satisfying the following: + +• $f ( x )$ is $L$ -Lipschitz smooth: meaning $\| \nabla f ( x ) - \nabla f ( y ) \| \leq L \| x - y \|$ for any $x , y \in \mathbb { R } ^ { d }$ ; +• $f ( x )$ is $L _ { 2 }$ -second-order smooth: $\| \nabla ^ { 2 } f ( x ) - \nabla ^ { 2 } f ( y ) \| _ { 2 } \leq L _ { 2 } \cdot \| x - y \|$ for any $x , y \in \mathbb { R } ^ { d }$ ; + +For notational simplicity of the proofs, we assume $L = L _ { 2 } = \mathcal { V } = 1$ .2 Note that we have also assumed the domain of $f$ is the entire space $\mathbb { R } ^ { d }$ . If instead there is a compact domain $\mathcal { X } \subset \mathbb { R } ^ { d }$ , then one can use projected SGD and re-derive similar results of this paper. We choose to present our result in the simplest setting to convey our main ideas. + +Byzantine non-convex stochastic distributed optimization. We let $m$ be the number of worker machines and assume at most an $\alpha$ fraction of them are Byzantine for $\alpha \in [ 0 , \frac { 1 } { 2 } )$ . We denote by good $\subseteq [ m ]$ the set of good (i.e. non-Byzantine) machines, and the algorithm does not know good. + +Assumption 2.1. In each iteration $t _ { : }$ , the algorithm (on the master) is allowed to specify a point $x _ { t }$ and query m machines. Each machine $i \in [ m ]$ gives back a vector $\nabla _ { t , i } \in \mathbb { R } ^ { d }$ satisfying + +• If $i \in \mathtt { g o o d }$ , the stochastic gradient $\nabla _ { t , i }$ satisfies $\mathbb { E } [ \nabla _ { t , i } ] = \nabla f ( x _ { t } )$ and $\lVert \nabla f ( x _ { t } ) - \nabla _ { t , i } \rVert \leq \mathcal { V }$ • If $i \in [ m ]$ \ good, then $\nabla _ { t , i }$ can be arbitrary (w.l.o.g. we assume $\lVert \nabla f ( x _ { t } ) - \nabla _ { t , i } \rVert \leq \mathcal { V } )$ .4 + +Remark 2.2. For each $t$ and $i \not \in { \mathfrak { g o o d } }$ , the vector $\nabla _ { t , i }$ can be adversarially chosen and may depend + +Input: point $x _ { 0 } \in \mathbb { R } ^ { d }$ , rate $\eta > 0$ , lengths $T \geq T _ { 1 } \geq T _ { 0 } \geq 1$ , threshold $\mathfrak { T } _ { 1 } > \mathfrak { T } _ { 0 } > 0$ ; +1: $\mathbf { g } _ { 0 0 } \mathsf { d } _ { 0 } \gets [ m ]$ ; +2: for $t \gets 0$ to $T - 1$ do +3: $l a s t _ { 1 } \gets \operatorname* { m a x } \{ t _ { 1 } \in [ t ] : t _ { 1 }$ is a multiple of $T _ { 1 } \}$ ; +4: $l a s t _ { 0 } \gets \operatorname* { m a x } \{ t _ { 0 } \in [ t ] : t _ { 0 }$ is a multiple of $T _ { 0 } \}$ +5: for each $i \in \mathtt { g o o d } _ { t }$ do +6: 7: $\nabla _ { t , i } \in \mathbb { R } ^ { d }$ macand $i$ +Ai ← Ptk=last1 ∇k,i|good | $\begin{array} { r } { B _ { i } \gets \sum _ { k = l a s t _ { 0 } } ^ { t } \frac { \nabla _ { k , i } } { | \mathbf { g } \circ \mathbf { o } \mathbf { d } _ { k } | } } \end{array}$ +8: $A _ { \sf m e d } A _ { i }$ where $i \in { \mathfrak { g o o d } } _ { t }$ is any machine s.t. $\left| \left\{ j \in \operatorname { g o o d } _ { t } \colon \left\| A _ { j } - A _ { i } \right\| \leq \mathfrak { T } _ { 1 } \right\} \right| > m / 2$ . +9: $B _ { \sf m e d } B _ { i }$ where $i \in \mathtt { g o o d } _ { t }$ is any machine s.t. $\left| \left\{ j \in \mathbf { g 0 0 d } _ { t } \colon \left\| B _ { j } - B _ { i } \right\| \le \mathfrak { T } _ { 0 } \right\} \right| > m / 2$ . +10: $\begin{array} { r l r } & { \mathsf { g o o d } _ { t + 1 } \gets \big \{ i \in \mathsf { g o o d } _ { t } : \| A _ { i } - A _ { \mathsf { m e d } } \| \le 2 \mathfrak { T } _ { 1 } \bigwedge \| B _ { i } - B _ { \mathsf { m e d } } \| \le 2 \mathfrak { T } _ { 0 } \big \} } & \\ & { x _ { t + 1 } = x _ { t } - \eta \Big ( \xi _ { t } + \frac { 1 } { \vert \mathsf { g o o d } _ { t } \vert } \sum _ { i \in \mathsf { g o o d } _ { t } } \nabla _ { t , i } \Big ) ; } & { \diamondsuit a u s s i a n } \end{array}$ +11: noise $\xi _ { t } \sim \mathcal { N } ( 0 , \nu ^ { 2 } \mathbf { I } )$ + +on $\{ \nabla _ { t ^ { \prime } , i } \} _ { t ^ { \prime } \leq t , i \in [ m ] }$ . In particular, the Byzantine machines can even collude during an iteration. + +# 2.1 OUR ALGORITHM AND THEOREM + +Our algorithm is based on arguably the simplest possible method for achieving this goal, (perturbed) stochastic gradient descent $( S G D )$ (Ge et al., 2015). Our techniques more broadly apply to more complicated methods (e.g. at least to Allen-Zhu (2018a;b)), but we choose to analyze the simplest variant of SGD, since it is the most widely applied method in modern non-convex machine learning. + +As illustrated in Algorithm 1, in each iteration $t = 0 , 1 , \dots , T - 1$ , we maintain a set of (allegedly) good machines $\mathsf { g o o d } _ { t } \subseteq [ m ]$ . We begin with $\mathtt { g o o d } _ { 0 } = [ m ]$ and start to detect malicious machines and remove them from the set. We choose a learning rate $\eta > 0$ , and perform the SGD update + +$$ +\begin{array} { r } { x _ { t + 1 } = x _ { t } + \xi _ { t } - \eta \frac { 1 } { | \mathrm { g o o d } _ { t } | } \sum _ { i \in \mathrm { g o o d } _ { t } } \nabla _ { t , i } } \end{array} +$$ + +where $\xi _ { t } \sim \mathcal { N } ( 0 , \nu ^ { 2 } \mathbf { I } )$ is a random Gaussian perturbation that is added for theoretical purpose. + +For each machine $i \in [ m ]$ , we keep track of the history of its stochastic gradients up to two windows. Namely, Ai ← Ptk=last1 k,i|good | and Bi ← P k=last0 k,i|good | for windows sizes $T _ { 0 } \leq T _ { 1 } \leq T$ . We compare among remaining machines in $\mathtt { g o o d } _ { t }$ , and kick out those ones whose $A _ { i }$ or $B _ { i }$ deviate “more than usual” to construct $\mathtt { g o o d } _ { t + 1 }$ . Conceptually, we view these two as safe guards. + +Our theory makes sure that, when the “window sizes” and the thresholds for “more than usual” are defined properly, then $\mathtt { g o o d } _ { t }$ shall always include good, and the algorithm shall proceed to find approximate local minima. Formally, we have (letting the $\widetilde O$ notion to hide polylogarithmic factors) + +Theorem 2.3. Let $\begin{array} { r } { C _ { 3 } = \alpha ^ { 2 } + \frac { 1 } { m } } \end{array}$ . Suppose we choose $\nu ^ { 2 } = \widetilde { \Theta } ( C _ { 3 } )$ , $\begin{array} { r } { \eta = \widetilde { \Theta } \big ( \frac { \varepsilon ^ { 2 } } { d C _ { 3 } } \big ) } \end{array}$ , $T _ { 0 } = \widetilde { \Theta } ( \textstyle { \frac { 1 } { \eta } } )$ $\begin{array} { r } { T _ { 1 } = \widetilde { \Theta } ( \frac { 1 } { \eta \sqrt { \varepsilon } } ) } \end{array}$ , ${ \mathfrak { T } } _ { 0 } = { \widetilde { \Theta } } ( { \sqrt { T _ { 0 } } } )$ , and $\mathfrak { T } _ { 1 } = \widetilde { \Theta } ( \sqrt { T _ { 1 } } )$ , then after + +$$ +\begin{array} { r } { T = \widetilde { O } \left( \frac { ( f ( x _ { 0 } ) - \operatorname* { m i n } f ( x ) ) d } { \varepsilon ^ { 4 } } ( \alpha ^ { 2 } + \frac { 1 } { m } ) \right) } \end{array} +$$ + +iterations, with high probability, for at least constant fraction of the indices $t \in [ T ]$ , they satisfy + +$$ +\| \nabla f ( x _ { t } ) \| \le \varepsilon \quad a n d \quad \nabla ^ { 2 } f ( x _ { t } ) \succeq - \sqrt { \varepsilon } { \bf I } . +$$ + +Remark 2.4. If one only wishes to achieve a significantly simpler goal — finding first-order critical points $\| \nabla f ( x _ { t } ) \| \leq \varepsilon$ — the analysis becomes much easier (see Section 3.1). In particular, having one safe guard without perturbation (i.e. $\nu = 0$ ) suffices, and the iteration complexity reduces to T = Oe f(x0)−min f(x)ε4 (α2 + 1m ). Bulusu et al. (2020) achieves this easier goal but requires an additional assumption: there is one guaranteed good worker known by the master. + +Our contribution. We reiterate our theoretical contributions from three perspectives. 1) When $\alpha < 1 / \sqrt { m }$ , our algorithm requires $\begin{array} { r } { m T = \widetilde { O } \big ( \frac { ( f ( x _ { 0 } ) - \operatorname* { m i n } f ( x ) ) d } { \varepsilon ^ { 4 } } \big ) } \end{array}$ stochastic gradient computations. This matches the best known result (Jin et al., 2019) under our minimal assumptions of the nonconvex objective. (There exist other works in the stochastic setting that break the $\varepsilon ^ { - 4 }$ barrier and get rid of the dimension dependence $d$ under stronger assumptions.)5. 2) When $\alpha < 1 / \sqrt { m }$ , our algorithm enjoys linear parallel speed-up: the parallel time complexity reduces by a factor of $\Theta ( m )$ . When $\alpha \in [ 1 / \sqrt { m } , 1 / 2 )$ , our parallel time complexity is $\widetilde { O } ( \alpha ^ { 2 } )$ times that needed when no parallelism is used, still giving noticeable speedup. The $\textstyle { \hat { \alpha } } ^ { 2 }$ factor also appeared in convex Byzantine distributed optimization (and is known to be tight there) (Yin et al., 2018; Alistarh et al., 2018). + +Comparison to (Yin et al., 2019). Yin et al. (2019) derived three gradient descent-type algorithms to find points with a weaker (and less standard) guarantee: $\| \nabla f ( x ) \| ~ \leq ~ \varepsilon$ and $\bar { \nabla ^ { 2 } } f ( x ) ~ \succeq ~ - ( \varepsilon ^ { 2 } d ) ^ { \bar { 1 } / 5 } \mathbf { I }$ . Despite practical differences (namely, gradient descent may be less favorable comparing to stochastic gradient descent especially in deep learning applications), the parallel time complexities derived from their result are also generally larger than ours. + +Their paper focuses on bounding the number of sampled stochastic functions, as opposed to the number of stochastic gradient evaluations like we do. When translated to our language, each of the workers in their setting needs to evaluate T stochastic gradients, where (1) T = Oe α2dε4 $\begin{array} { r } { 1 ) T = \widetilde { O } \big ( \frac { \alpha ^ { 2 } \overline { { d } } } { \varepsilon ^ { 4 } } + \frac { d ^ { 2 } } { \varepsilon ^ { 4 } m } + \frac { \sqrt { d } } { \varepsilon ^ { 3 } } \big ) } \end{array}$ if using coordinate-wise median, (2) T = Oe α2d2ε4 $\begin{array} { r } { T = \widetilde { O } \big ( \frac { \alpha } { \varepsilon ^ { 4 } } + \frac { d } { \varepsilon ^ { 4 } m } \big ) } \end{array}$ if using iterative filt if using trimmed mean, and (3)(1) and (2) are larger than ours (also with a weaker guarantee); the complexity (3) seems incomparable to ours, but when translating to the more standard (ε, √ε) guarantee, becomes T = Oe αd2ε5 $\begin{array} { r } { T = \widetilde { \cal O } \big ( \frac { \alpha d ^ { 2 } } { \varepsilon ^ { 5 } } + \frac { d ^ { 3 } } { \varepsilon ^ { 5 } m } \big ) } \end{array}$ so is also larger than ours. It is worth noting that (3) requires $\alpha < 1 / 4$ so cannot withstand half of the machines being Byzantine. + +Resilience against practical attacks. Our algorithm’s filtering is based upon tracking $B _ { i }$ (resp. $A _ { i , { \bf \Pi } }$ ), the stochastic gradients of each machine $i$ averaged over a window of $T _ { 0 }$ (resp. $T _ { 1 . }$ ) iterations. This is a departure from previous defenses, most of which are history-less, and enables us to be provably Byzantine-resilient against state-of-the-art attacks (Baruch et al., 2019; Xie et al., 2020). + +In Baruch et al. (2019), Byzantine workers collude to shift the gradient mean by a factor $\beta$ times the standard deviation of the (true stochastic) gradient, while staying within population variance. They noticed $\beta$ can be quite large especially in neural network training. Their attack circumvent existing defenses because those defense algorithms are “historyless”, while their attack is statistically indistinguishable from an honest execution in any single iteration. However, our algorithm can provably defend against this attack since it has memory: Byzantine workers following their strategy will progressively diverge from the (honest) “median” √ $B _ { \mathrm { m e d } }$ (by an amount proportional to $\Omega ( T )$ in $T$ iterations as opposed to $\sqrt { T }$ ), and be marked as malicious by our algorithm. (See Figure 2(a).) In Xie et al. (2020), Byzantine workers deviate in the negative direction of the gradient. However, to avoid being caught by our algorithm, the maximum “magnitude” of this attack has to stay within our thresholds. We implemented both attacks and showed our algorithm’s robustness experimentally. + +Finally, we note that prior “historyless” schemes, such as Krum or median-based schemes, could be thought of as providing stronger guarantees, as they in theory allow Byzantine nodes to change IDs during the computation: such schemes only require an upper bound on the number of Byzantine agents in each round. However, the attack of Baruch et al. (2019) essentially shows that all such schemes are vulnerable to variance attacks, and that such attacks are eminently plausible in practice. Thus, this suggests that the use of historical information, which requires that Byzantine nodes cannot change their IDs during the execution, may be necessary for Byzantine resilience. + +Tolerating transient failures and node ID relabeling. Our algorithm can also withstand transient node failures and some degrees of ID relabeling, by resetting the set of good nodes $\mathsf { g o o d } _ { t }$ to include all nodes every $T _ { 1 }$ steps. The algorithm then proceeds as usual. The key observation behind this relaxation is the fact that our analysis only requires that the attack conditions hold inside the current window. (Please see the Theorem B.1 for details.) We validate this experimentally in Section 5. + +# 3 WARMUP: SINGLE SAFE GUARD + +As a warmup, let us first analyze the behavior of perturbed SGD with a single safe guard. Consider Algorithm 2, where we start with a point $w _ { 0 }$ , a set $\mathtt { g o o d } _ { 0 } \supseteq \mathtt { g o o d }$ , and perform $T$ steps of perturbed SGD. (We use the $w _ { t }$ sequence instead of the $x _ { t }$ sequence to emphasize that we are in Algorithm 2.) + +Input: point $w _ { 0 } \in \mathbb { R } ^ { d }$ , set $\mathtt { g o o d } _ { 0 } \supseteq \mathtt { g o o d }$ , rate $\eta > 0$ , length $T \geq 1$ , threshold $\mathfrak T > 0$ ; +1: for $t \gets 0$ to $T - 1$ do +2: for each $i \in \mathtt { g o o d } _ { t }$ do +3: receive $\nabla _ { t , i } \in \mathbb { R } ^ { d }$ from machine $i$ ; +4: Bi ← Ptk=0 ∇k,i|goodk| ; +5: $B _ { \sf m e d } B _ { i }$ where $i \in \mathtt { g o o d } _ { t }$ is any machine s.t. $\left| \left\{ j \in \mathbf { g 0 0 d } _ { t } \colon \left\| B _ { j } - B _ { i } \right\| \le \mathfrak { T } \right\} \right| > m / 2 .$ . +6: $\begin{array} { r l } & { \mathsf { g o o d } _ { t + 1 } \gets \big \{ i \in \mathsf { g o o d } _ { t } \colon \| B _ { i } - B _ { \mathsf { m e d } } \| \le 2 \mathfrak { T } \big \} } \\ & { w _ { t + 1 } = w _ { t } - \eta \Big ( \xi _ { t } + \frac { 1 } { | \mathsf { g o o d } _ { t } | } \sum _ { i \in \mathsf { g o o d } _ { t } } \nabla _ { t , i } \Big ) ; } \end{array}$ ; +7:  Gaussian noise $\xi _ { t } \sim \mathcal { N } ( 0 , \nu ^ { 2 } \mathbf { I } )$ + +Definition 3.1. We make the following definition to simplify notations: let $\Xi _ { t } : = \sigma _ { t } + \Delta _ { t }$ where + +$$ +\begin{array} { r l } & { \sigma _ { t } : = \frac { 1 } { | \mathrm { g o o d } _ { t } | } \sum _ { i \in \mathrm { g o o d } } \left( \nabla _ { t , i } - \nabla f ( w _ { t } ) \right) } \\ & { \Delta _ { t } : = \frac { 1 } { | \mathrm { g o o d } _ { t } | } \sum _ { i \in \mathrm { g o o d } _ { t } \backslash \mathrm { g o o d } } \left( \nabla _ { t , i } - \nabla f ( w _ { t } ) \right) } \end{array} +$$ + +Therefore, we can re-write the SGD update as $w _ { t + 1 } = w _ { t } - \eta ( \nabla f ( w _ { t } ) + \xi _ { t } + \Xi _ { t } )$ + +The following lemma is fairly immediate to prove: + +Lemma 3.2 (single safe guard). In Algorithm 2, suppose we choose $\mathfrak { T } = 8 \sqrt { T \log ( 1 6 m T / p ) }$ . Then, with probability at least $1 - p / 4$ , for every $t = 0 , \ldots , T - 1$ , + +$$ +\begin{array} { r l } & { \bullet \Vert \sigma _ { t } \Vert ^ { 2 } \leq O ( \frac { \log ( T / p ) } { m } ) a n d \Vert \sigma _ { 0 } + \cdot \cdot \cdot + \sigma _ { t - 1 } \Vert ^ { 2 } \leq O ( \frac { T \log ( T / p ) } { m } ) } \\ & { \bullet \Vert \Delta _ { t } \Vert ^ { 2 } \leq \alpha ^ { 2 } a n d \Vert \Delta _ { 0 } + \cdot \cdot \cdot + \Delta _ { t - 1 } \Vert ^ { 2 } \leq O ( \alpha ^ { 2 } T \log ( m T / p ) ) } \\ & { \bullet \vert \langle \nabla f ( w _ { t } ) , \xi _ { t } \rangle \vert \leq \Vert \nabla f ( w _ { t } ) \Vert \cdot O ( \nu \sqrt { \log ( T / p ) } ) , } \\ & { \bullet \Vert \xi _ { t } \Vert ^ { 2 } \leq O ( \nu ^ { 2 } d \log ( T / p ) ) , \Vert \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } \Vert ^ { 2 } \leq O ( \nu ^ { 2 } d T \log ( T / p ) ) } \\ & { \textit { e c a l l t h i s p r o b a b i l i s t i c e n t \ 5 \mathrm { v e n t } } \mathrm { t v e n t } _ { T } ^ { \mathrm { s i n g l e } } ( w _ { 0 } ) a n d \operatorname* { P r } [ \mathrm { E v e n t } _ { T } ^ { \mathrm { s i n g l e } } ( w _ { 0 } ) ] \geq 1 - p / 4 . } \end{array} +$$ + +(The third property above is ensured by our choice of $\mathfrak { T }$ and the use of safe guard, and the rest of the properties follow from simple martingale concentration arguments. Details are in Appendix A.1.) + +# 3.1 CORE TECHNICAL LEMMA 1: OBJECTIVE DECREASE + +Our first main technical lemma is the following: + +Lemma 3.3. Suppose we choose $\mathfrak { T }$ as in Lemma 3.2. Denote by $C _ { 1 } = \log ( T / p )$ and $C _ { 2 } \ =$ $\begin{array} { r } { \alpha ^ { 2 } \log { \frac { m T } { p } } + { \frac { \log ( T / p ) } { m } } } \end{array}$ . Suppose $\eta \leq 0 . 0 1 \operatorname* { m i n } \{ 1 , \frac { 1 } { C _ { 2 } } \}$ , 1100η(1+√C2) and we start from w0 and apply Algorithm 2. Under event Even $\mathbf { \Pi } _ { T } ^ { \mathsf { s i n g l e } } ( w _ { 0 } )$ , it satisfies + +$$ +\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \ge 0 . 7 \eta \sum _ { t = 0 } ^ { T - 1 } \left( \| \nabla f ( w _ { t } ) \| ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) } \end{array} +$$ + +Lemma 3.3 says after $\begin{array} { r } { T \approx \frac { 1 } { \eta } } \end{array}$ steps of perturbed SGD, the objective value decreases by, up to some small additive error and up to logarithmic factors, $\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \geq 0 . 7 \eta \sum _ { t = 0 } ^ { T - 1 } ( \Vert \nabla f ( w _ { t } ) \Vert ^ { 2 } - } \end{array}$ $\eta C _ { 2 } )$ . This immediately implies, if we choose $\begin{array} { r } { \eta \approx \frac { \varepsilon ^ { 2 } } { C _ { 2 } } } \end{array}$ , then by repeating this analysis for $\begin{array} { r } { O ( \frac { C _ { 2 } } { \varepsilon ^ { 4 } } ) = O ( \frac { \alpha ^ { 2 } + 1 / m } { \varepsilon ^ { 4 } } ) } \end{array}$ ( α2+1/mε4 ) iterations, we can find approximate critical point x with k∇f (x)k ≤ ε. + +Proof sketch of Lemma 3.3. The full proof is in Appendix A.2 but we illustrate the main idea and difficulties below. After simple manipulations, it is not hard to derive that + +$$ +\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \gtrapprox 0 . 9 \eta \sum _ { t = 0 } ^ { T - 1 } \left( \Vert \nabla f ( w _ { t } ) \Vert ^ { 2 } - \eta \right) + \underbrace { \eta \sum _ { t = 0 } ^ { T - 1 } \langle \nabla f ( w _ { t } ) , \Xi _ { t } \rangle } _ { \mathrm { r e m a i n d e r ~ t e r m s } } } \end{array} +$$ + +where recall that $\Xi _ { t } = \sigma _ { t } + \Delta _ { t }$ . When there are no Byzantine machines, we have $\mathbb { E } [ \Xi _ { t } ] = \mathbb { E } [ \sigma _ { t } ] = 0$ so the remainder terms must be small by martingale concentration. Therefore, the main technical difficulty arises to deal with those Byzantine machines, who can adversarially design their $\nabla _ { t }$ (even by collusion) so as to negatively correlate with $\nabla f ( w _ { t } )$ to “maximally destroy” the above inequality. + +Our main idea is to use second-order smoothness to write $\nabla f ( w _ { t } ) \approx \nabla f ( w _ { 0 } ) + \nabla ^ { 2 } f ( w _ { 0 } ) \cdot ( w _ { t } - w _ { 0 } )$ . To illustrate our idea, let us ignore the constant vector and assume that the Hessian is the identity: that is, imagine as if $\nabla f ( w _ { t } ) \mathbf { \bar { \approx } } w _ { t } - w _ { 0 }$ . Using $\begin{array} { r } { w _ { t } - w _ { 0 } = - \sum _ { k < t } \Xi _ { t } + \xi _ { t } } \end{array}$ , we immediately have + +$$ +\begin{array} { r } { - \langle \nabla f ( w _ { t } ) , \Xi _ { t } \rangle \approx - \langle w _ { t } - w _ { 0 } , \Xi _ { t } \rangle = \sum _ { k < t } \langle \Xi _ { k } , \Xi _ { t } \rangle + \sum _ { k < t } \langle \xi _ { k } , \Xi _ { t } \rangle } \end{array} +$$ + +For the first partial sum $\textstyle \langle \sum _ { k < t } \Xi _ { k } , \Xi _ { t } \rangle$ in (3.1), it is easy to bound its magnitude using our safeguard. Indeed, we have $\begin{array} { r } { \big | \sum _ { t } \langle \sum _ { k < t } \Xi _ { k } , \Xi _ { t } \rangle \big | \leq \| \sum _ { t } \Xi _ { t } \| ^ { 2 } + \sum _ { t } \| \Xi _ { t } \| ^ { 2 } } \end{array}$ so we can apply Lemma 3.2. For the second partial sum $\begin{array} { r } { \sum _ { t } \sum _ { k < t } \langle \xi _ { k } , \Xi _ { t } \rangle } \end{array}$ , we can apply the concentration Proposition 3.4 below.  + +Proposition 3.4. Fix the dimension parameter $d \ge 1$ . Suppose $\xi _ { 0 \underline { { : } } } \dots , \xi _ { T - 1 } \ \in \ \mathbb { R } ^ { d }$ are i.i.d. drawn from $\mathcal { N } ( 0 , \bf { I } )$ , and that $\Delta _ { 1 } , \ldots , \Delta _ { T - 1 }$ are arbitrary vectors in $\mathbb { R } ^ { d }$ . Here, each vector $\Delta _ { t }$ with $t = 1 , \dots , T - 1$ can depend on $\xi _ { 0 } , \ldots , \xi _ { t - 1 }$ but not on $\xi _ { t } , \dots , \xi _ { T - 1 }$ . Suppose that these vectors satisfy $\| \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \| ^ { 2 } \leq \mathfrak { T }$ for every $t = 1 , \dots , T - 1$ . Then, with probability at least $1 - p$ , + +$$ +\begin{array} { r } { \left| \sum _ { t = 1 } ^ { T - 1 } \langle \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } , \Delta _ { t } \rangle \right| \leq O ( \sqrt { d T \mathfrak { T } \log ( T / p ) } ) } \end{array} . +$$ + +3.2 CORE TECHNICAL LEMMA 2: RANDOMNESS COUPLING + +Our next technical lemma studies that, if run Algorithm 2 from a point $w _ { 0 }$ so that the Hessian $\nabla ^ { 2 } f ( w _ { 0 } )$ has a eigenvalue which is less than $- \delta$ (think of $w _ { 0 }$ as a saddle point), then with good probability, after sufficiently many iterations, the sequence $w _ { 1 } , w _ { 2 } , \dots , w _ { T }$ shall escape from $w _ { 0 }$ to distance at least $R$ for some parameter $R \approx \delta$ . To prove this, motivated by Jin et al. (2017), we study two executions of Algorithm 2 where their randomness are coupled. We then argue that at least one of them has to escape from $w _ { 0 }$ . For any vector $v$ , let $[ v ] _ { i }$ denote the $i$ -th coordinate of $v$ . + +Lemma 3.5. Suppose we choose $\mathfrak { T }$ as in Lemma 3.2 and $C _ { 1 } , C _ { 2 }$ as in Lemma 3.3. Suppose $\boldsymbol { w } _ { 0 } \in \mathbb { R } ^ { d }$ satisfies $\dot { \lambda } _ { \mathrm { m i n } } ( \nabla ^ { 2 } f ( w _ { 0 } ) ) = - \delta$ for some $\delta \geq 0$ . Without loss of generality let $\mathbf { e } _ { 1 }$ be the eigenvector of $\nabla ^ { 2 } f ( w _ { 0 } )$ with smallest eigenvalue. Consider now two executions of Algorithm 2, both starting from $w _ { 0 } ^ { \mathsf { a } } = w _ { 0 } ^ { \mathsf { b } } = w _ { 0 }$ , and suppose their randomness $\{ \xi _ { t } ^ { \mathsf { a } } \} _ { t }$ and $\{ \xi _ { t } ^ { \mathrm { b } } \} _ { t }$ are coupled so that $[ \xi _ { t } ^ { \mathsf { a } } ] _ { 1 } = - [ \xi _ { t } ^ { \mathsf { b } } ] _ { 1 }$ but $[ \xi _ { t } ^ { \mathsf { a } } ] _ { i } = [ \xi _ { t } ^ { \mathsf { b } } ] _ { i }$ for $i > 1$ . In words, the randomness is the same orthogonal to $\mathbf { e } _ { 1 }$ , but along $\mathbf { e } _ { 1 }$ , the two have opposite signs. Now, suppose we perform $\begin{array} { r } { T = \Theta \big ( \frac { 1 } { \eta \delta } \log \frac { R ^ { 2 } \delta } { \eta \nu ^ { 2 } } \big ) } \end{array}$ steps of perturbed SGD from $w _ { 0 } ^ { \mathsf { a } } , w _ { 0 } ^ { \mathsf { b } }$ respectively using Algorithm 2. Suppose + +$$ +\begin{array} { r } { R \leq { \cal O } \big ( { \frac { \delta } { \sqrt { C _ { 1 } } \log ( R ^ { 2 } \delta / \eta \nu ^ { 2 } ) } } \big ) \quad a n d \quad \nu ^ { 2 } \geq \Omega \big ( C _ { 2 } \log { \frac { R ^ { 2 } \delta } { \eta \nu } } \big ) . } \end{array} +$$ + +Then, under events Even $\mathbf { t } _ { T } ^ { \mathsf { s i n g l e } } ( w _ { 0 } ^ { \mathsf { a } } )$ and $\mathsf {bar { c } v e n t } _ { T } ^ { \mathsf { s i n g l e } } ( w _ { 0 } ^ { \mathsf { b } } )$ , with probability at least 0.98, either $\| w _ { t } ^ { \mathsf { a } } - w _ { 0 } \| > R$ or $\| w _ { t } ^ { \mathsf { b } } - w _ { 0 } \| > R$ for some $t \in [ T ]$ . + +Proof details in Appendix A.4. The main proof difficulty is to analyze a noisy version of the power method, where the noise comes from (1) Gaussian perturbation (which is the good noise), (2) stochastic gradients (which has zero mean), and (3) Byzantine workers (which can be adversarial). + +# 4 FROM WARMUP TO FINAL THEOREM WITH DOUBLE SAFE GUARDS + +At a high level, Lemma 3.3 ensures that if we keep encountering points with large gradient $\| \nabla f ( w _ { t } ) \|$ , then the objective should sufficiently decrease; in contrast, Lemma 3.5 says that if we keep encountering points with negative Hessian directions (i.e., $\lambda _ { \operatorname* { m i n } } ( \nabla ^ { 2 } f ( w _ { t } ) ) < \dot { - } \delta )$ , then the points must move a lot (i.e., by more than $R$ in $T$ iterations, which can also lead to sufficient objective decrease, see Lemma B.4). Therefore, at a high level, when the two lemmas are combined, they tell that we must not encounter points with $\| \nabla f ( x ) \|$ being large, or $\lambda _ { \operatorname* { m i n } } ( \nabla ^ { 2 } f ( x ) )$ being very negative, for too many iterations. Therefore, the algorithm can find approximate local minima. + +The reason we need two safe guards, is because the number of rounds $T$ for Lemma 3.3 and Lemma 3.5 differ by a factor. We need two safe guards with different window sizes to ensure the two lemmas simultaneously hold. We encourage the reader to examine the full analysis in Appendix B. + +# 5 EXPERIMENTAL VALIDATION + +We evaluate the convergence of SafeguardSGD to examine its practical performance against prior works. We perform the non-convex task of training a residual network ResNet-20 (He et al., 2016) on the CIFAR-10/100 datasets (Krizhevsky et al., 2014). More details are given in Appendix C. + +![](images/fd2edb2655662c1193ffe6a690559d8147cb61fcf659521ef7f5a1fb281c36fa.jpg) +Figure 1: Convergence comparison (CIFAR-10 test accuracy) under different attacks. (In Appendix C.2, one can find additional CIFAR-100 experiments, more discussions, and bigger plots.) + +We instantiate $m = 1 0$ workers and one master node executing data-parallel SGD for 140 passes (i.e. epochs) over the training dataset. The results for higher number of workers and epochs are similar, and therefore omitted. We compare against Geometric Median (Chen et al., 2017), Coordinatewise Median (Yin et al., 2018; 2019), Krum (Blanchard et al., 2017), and Zeno (Xie et al., 2018b). Overall, our experimental setup is very similar to Zeno (Xie et al., 2018b) but with additional attacks. + +We implemented the approach of Yang et al. (2019), but found it very sensitive to hyper-parameter values and were unable to make it converge across all attacks even after significant tuning of its $\gamma$ parameter. We also implemented the convex algorithm of Alistarh et al. (2018), and executed it in our non-convex setting. We found their algorithm can be easily attacked on our ResNet training tasks. There exists a simple attack, described in Appendix C.4 which causes their algorithm to either mislabel most good workers as Byzantine, or diverge, or converge to very poor solutions. This is not surprising, since their algorithm is designed for, and only guaranteed to work in, the convex setting. + +To make the comparison stronger, when implementing SafeguardSGD, we have chosen fixed window sizes $T _ { 0 } = 1$ epoch and $T _ { 1 } = 6$ epochs across all experiments, and adopted an automated process to select $\mathfrak { T } _ { 0 } , \mathfrak { T } _ { 1 }$ . Determining these thresholds requires being able to pre-run the task on an honest worker. We have also implemented a single safeguard variant of SafeguardSGD, with window size $T = 3$ epochs. + +Attacks. We set $\alpha = 0 . 4$ , which means that there are 4 Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.) + +• LABEL-FLIPPING ATTACK: each Byzantine worker computes its gradient based on the crossentropy loss with flipped labels: for CIFAR-10, label $\ell \in \{ 0 , . . . , 9 \}$ is flipped to $9 - \ell$ . • DELAYED-GRADIENT ATTACK: each Byzantine worker sends an old gradient to master. In our experiments, the delay is of $D = 1 0 0 0$ iterations. • VARIANCE ATTACK (Baruch et al., 2019): Byzantine workers measure the mean and the standard-deviation of gradients at each round, and collude to move the mean by the largest value which still operates within population variance. (For our parameter settings, this is 0.3 times the standard deviation. We discuss results for additional parameter values in the Appendix.) • SIGN-FLIPPING ATTACK: each Byzantine worker sends the negative gradient to the master. • SAFEGUARD ATTACK: each Byzantine workers sends a negative but re-scaled gradient to the master. We use re-scale factors 0.6 and 0.7 in our experiments. The re-scale factor 0.6 avoids triggering the safe-guard conditions at the master, and the re-scale factor 0.7 occasionally triggers the safe-guard conditions. This attack is an instantiation of the inner-product attack (Xie et al., 2020), customized specifically to maximally affect our SafeguardSGD algorithm. + +Main experimental results. The ideal test accuracy is $9 1 . 7 \%$ , which corresponds to applying SGD using only the stochastic gradients from the honest workers. Figure 1 compares the performances in test accuracy. Below we summarize our main findings for the experiments, and we defer detailed discussions (and additional experiments for CIFAR-100) to Appendix C. + +![](images/383ad3190abfe8f2cfb2caf94b401335009e80d8bed41ff9a0c8c89b12115762.jpg) +(a) $\| B _ { i } - B _ { \mathrm { m e d } } \|$ between a good node (blue), and a bad node (red) which pretends to be honest and then starts to apply the variance attack. + +![](images/cae7626417996e121b4e9ed1583ac1e9c08ef0c2c60549476e6ad5df3383dbb6.jpg) +(b) Convergence for our safeguard algorithms under the variance attack, after periodically resetting the set of good nodes. +Figure 2 + +• SafeguardSGD generally outperforms all the previous methods in test accuracy. The test accuracy difference can be ${ } ^ { 6 6 } 9 0 \%$ vs. $< 4 0 \%$ ” between our algorithm and the best prior work. • The variance attack is indeed very strong, in that it severely affects the accuracy of all prior works (test accuracy $< 3 5 \%$ ). This is because thesese defenses are “historyless.” By contrast, our algorithm not only provably but also empirically defends against it. • Our safeguard attack (especially with re-scale factor 0.7) is as strong as the variance attack, and even stronger on the CIFAR-100 dataset; please see the results in Appendix C.2.5. • The label-flipping attack is rather weak: although some defenses, such as Zeno, did not determine which of the workers are malicious, they still converge well under this attack. • The sign-flipping and delayed-gradient attacks are moderate: the best prior works can achieve accuracy $6 0 \% \sim 7 0 \%$ . It is worth noting that the sign-flipping attack can already nullify the Zeno defence (test accuracy $2 0 \%$ ). The issue seems to be that it can be very hard for Zeno to use relatively few samples to determine if the gradient direction is flipped to negative. • SafeguardSGD can easily catch all the bad workers under sign-flipping and variance attacks, and thus leads to gives ideal performance. It cannot catch any bad worker for label-flipping and delayed-gradient attacks, but there is no performance loss anyways if we use such bad gradients. • The safeguard attacks, designed to maximally impact the performance of our SafeguardSGD, can indeed affect our performance. Specifically, under re-scale factor 0.6, the test accuracy drops from $9 1 . 7 \%$ to $8 9 . 3 \%$ because SafeguardSGD cannot catch any bad worker; however, under rescale factor 0.7, the test accuracy no longer drops because SafeguardSGD can begin to catch some bad workers (it can catch between 0 and 4 bad workers depending on the randomness.) • In most cases, the single-safeguard algorithm is close to double-safeguard, except for the safeguard $( \mathrm { x 0 . 7 } )$ attack, in which using double-safeguard one can more easily catch bad workers. (This is more apparent in the CIFAR-100 experiment, see Appendix C.2.5.) + +We conclude that SafeguardSGD can be practical, and outperforms previous approaches. + +A deeper dive: how the algorithm works. Let us explain the inner workings of our algorithm in the context of a “delayed” attack, where the Byzantine nodes collude to execute an attack only after a specific, given point in the execution (in this case, the first half-epoch). Figure 2(a) presents the results from the perspective of the value of $\| B _ { i } - B _ { \mathrm { m e d } } \|$ registered at the master server, for two nodes, an honest one, and a Byzantine one. The value of √ $\| B _ { i } - B _ { \sf m e d } \|$ increases for all the nodes (at a rate of roughly $\sqrt { t }$ at step $t$ ); but, once the attack starts, the statistic for the Byzantine node grows linearly in $t$ , leading to fast detection. + +Transient attacks and node ID relabeling. Finally, in Figure 2(b) we analyze the behaviour of our algorithm when it periodically (every 3 epochs for single safeguard and 6 epochs for double safeguard) resets the set of good nodes to include all nodes, restarting the detection process from scratch. Our theoretical result still applies after this relaxation. This relaxation has two benefits. First, it benefits from bad workers that under transient failures (e.g., the node fails for 10 epochs but resumes to work correctly after a while), and thus benefits from the data stored on this worker. Second, it can defend against certain degree of node ID relabeling: it supports the case when good and bad workers exchange their IDs every 6 epochs. In Figure 2(b), we see even under the (very strong) variance attack, relaxed safeguard maintains good performance. + +# ACKNOWLEDGMENTS + +F. E. and D. A. were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 805223 ScaleML). + +REFERENCES +Dan Alistarh, Zeyuan Allen-Zhu, and Jerry Li. Byzantine stochastic gradient descent. In Advances in Neural Information Processing Systems, pp. 4613–4623, 2018. +Zeyuan Allen-Zhu. Natasha 2: Faster Non-Convex Optimization Than SGD. In NeurIPS, 2018a. 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Defending against saddle point attack in byzantine-robust distributed learning. In International Conference on Machine Learning, pp. 7074–7084, 2019. + +# APPENDIX + +# A MISSING PROOFS FOR SECTION 3 + +A.1 PROOF OF LEMMA 3.2 + +Recall the following, useful inequality. + +Lemma A.1 (Pinelis’ 1994 inequality (Pinelis, 1994)). Let $X _ { 1 } , \dots , X _ { T } \in \mathbb { R } ^ { d }$ be a random process satisfying $\mathbb { E } [ X _ { t } | X _ { 1 } , \dots , X _ { t - 1 } ] = 0$ and $\| X _ { t } \| \leq M$ . Then, $\operatorname* { P r } \left[ \| X _ { 1 } + \cdot \cdot \cdot + X _ { T } \| ^ { 2 } > 2 \log ( 2 / \delta ) M ^ { \hat { 2 } } T \right] \leq \delta .$ . + +Lemma 3.2 is in fact a direct corollary of the following claim, whose proof is quite classical. Denote by $C = \log ( 1 6 m T / p )$ . Denote by + +$$ +B _ { i } ^ { ( t ) } : = \frac { \nabla _ { 0 , i } } { | \mathbf { g } \mathsf { o o d } _ { 0 } | } + \cdot \cdot \cdot + \frac { \nabla _ { t - 1 , i } } { | \mathbf { g } \mathsf { o o d } _ { t - 1 } | } \quad \mathrm { a n d } \quad B _ { \star } ^ { ( t ) } : = \frac { \nabla f ( w _ { 0 } ) } { | \mathbf { g } \mathsf { o o d } _ { 0 } | } + \cdot \cdot \cdot + \frac { \nabla f ( w _ { t - 1 } ) } { | \mathbf { g } \mathsf { o o d } _ { t - 1 } | } . +$$ + +Recall at iteration $t - 1$ , Algorithm 2 computes $\{ B _ { 1 } ^ { ( t ) } , \ldots , B _ { m } ^ { ( t ) } \}$ as well as some $B _ { \mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ where $i$ is any machine in $\tt g o o d _ { t - 1 }$ such that at least half of $j \in [ m ]$ satisfies $\lVert B _ { j } ^ { ( t ) } - B _ { i } ^ { ( t ) } \rVert \leq 8 \sqrt { t C } / m$ . + +Claim A.2. Let $C = \log ( 1 6 m T / p )$ . Then, with probability at least $1 - p / 4$ , we have + +(a) for all $i \in$ good and $\in [ T ] , \| B _ { i } ^ { ( t ) } - B _ { \star } ^ { ( t ) } \| \leq 4 \sqrt { t C } / m .$ . +(b) for all $t \in [ T ]$ , each $i \in$ good is a valid choice for $B _ { \mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ = B ( t )i . +(c) for all $i \in$ good and $t \in [ T ] .$ , $\lVert B _ { i } ^ { ( t ) } - B _ { \mathrm { m e d } } ^ { ( t ) } \rVert \leq 1 6 \sqrt { t C } / m$ and $\| B _ { \star } ^ { ( t ) } - B _ { \sf m e d } ^ { ( t ) } \| \le 1 2 \sqrt { t C } / m$ (d) for all $i \in$ good and $t \in [ T ]$ , we also have $i \in \mathbf { g } \circ \circ \mathsf { d } _ { t + 1 }$ . +(e) $\begin{array} { r } { \big \| \sum _ { i \in \mathrm { g o o d } } \big ( B _ { i } ^ { ( t ) } - B _ { \star } ^ { ( t ) } \big ) \big \| \leq O ( \sqrt { t \log ( T / p ) } / \sqrt { m } ) . } \end{array}$ + +Proof of Claim A.2. We prove by induction. Suppose the statements hold for $t - 1$ and we now move to $t$ . + +(a) $\textit { i } \in$ $\mathbb { E } [ \nabla _ { t , i } ] ~ = ~ \nabla _ { t }$ and n thu $\| \nabla _ { t , i } - \nabla _ { t } \| \leq 1$ . Let to the $\begin{array} { r } { { \boldsymbol { X } } _ { t } ~ = ~ \frac { \nabla _ { t , i } - \nabla _ { t } } { | { \bf g } \circ { \bf o d } _ { t } | } } \end{array}$ , so that a union $\begin{array} { r } { \Vert X _ { t } \Vert ~ \leq ~ \frac { 1 } { \vert \mathbf { g } \circ \mathbf { o } \mathbf { d } _ { t } \vert } ~ \leq ~ \frac { 1 } { ( 1 - \alpha ) m } ~ \leq ~ \frac { 2 } { m } } \end{array}$ $X _ { t }$ bound over all $i \in \mathtt { g o o d }$ . Thus, with probability at least $1 - { \frac { p } { 8 T } }$ we have $\lVert B _ { i } ^ { ( t ) } - B _ { \star } ^ { ( t ) } \rVert \leq 4 \sqrt { t C } / m$ for all $i \in { \bf g } { \sf o o d }$ . The result follows from a further union bound over $t \in [ T ]$ . + +(b) Claim A.2a implies for every $i , j \in \mathtt { g o o d }$ we have $\| B _ { i } ^ { ( t ) } - B _ { j } ^ { ( t ) } \| \leq 8 \sqrt { t C } / m$ . Therefore each $i \in$ good is a valid choice for setting $B _ { \mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ . + +(c) This is a consequence of the previous items and the definition of $B _ { \mathrm { m e d } } ^ { ( t ) }$ . + +(d) This is a consequence of the previous item. + +(e) We can apply Lemma A.1 with $\begin{array} { r } { \{ X _ { 1 } , X _ { 2 } , \dots , X _ { t \mid \operatorname { g o o d } \mid } \} ~ = ~ \{ { \frac { \nabla _ { k , i } - \nabla f ( w _ { k } ) } { | \operatorname { g o o d } _ { k } | } } \} _ { k \in [ t ] , i \in \operatorname { g o o d } } } \end{array}$ . It holds with probability at least $\textstyle 1 - { \frac { p } { 8 T } }$ that $\begin{array} { r } { \big \| \sum _ { i \in \mathrm { g o o d } } \big ( B _ { i } ^ { ( t ) } - B _ { \star } ^ { ( t ) } \big ) \big \| \leq O ( \sqrt { t \log ( T / p ) } / \sqrt { m } ) } \end{array}$ . + +Proof of Lemma 3.2. The property $\mathbf { g } _ { 0 0 } \mathsf { d } _ { t } \supseteq \mathsf { g } _ { 0 0 } \mathsf { d }$ is from Claim A.2d. + +The property $\begin{array} { r } { \| \sigma _ { t } \| ^ { 2 } \le { \cal O } ( \frac { \log ( T / p ) } { m } ) } \end{array}$ is by standard concentration inequalities for sums of bounded random vectors. + +The property $\| \sigma _ { 0 } + \cdot \cdot \cdot + \sigma _ { t - 1 } \| ^ { 2 } \leq O ( \frac { T \log ( T / p ) } { m } )$ is from Claim A.2e. + +The property $\| \Delta _ { t } \| \le \alpha$ is obvious as we have at most $\alpha$ fraction of the bad machines. + +The bound on $\| \Delta _ { 0 } + \cdots + \Delta _ { t - 1 } \| ^ { 2 }$ can be derived as follows. For every $i \in [ m ] \ \backslash \ \mathtt { g o o d }$ , let $t$ be the last iteration $_ { i }$ satisfies $i \in \mathtt { g o o d } _ { t }$ . Then, by the triangle inequality, + +$$ +\| B _ { i } ^ { ( t + 1 ) } - B _ { \star } ^ { ( t + 1 ) } \| \le \frac { 2 } { m } + \| B _ { i } ^ { ( t ) } - B _ { \star } ^ { ( t ) } \| +$$ + +On the other hand, $t \in { \mathsf { g o o d } } _ { t }$ implies $\lVert B _ { i } ^ { ( t ) } - B _ { \mathrm { m e d } } ^ { ( t ) } \rVert \leq 1 6 \sqrt { t C } / m$ by the algorithm; combining this with $\| B _ { \star } ^ { ( t ) } - B _ { \sf m e d } ^ { ( t ) } \| \le 1 2 \sqrt { t C } / m$ , and summing up over all such bad machines $_ { i }$ finishes the proof. + +The final two properties follow from standard facts about Gaussian random vectors. + +# A.2 PROOF OF LEMMA 3.3 + +Proof of Lemma 3.3. Using the Lipschitz smoothness of $f ( \cdot )$ , we have + +$$ +\begin{array} { l } { f ( w _ { t } ) - f ( w _ { t + 1 } ) \geq \langle \nabla f ( w _ { t } ) , w _ { t } - w _ { t + 1 } \rangle - \displaystyle \frac { 1 } { 2 } \| w _ { t } - w _ { t + 1 } \| ^ { 2 } } \\ { \displaystyle \qquad = \eta \| \nabla f ( w _ { t } ) \| ^ { 2 } + \eta \langle \nabla f ( w _ { t } ) , \Xi _ { t } \rangle - \displaystyle \frac { 1 } { 2 } \| w _ { t } - w _ { t + 1 } \| ^ { 2 } + \eta \langle \nabla f ( w _ { t } ) , \xi _ { t } \rangle } \end{array} +$$ + +We first show: + +$$ +\begin{array} { c } { { \displaystyle \| w _ { t } - w _ { t + 1 } \| ^ { 2 } = \eta ^ { 2 } \| \nabla f ( w _ { t } ) + \Xi _ { t } - \xi _ { t } \| ^ { 2 } \le 3 \eta ^ { 2 } ( \| \nabla f ( w _ { t } ) \| ^ { 2 } + \| \Xi _ { t } \| ^ { 2 } + \| \xi _ { t } \| ^ { 2 } ) } } \\ { { \displaystyle \sum _ { t = 0 } ^ { T - 1 } \eta \langle \nabla f ( w _ { t } ) , \xi _ { t } \rangle | \le \eta \sqrt { \displaystyle \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } \cdot O ( \nu \sqrt { C _ { 1 } } ) } \le \left( 0 . 0 5 \eta \displaystyle \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } \right) + O ( C _ { 1 } \nu ^ { 2 } \eta ) } } \end{array} +$$ + +The first follows since $( a + b + c ) ^ { 2 } \leq 3 ( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } )$ for any $a , b , c \in { \mathsf { R } }$ , and the second follows from Lemma 3.2. Combining them, and also using that $\left. \Xi _ { t } \right. ^ { 2 } \leq O ( \dot { C } _ { 2 } )$ , $\| \dot { \xi } _ { t } \| ^ { 2 } \le { \cal O } ( d \nu ^ { 2 } C _ { 1 } )$ , and $\eta \leq 0 . 0 1$ , we have + +$$ +f ( w _ { 0 } ) - f ( w _ { T } ) \ge 0 . 9 \eta \sum _ { t = 0 } ^ { T - 1 } \left( \| \nabla f ( w _ { t } ) \| ^ { 2 } - O ( \eta C _ { 2 } ) \right) + \eta \sum _ { t = 0 } ^ { T - 1 } \langle \nabla f ( w _ { t } ) , \Xi _ { t } \rangle - O ( \eta T \nu ^ { 2 } C _ { 1 } ( \eta d + \frac { 1 } { T } ) ) +$$ + +For the inner product on the right hand of (A.1), we have that + +$$ +\eta \sum _ { t = 0 } ^ { T - 1 } \langle \nabla f ( w _ { t } ) , \Xi _ { t } \rangle = \underbrace { \frac { \eta } { T } \sum _ { q = 0 } ^ { T - 1 } \left. \nabla f ( w _ { q } ) , \sum _ { t = 0 } ^ { T - 1 } \Xi _ { t } \right. } _ { \bullet } + \underbrace { \frac { \eta } { T } \sum _ { q = 0 } ^ { T - 1 } \sum _ { t = 0 } ^ { T - 1 } \langle \nabla f ( w _ { t } ) - \nabla f ( w _ { q } ) , \Xi _ { t } \rangle } _ { \bullet } +$$ + +For the first term $\spadesuit$ , we have + +$$ +\begin{array} { l } { | \bullet | \leq \displaystyle \frac { \eta } { T } \sum _ { q = 0 } ^ { T - 1 } \left| \left. \nabla f ( w _ { q } ) , \sum _ { t = 0 } ^ { T - 1 } \Xi _ { t } \right. \right| \leq \displaystyle \frac { \eta } { T } \sum _ { q = 0 } ^ { T - 1 } \| \nabla f ( w _ { q } ) \| \cdot \left\| \sum _ { t = 0 } ^ { T - 1 } \Xi _ { t } \right\| } \\ { \leq 0 . 1 \eta \displaystyle \sum _ { q = 0 } ^ { T - 1 } \| \nabla f ( w _ { q } ) \| ^ { 2 } + \displaystyle \frac { O ( \eta ) } { T ^ { 2 } } \sum _ { q = 0 } ^ { T - 1 } \left\| \sum _ { t = 0 } ^ { T - 1 } \Xi _ { t } \right\| ^ { 2 } } \\ { \leq 0 . 1 \eta \displaystyle \sum _ { q = 0 } ^ { T - 1 } \| \nabla f ( w _ { q } ) \| ^ { 2 } + O \left( \eta C _ { 2 } \right) } \end{array} +$$ + +where the last inequality follows from Lemma 3.2. + +For the second term $\clubsuit$ , we have + +$$ +\begin{array} { r } { | \bullet | \le \underbrace { \eta } _ { T } \displaystyle \sum _ { q = 0 } ^ { T - 1 } \Big | \displaystyle \sum _ { t = 0 } ^ { T - 1 } \langle \nabla f ( w _ { t } ) - \nabla f ( w _ { q } ) , \Xi _ { t } \rangle \Big | \le \underbrace { \frac { \eta } { T } \displaystyle \sum _ { q = 0 } ^ { T - 1 } \Big | \displaystyle \sum _ { t = 0 } ^ { T - 1 } \langle \nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \Xi _ { t } \rangle \Big | } _ { \diamond } } \\ { + \underbrace { \eta } _ { T } \displaystyle \sum _ { q = 0 } ^ { T - 1 } \sum _ { t = 0 } ^ { T - 1 } ( \| w _ { t } - w _ { 0 } \| + \| w _ { q } - w _ { 0 } \| ) \| w _ { t } - w _ { q } \| \| \Xi _ { t } \| } _ { \displaystyle \int _ { 0 } ^ { T } \eta = 0 } , \end{array} +$$ + +Using $\left\| w _ { t } - w _ { q } \right\| \leq \left\| w _ { t } - w _ { 0 } \right\| + \left\| w _ { q } - w _ { 0 } \right\|$ , one can derive + +$$ +\begin{array} { r l } & { \displaystyle \bigtriangledown \le \frac { \eta } { T } \sum _ { q = 0 } ^ { T - 1 } \sum _ { t = 0 } ^ { T - 1 } ( \| w _ { t } - w _ { 0 } \| + \| w _ { q } - w _ { 0 } \| ) ^ { 2 } \cdot O ( \sqrt { C _ { 2 } } ) } \\ & { \displaystyle \le \eta \sum _ { t = 0 } ^ { T - 1 } \| w _ { t } - w _ { 0 } \| ^ { 2 } \cdot O ( \sqrt { C _ { 2 } } ) } \\ & { \displaystyle \le \eta ^ { 3 } \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { 0 } ) + \cdot \cdot \cdot + \nabla f ( w _ { t - 1 } ) + \Xi _ { 0 } + \cdot \cdot \cdot + \Xi _ { t - 1 } + \xi _ { 0 } + \cdot \cdot + \xi _ { t - 1 } \| ^ { 2 } \cdot O ( \sqrt { C _ { 2 } } ) } \\ & { \displaystyle \le O ( \sqrt { C _ { 2 } } \eta ^ { 3 } T ^ { 2 } ) \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } + O ( \sqrt { C _ { 2 } } C \eta ^ { 3 } T ^ { 2 } ) + O ( \eta ^ { 3 } \nu ^ { 2 } T ^ { 2 } d C _ { 1 } \sqrt { C _ { 2 } } ) } \end{array} +$$ + +As for $\diamondsuit$ + +$$ +\Big | \sum _ { t = 0 } ^ { T - 1 } \langle \nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \Xi _ { t } \rangle \Big | \leq \Big | \sum _ { t = q + 1 } ^ { T - 1 } \langle \nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \Xi _ { t } \rangle \Big | + \Big | \sum _ { t = 0 } ^ { q - 1 } \langle \nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \Xi _ { t } \rangle \Big | +$$ + +For the first term (and the second term is analogous), we have + +$$ +\begin{array} { r l } & { \begin{array} { r l } & { \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( j ) ( x , \theta ) - x _ { j } \right| , \mathcal { E } _ { k } \Biggr | } \\ & { = - \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) \right| , } \end{array} } \\ & { \begin{array} { r l } & { \le - \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) - x _ { j } - \mathbb { F } ^ { j } ( x ( x , x _ { j } ) ) + \mathbb { E } _ { \eta _ { k } + 1 } - \mathbb { E } _ { \eta _ { k - 1 } + 1 } + ( \epsilon + \epsilon - \epsilon + \epsilon - \epsilon ) \mathbb { E } _ { k } \right| , \xi _ { 1 } } \\ & { \le \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) \right| , } \end{array} } \\ & \begin{array} { r l } & { \ge \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) \right| , \mathcal { E } _ { k } - \eta - \frac { 1 } { \gamma } \mathcal { F } _ { k } \Big | , } \\ & { \le \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) - x _ { j } \right| , \mathcal { E } _ { k } \Big | - \eta \left| \displaystyle \sum _ { j = 1 } ^ { N } \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) \right| , \mathcal { E } _ { k } + \cdots - \mathbb { E } _ { \eta _ { k - 1 } - 1 } ) \mathbb { E } _ { \eta _ { k - 1 } + 1 } ^ { j } } \\ & { \le \eta \left( \frac { \gamma ^ { j } } { \gamma } \mathcal { F } ^ { j } ( x ) ( x , \theta ) - x _ { j } \right) , } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array} \end{array} +$$ + +Above, inequality $\textcircled{1}$ uses $\| \Xi _ { 0 } + \cdot \cdot \cdot \Xi _ { t } \| \le { \cal O } ( \sqrt { T C _ { 2 } } )$ for $\begin{array} { r } { C _ { 2 } = \alpha ^ { 2 } \log \frac { m T } { p } + \frac { \log ( T / p ) } { m } } \end{array}$ (see Lemma 3.2) and a delicate application of Azuma’s inequality that we state at the end of this subsection (see Proposition 3.4); Inequality $\textcircled{2}$ uses Young’s inequality and Lemma 3.2. + +Putting this back to the formula of $\diamondsuit$ , we have + +$$ +\begin{array} { l } { \displaystyle \diamond \lambda \leq O ( \eta ^ { 2 } \sqrt { T C _ { 2 } } ) \cdot \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| + O ( T \eta ^ { 2 } C _ { 2 } + T \eta ^ { 2 } \nu ^ { 2 } d C _ { 1 } ) } \\ { \displaystyle \qquad \leq 0 . 1 \eta \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } + O ( \eta ^ { 3 } T ^ { 2 } C _ { 2 } + T \eta ^ { 2 } C _ { 2 } + T \eta ^ { 2 } \nu ^ { 2 } d C _ { 1 } ) } \end{array} +$$ + +Finally, putting $\diamondsuit$ and $\heartsuit$ back to $\clubsuit$ , and putting $\clubsuit$ and $\spadesuit$ back to (A.2) and (A.1), we have + +$$ +\begin{array} { l } \displaystyle { \mathrm { ~ \gamma ~ } _ { \mathrm { { \Theta } } } ) - f ( w _ { T } ) \geq 0 . 8 \eta \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } - O ( \sqrt { C _ { 2 } } \eta ^ { 3 } T ^ { 2 } ) \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } } \\ { \displaystyle \qquad - C _ { 2 } \cdot O ( \eta + \eta ^ { 2 } T + \eta ^ { 3 } T ^ { 2 } + \sqrt { C _ { 2 } } \eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \cdot O ( T \eta ^ { 2 } \nu ^ { 2 } d + T ^ { 2 } \eta ^ { 3 } \nu ^ { 2 } \sqrt { C _ { 2 } } + \eta T \nu ^ { 2 } ( \eta d + \frac { 1 } { T } ) ^ { 2 } } \end{array} +$$ + +$\begin{array} { r } { T = \frac { 1 } { 1 0 0 \eta ( 1 + \sqrt { C _ { 2 } } ) } } \end{array}$ and $\eta \leq 0 . 0 1 \operatorname* { m i n } \{ 1 , \frac { 1 } { C _ { 2 } } \}$ + +$$ +\begin{array} { l } { { \displaystyle { \dot { \tau } } ( w _ { 0 } ) - f ( w _ { T } ) \geq 0 . 7 \eta \sum _ { t = 0 } ^ { T - 1 } \| \nabla f ( w _ { t } ) \| ^ { 2 } - C _ { 2 } \cdot O ( \eta + \eta ^ { 2 } T + \eta ^ { 3 } T ^ { 2 } + \sqrt { C _ { 2 } } \eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \cdot O ( T \eta \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ^ { 2 } ) } } \\ { { \displaystyle \qquad = 0 . 7 \eta \sum _ { t = 0 } ^ { T - 1 } \left( \| \nabla f ( w _ { t } ) \| ^ { 2 } - C _ { 2 } \cdot O ( \frac { 1 } { T } + \eta + \eta ^ { 2 } T + \sqrt { C _ { 2 } } \eta ^ { 2 } T ) - C _ { 1 } \cdot O ( \eta T \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) } } \\ { { \displaystyle \qquad \geq 0 . 7 \eta \sum _ { t = 0 } ^ { T - 1 } \left( \| \nabla f ( w _ { t } ) \| ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) \ . } } \end{array} +$$ + +# A.3 PROOF OF PROPOSITION 3.4 + +Proposition 3.4. Fix the dimension parameter $d \geq 1$ . Suppose $\xi _ { 0 } , \ldots , \xi _ { T - 1 } \in \mathbb { R } ^ { d }$ are i.i.d. drawn from $\mathcal { N } ( 0 , \bf { I } )$ , and that $\Delta _ { 1 } , \ldots , \Delta _ { T - 1 }$ are arbitrary vectors in $\mathbb { R } ^ { d }$ . Here, each vector $\Delta _ { t }$ with $t = 1 , \dots , T - 1$ can depend on $\xi _ { 0 } , \ldots , \xi _ { t - 1 }$ but not on $\xi _ { t } , \dots , \xi _ { T - 1 }$ . Suppose that these vectors satisfy $\| \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \| ^ { 2 } \leq \mathfrak { T } _ { j }$ or every $t = 1 , \dots , T - 1$ . Then, with probability at least $1 - p ,$ , + +$$ +\begin{array} { r } { \left| \sum _ { t = 1 } ^ { T - 1 } \langle \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } , \Delta _ { t } \rangle \right| \leq O ( \sqrt { d T \mathfrak { T } \log ( T / p ) } ) } \end{array} . +$$ + +Proof of Proposition 3.4. Using the identity formula6 + +$$ +\begin{array} { r } { \sum _ { t = 1 } ^ { T - 1 } \langle \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } , \Delta _ { t } \rangle = \left( \sum _ { t = 0 } ^ { T - 2 } \xi _ { t } \right) \left( \sum _ { t = 1 } ^ { T - 1 } \Delta _ { t } \right) - \sum _ { t = 1 } ^ { T - 2 } \langle \xi _ { t } , \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \rangle = \Delta _ { 0 } ^ { 2 } \Delta _ { t } . } \end{array} +$$ + +we have + +$$ +\begin{array} { r l } & { \left| \sum _ { t = 1 } ^ { T - 1 } \langle \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } , \Delta t \rangle \right| \leq \left\| \sum _ { t = 0 } ^ { T - 2 } \xi _ { t } \right\| \cdot \left\| \sum _ { t = 1 } ^ { T - 1 } \Delta _ { t } \right\| + \left| \sum _ { t = 1 } ^ { T - 2 } \langle \xi _ { t } , \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \rangle \right| } \\ & { \qquad \leq O ( \sqrt { d T ^ { \sum } \log ( T / p ) } ) + \left| \sum _ { t = 1 } ^ { T - 2 } \langle \xi _ { t } , \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \rangle \right| \ . } \end{array} +$$ + +where the last inequality uses $\| \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { T - 2 } \| \le O ( \sqrt { d T \log ( 1 / p ) } )$ with probability at least $1 - p / 2$ . Furthermore, we note that $\xi _ { t }$ is independent of $\xi _ { 0 } , \ldots , \xi _ { t - 1 } , \Delta _ { 1 } , \ldots , \Delta _ { t }$ and $\mathbb { E } [ \xi _ { t } ] = 0$ . Therefore, letting $S _ { t } = \langle \xi _ { t } , \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \rangle$ , we have $\mathbb { E } [ S _ { t } | \xi _ { 0 } , \dots , \xi _ { t - 1 } ] = 0$ ; furthermore, with probability at least $1 - p / 2$ , it satisfies $| S _ { t } | \leq O ( \sqrt { d \mathfrak { T } \log ( T / p ) } )$ for every $t$ . Finally, by Azuma’s inequality, we have + +$$ +\begin{array} { r } { \left| \sum _ { t = 1 } ^ { T - 2 } \langle \xi _ { t } , \Delta _ { 1 } + \cdot \cdot \cdot + \Delta _ { t } \rangle \right| \leq O ( \sqrt { d T \mathfrak { T } \log ( T / p ) } ) } \end{array} . +$$ + +# A.4 PROOF OF LEMMA 3.5 + +Proof of Lemma 3.5. Let us denote by $\begin{array} { r } { r _ { t } = \frac { [ \xi _ { t } ^ { \mathsf { a } } ] _ { 1 } } { 2 } = - \frac { [ \xi _ { t } ^ { \mathsf { b } } ] _ { 1 } } { 2 } } \end{array}$ and we know $\begin{array} { r } { r _ { t } \sim \mathcal { N } ( 0 , \frac { \nu ^ { 2 } } { 4 } ) } \end{array}$ . We can write + +$$ +w _ { t + 1 } ^ { \mathsf { a } } - w _ { t + 1 } ^ { \mathsf { b } } = \eta r _ { t } \mathbf { e } _ { 1 } + w _ { t } ^ { \mathsf { a } } - w _ { t } ^ { \mathsf { b } } - \eta ( \nabla f ( w _ { t } ^ { \mathsf { a } } ) - \nabla f ( w _ { t } ^ { \mathsf { b } } ) ) - \eta ( \Xi _ { t } ^ { \mathsf { a } } - \Xi _ { t } ^ { \mathsf { b } } ) +$$ + +Using the second-order smoothness, we have + +$$ +\begin{array} { r l } & { \nabla f ( w _ { t } ^ { \mathsf { a } } ) - \nabla f ( w _ { t } ^ { \mathsf { b } } ) = \displaystyle \int _ { \tau = 0 } ^ { 1 } \nabla ^ { 2 } f \big ( { w } _ { t } ^ { \mathsf { a } } + \tau ( w _ { t } ^ { \mathsf { b } } - w _ { t } ^ { \mathsf { a } } ) \big ) \big ( w _ { t } ^ { \mathsf { a } } - w _ { t } ^ { \mathsf { b } } \big ) d \tau } \\ & { \qquad = \nabla ^ { 2 } f \big ( { w } _ { 0 } \big ) \cdot \big ( { w } _ { t } ^ { \mathsf { a } } - { w } _ { t } ^ { \mathsf { b } } \big ) + \theta _ { t } } \end{array} +$$ + +for some vector $\| \theta _ { t } \| \leq \operatorname* { m a x } \{ \| w _ { 0 } ^ { \mathsf { a } } - w _ { t } ^ { \mathsf { a } } \| , \| w _ { 0 } ^ { \mathsf { b } } - w _ { t } ^ { \mathsf { b } } \| \} \cdot \| w _ { t } ^ { \mathsf { a } } - w _ { t } ^ { \mathsf { b } } \|$ . Therefore, we have + +$$ +w _ { t + 1 } ^ { \mathsf { a } } - w _ { t + 1 } ^ { \mathsf { b } } = \eta r _ { t } \mathbf { e } _ { 1 } + \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) ( w _ { t } ^ { \mathsf { a } } - w _ { t } ^ { \mathsf { b } } ) - \eta ( \Xi _ { t } ^ { \mathsf { a } } - \Xi _ { t } ^ { \mathsf { b } } + \theta _ { t } ) +$$ + +Now, giving $\psi _ { 0 } = \overline { { \psi } } _ { 0 } = 0$ , imagine two sequences + +• $\psi _ { t + 1 } = \eta r _ { t } \mathbf { e } _ { 1 } + \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) \psi _ { t }$ and • $\overline { { \psi } } _ { t + 1 } = \eta r _ { t } \mathbf { e } _ { 1 } + \big ( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \big ) \overline { { \psi } } _ { t } - \eta \big ( \Xi _ { t } ^ { 3 } - \Xi _ { t } ^ { \mathsf { b } } + \theta _ { t } \big ) = w _ { t + 1 } ^ { \mathsf { a } } - w _ { t + 1 } ^ { \mathsf { b } }$ + +We will inductively prove $\lVert \psi _ { t } - \overline { { \psi } } _ { t } \rVert \leq \frac 1 2 \lVert \psi _ { t } \rVert$ . On one hand, it is easy to see that $\psi _ { t }$ is zero except in the first coordinate, in which it behaves as a Gaussian with zero mean and variance Pt−1k=0(1 + ηδ)2k · η2ν24 Θ  (1+ηδ)2tηδ · η2ν2. By Gaussian tail bounds, we know that + +• with probability at least 0.99, it satisfies $\begin{array} { r } { \| \psi _ { t } \| \leq O ( \frac { \sqrt { \eta C _ { 1 } } \nu ( 1 + \eta \delta ) ^ { t } } { \sqrt { \delta } } ) } \end{array}$ for every $t$ • with probability at least 0.99, it satisfies $\begin{array} { r } { \| \psi _ { T } \| \ge \frac { 1 } { 1 0 0 0 } \big ( \frac { \sqrt { \eta } \nu ( 1 + \eta \delta ) ^ { T } } { \sqrt { \delta } } \big ) } \end{array}$ + +In the rest of the proof, we condition on this event happens. We prove towards contradiction by assuming $\| w _ { t } ^ { \mathsf { a } } - w _ { 0 } ^ { \mathsf { a } } \| \leq R$ and $\| w _ { t } ^ { \mathsf { b } } - w _ { 0 } ^ { \mathsf { b } } \| \leq R$ for all $t \in [ T ]$ . + +We will inductively prove that $\begin{array} { r } { \| \psi _ { t } - \overline { { \psi } } _ { t } \| \leq \frac { 1 } { 2 0 0 0 } \big ( \frac { \sqrt { \eta } \nu ( 1 + \eta \delta ) ^ { t } } { \sqrt { \delta } } \big ) } \end{array}$ . We calculate the difference + +$$ +\psi _ { t } - { \overline { { \psi } } } _ { t } = \eta \sum _ { i = 0 } ^ { t - 1 } \left( \mathbf { I } - \eta { \boldsymbol \nabla } ^ { 2 } f ( w _ { 0 } ) \right) ^ { t - 1 - i } ( { \boldsymbol \Xi } _ { i } ^ { \mathrm { a } } - { \boldsymbol \Xi } _ { i } ^ { \mathrm { b } } + { \boldsymbol \theta } _ { i } ) +$$ + +Let g = $\begin{array} { r } { g = \frac { \psi _ { t } - \overline { { \psi } } _ { t } } { \| \psi _ { t } - \overline { { \psi } } _ { t } \| } } \end{array}$ , then we can inner-product the above equation by vector $g$ , which gives + +$$ +\| \psi _ { t } - \overline { { \psi } } _ { t } \| = \eta \sum _ { i = 0 } ^ { t - 1 } \Big \langle \Xi _ { i } ^ { \mathrm { a } } - \Xi _ { i } ^ { \mathrm { b } } + \theta _ { i } , \big ( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \big ) ^ { t - 1 - i } g \Big \rangle +$$ + +6We thank an anonymous reviewer on openreview who pointed out this simpler proof to us. + +$$ +\begin{array} { r l } & { \overset { \circledast } { \leq } \eta \displaystyle \sum _ { i = 0 } ^ { t - 1 } \left( \left. \Xi _ { i } ^ { \mathsf { a } } - \Xi _ { i } ^ { \mathsf { b } } , \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) ^ { t - 1 - i } g \right. + R \cdot O ( \frac { \sqrt { \eta C _ { 1 } } \nu \left( 1 + \eta \delta \right) ^ { i } } { \sqrt { \delta } } ) \cdot ( 1 + \eta \delta ) ^ { t - 1 - i } \right) } \\ & { \leq \eta \displaystyle \sum _ { i = 0 } ^ { t - 1 } \left. \Xi _ { i } ^ { \mathsf { a } } - \Xi _ { i } ^ { \mathsf { b } } , \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) ^ { t - 1 - i } g \right. + O ( R \eta T \frac { \sqrt { \eta C _ { 1 } } } { \sqrt { \delta } } \nu ( 1 + \eta \delta ) ^ { t } ) } \end{array} +$$ + +where the inequality $\textcircled{1}$ uses $\| \theta _ { i } \| \leq R \cdot \| \overline { { \psi } } _ { i } \| \leq R \cdot ( \| \psi _ { i } \| + \| \psi _ { i } - \overline { { \psi } } _ { i } \| )$ , $\begin{array} { r l } { \Vert \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) ^ { t - 1 - i } g \Vert } & { { } \leq } \end{array}$ $( 1 + \eta \delta ) ^ { t - 1 - i }$ , and the inductive assumption. Let us call $M = \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right)$ , and focus on + +$$ +\begin{array} { r l } & { \left| \displaystyle \sum _ { i = 0 } ^ { \ell - 1 } \left. \Xi _ { i } ^ { 3 } , \left( \mathbf { I } - \eta \nabla ^ { 2 } f ( w _ { 0 } ) \right) ^ { t - 1 - i } g \right. \right| } \\ & { = \left| \displaystyle \langle \Xi _ { 0 } ^ { 3 } + \dots + \Xi _ { t - 1 } ^ { 3 } , g \rangle + \displaystyle \sum _ { i = 0 } ^ { t - 2 } \left. \Xi _ { 0 } ^ { 3 } + \dots + \Xi _ { i } ^ { 2 } , M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \right. \right| } \\ & { \leq \| \Xi _ { 0 } ^ { 3 } + \dots + \Xi _ { t - 1 } ^ { 3 } \| \cdot \| g \| + \displaystyle \sum _ { i = 0 } ^ { t - 2 } \| \Xi _ { 0 } ^ { 3 } + \dots + \Xi _ { i } ^ { 2 } \| \cdot \| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \| } \\ & { \leq O ( \sqrt { T C _ { 2 } } \left( \| g \| + \displaystyle \sum _ { i = 0 } ^ { t - 2 } \| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \| \right) } \\ & { \leq O ( \sqrt { T C _ { 2 } } \cdot ( 1 + \eta \delta ) ^ { t - 1 } ) } \end{array} +$$ + +Together, we have + +$$ +\| \psi _ { t } - \overline { { \psi } } _ { t } \| \leq O ( \eta \sqrt { T C _ { 2 } } ) \cdot ( 1 + \eta \delta ) ^ { t } + O ( R \eta T \frac { \sqrt { \eta C _ { 1 } } } { \sqrt { \delta } } \nu ( 1 + \eta \delta ) ^ { t } ) +$$ + +Under our assumption, we have $\begin{array} { r } { \| \psi _ { t } - \overline { { \psi } } _ { t } \| < \frac { 1 } { 2 0 0 0 } \big ( \frac { \sqrt { \eta } \nu ( 1 + \eta \delta ) ^ { t } } { \sqrt { \delta } } \big ) } \end{array}$ and therefore $\lVert \overline { { \psi } } _ { T } \rVert \ge \lVert \psi _ { T } \rVert - \lVert \psi _ { T } - \overline { { \psi } } _ { T } \rVert \ge$ 0 ( √ην(1+ηδ)t√ ). Thus, within T iterations, we have kψtk > R and this gives a contradiction.  + +# B FINAL: DOUBLE SAFE GUARD + +We now come to our final Algorithm 3 which is our perturbed SGD algorithm with two safeguards. The two safeguard algorithm naturally divides itself into epochs, each consisting of $T _ { 1 }$ iterations. We will demonstrate that within most epochs, we make good progress. Thus, consider some iterate $x _ { m T _ { 1 } }$ , for some $m < T / T _ { 1 }$ . Our goal will be to argue that we make good function value progress by iterate $x _ { ( m + 1 ) T _ { 1 } }$ , and that we do not settle into any saddle points. To slightly simplify notation, let $w _ { 0 } = x _ { m T _ { 1 } }$ , and let the sequence of iterates be $w _ { 0 } , \dots , w _ { T _ { 1 } - 1 }$ , so that $w _ { T _ { 1 } - 1 } = x _ { ( m + 1 ) T _ { 1 } - 1 }$ . For completeness’ sake we rewrite this as Algorithm 1. + +# Algorithm 3 Perturbed SGD with double safe guard (for analysis purpose) + +Input: $w _ { 0 } \in \mathbb { R } ^ { d }$ , set $\mathtt { g o o d } _ { 0 } \supseteq \mathtt { g o o d }$ , rate $\eta > 0$ , lengths $T _ { 1 } \ge T _ { 0 } \ge 1$ , threshold $\mathfrak { T } _ { 1 } > \mathfrak { T } _ { 0 } > 0$ ; +1: for $t \gets 0$ to $T _ { 1 } - 1$ do +2: $l a s t \gets \operatorname* { m a x } \{ t _ { 0 } \in [ t ] \colon t _ { 0 }$ is a multiple of $T _ { 0 } \}$ +3: for each $i \in \mathtt { g o o d } _ { t }$ do +4: receive $\nabla _ { t , i } \in \mathbb { R } ^ { d }$ from machine $i$ ; +5: Ai ← Ptk=0 ∇k,i|good | and Bi ← Ptk=last ∇k,i|good | ; +6: $A _ { \sf m e d } A _ { i }$ where $i \in { \mathfrak { g o o d } } _ { t }$ is any machine s.t. $\left| \left\{ j \in \operatorname { g o o d } _ { t } \colon \left\| A _ { j } - A _ { i } \right\| \leq \mathfrak { T } _ { 1 } \right\} \right| > m / 2$ . +7: $B _ { \sf m e d } B _ { i }$ where $i \in \mathtt { g o o d } _ { t }$ is any machine s.t. $\left| \left\{ j \in \mathbf { g 0 0 d } _ { t } \colon \left\| B _ { j } - B _ { i } \right\| \le \mathfrak { T } _ { 0 } \right\} \right| > m / 2$ . +8: $\begin{array} { r l } & { \mathsf { g o o d } _ { t + 1 } \gets \big \{ i \in \mathsf { g o o d } _ { t } : \| A _ { i } - A _ { \mathsf { m e d } } \| \le 2 \mathfrak { T } _ { 1 } \bigwedge \| B _ { i } - B _ { \mathsf { m e d } } \| \le 2 \mathfrak { T } _ { 0 } \big \} } \\ & { w _ { t + 1 } = w _ { t } - \eta \left( \xi _ { t } + \frac { 1 } { \vert \mathsf { g o o d } _ { t } \vert } \sum _ { i \in \mathsf { g o o d } _ { t } } \nabla _ { t , i } \right) ; } \end{array}$ ; +9: + +Our main result is the following theorem. + +Theorem B.1. Let $C _ { 3 } = \alpha ^ { 2 } + { \textstyle { \frac { 1 } { m } } }$ . Suppose we pick parameters $p , \delta \in ( 0 , 1 )$ , $\begin{array} { r } { \eta \le \widetilde { O } ( \frac { \delta ^ { 3 } } { C _ { 3 } } ) } \end{array}$ , $\nu ^ { 2 } = \widetilde { \Theta } ( C _ { 3 } )$ $\begin{array} { r } { T _ { 0 } = \widetilde { \Theta } \left( \frac { 1 } { \eta } \right) } \end{array}$ , $\begin{array} { r } { T _ { 1 } = \widetilde { \Theta } \big ( \frac { 1 } { \eta \delta } \big ) \geq T _ { 0 } } \end{array}$ , $\mathfrak { T } _ { 1 } = \widetilde { \Theta } ( \sqrt { T _ { 1 } } )$ , and ${ \mathfrak { T } } _ { 0 } = { \widetilde { \Theta } } ( { \sqrt { T _ { 0 } } } )$ . Then, starting from $w _ { 0 }$ , + +(a) with probability at least $1 - p$ we have + +$$ +f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \geq 0 . 7 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( \Vert \nabla f ( w _ { t } ) \Vert ^ { 2 } - \widetilde { O } ( \eta C _ { 3 } d ) \right) \ . +$$ + +(b) As long as $\| w _ { t } - w _ { 0 } \| \ge R$ for some $t \in \{ 1 , 2 , \dots , T _ { 1 } \}$ and for $\begin{array} { r } { R = \widetilde { \Theta } ( \delta ) \leq \frac { \delta } { 2 } } \end{array}$ , then with probability at least $1 - p$ we have then + +$$ +f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \geq 0 . 5 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \Big ( - \widetilde { O } ( \eta C _ { 3 } d ) \Big ) + \widetilde { \Omega } ( \delta ^ { 3 } ) +$$ + +(c) i $f \lambda _ { \operatorname* { m i n } } ( \nabla ^ { 2 } f ( w _ { 0 } ) ) \leq - \delta$ , we also have with probability at least 0.45, + +$$ +f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \geq 0 . 5 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \Big ( - \widetilde { O } ( \eta C _ { 3 } d ) \Big ) + \widetilde { \Omega } ( \delta ^ { 3 } ) +$$ + +# B.1 WHY THEOREM B.1 IMPLIES THEOREM 2.3 + +Using the parameter choice $\begin{array} { r } { \eta = \widetilde { \Theta } ( \frac { \varepsilon ^ { 2 } } { C _ { 3 } d } ) } \end{array}$ from Theorem 2.3, we know $\widetilde { \cal O } ( \eta C _ { 3 } d ) \le 0 . 1 \varepsilon ^ { 2 }$ . We claim two things: + +• For at least $9 0 \%$ of the epochs, they must satisfy (denoting by $w _ { 0 }$ and $w _ { T _ { 1 } }$ the beginning and ending points of this epoch) + +$$ +f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \leq 2 0 \frac { f ( x _ { 0 } ) - \operatorname* { m i n } f ( x ) } { T / T _ { 1 } } \leq \varepsilon ^ { 1 . 5 } +$$ + +The last inequality uses our choice of $T$ and $\delta = \widetilde { \Theta } ( \sqrt { \varepsilon } )$ . + +The reason for this is by way of contradiction. Suppose for at least $1 0 \%$ of the epochs it satisfies $\begin{array} { r } { f ( w _ { 0 } ) \ : - \ : f ( w _ { T _ { 1 } } ) > 2 0 \frac { \dot { f } ( x _ { 0 } ) - \operatorname* { m i n } f ( x ) } { T / T _ { 1 } } } \end{array}$ , then, for the remainder of the epochs, they must at least satisfy $f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \geq - 0 . 7 \eta T _ { 1 } \cdot 0 . 1 \varepsilon ^ { 2 }$ . Summing over all the epochs, we shall obtain $f ( x _ { 0 } ) - f ( x _ { T } ) >$ $f ( x _ { 0 } ) - \operatorname* { m i n } f ( x )$ but this gives a contradiction. + +• For at least $4 0 \%$ of the epochs, they must satisfy the three properties from Theorem B.1. + +In particular, for at least $3 0 \%$ of the epochs, they must satisfy both. Since $\varepsilon ^ { 1 . 5 }$ is so small that + +$$ +\varepsilon ^ { 1 . 5 } \ge f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \ge 0 . 5 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( - \widetilde { O } ( \eta C _ { 3 } d ) \right) + \widetilde { \Omega } ( \delta ^ { 3 } ) \ge \widetilde { \Omega } ( \delta ^ { 3 } ) - 0 . 0 5 \eta T _ { 1 } \varepsilon ^ { 2 } +$$ + +would give a contradiction (for instance, one can choose $\delta$ to be slightly larger than $\sqrt { \varepsilon }$ by some log factors), this means, for those $3 0 \%$ of the epochs, they must satisfy: + +• $\begin{array} { r } { \varepsilon ^ { 1 . 5 } \geq 0 . 7 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( \Vert \nabla f ( w _ { t } ) \Vert ^ { 2 } - 0 . 1 \varepsilon ^ { 2 } \right) \ , } \end{array}$ , • $\begin{array} { r } { \| w _ { t } - w _ { 0 } \| \leq \frac { \delta } { 2 } } \end{array}$ for every $t = 1 , 2 , \dots , T _ { 1 }$ , and • $\nabla ^ { 2 } f ( w _ { 0 } ) \succeq - \delta \mathbf { I } .$ . + +The latter two properties together implies $\nabla ^ { 2 } f ( w _ { t } ) \succeq - \frac { \delta } { 2 } \mathbf { I }$ for every $t = 1 , 2 , \ldots , T _ { 1 }$ (by the second-order smoothness). The first property implies for at least $9 0 \%$ of the iterations $t$ in this epoch, they must satisfy $\| \nabla f ( x ) \| \leq \varepsilon$ . This finishes the proof of Theorem 2.3.  + +# B.2 PROOF OF THEOREM B.1 + +We first have the following lemma + +Lemma B.2 (double safe guard). In Algorithm 3, suppose $\mathfrak { T } _ { 1 } ~ = ~ 8 \sqrt { T _ { 1 } \log ( 1 6 m T _ { 1 } / p ) }$ and $\mathfrak { T } _ { 0 } ~ =$ $8 \sqrt { T _ { 0 } \log ( 1 6 m T _ { 1 } / p ) }$ . Then, with probability at least $1 - p / 2 ,$ , for every $t = 0 , \ldots , T _ { 1 } - 1$ , + +• $\mathbf { g } _ { 0 0 } \mathbf { d } _ { t } \supseteq \mathbf { g } _ { 0 0 } \mathbf { d }$ . +• • $\begin{array} { r l } & { | \sigma _ { t } | | ^ { 2 } \leq O ( \frac { \log ( T _ { 1 } / p ) } { m } ) , \| \Delta _ { t } \| ^ { 2 } \leq \alpha ^ { 2 } , \| \xi _ { t } \| ^ { 2 } \leq O ( \nu ^ { 2 } d \log ( T _ { 1 } / p ) ) , } \\ & { | \sigma _ { 0 } + \cdot \cdot + \sigma _ { t - 1 } \| ^ { 2 } \leq O ( \frac { T _ { 1 } \log ( T _ { 1 } / p ) } { m } ) , \| \sigma _ { t a s t } + \cdot \cdot + \sigma _ { t - 1 } \| ^ { 2 } \leq O ( \frac { T _ { 0 } \log ( T _ { 1 } / p ) } { m } ) } \\ & { | \Delta _ { 0 } + \cdot \cdot \cdot + \Delta _ { t - 1 } \| ^ { 2 } \leq O ( \alpha ^ { 2 } T _ { 1 } \log ( m T _ { 1 } / p ) ) a n d \| \Delta _ { l a s t } + \cdot \cdot \cdot + \Delta _ { t - 1 } \| ^ { 2 } \leq O ( \alpha ^ { 2 } T _ { 0 } \log ( m T _ { 1 } / p ) ) } \\ & { | \xi _ { 0 } + \cdot \cdot \cdot + \xi _ { t - 1 } \| ^ { 2 } \leq O ( \nu ^ { 2 } d T _ { 1 } \log ( T _ { 1 } / p ) ) a n d \| \xi _ { l a s t } + \cdot \cdot \cdot + \xi _ { t - 1 } \| ^ { 2 } \leq O ( \nu ^ { 2 } d T _ { 0 } \log ( T _ { 1 } / p ) ) . } \\ & { \quad \cdot \quad \cdot \quad \cdot \quad \cdot } \end{array}$ • +• + +We call this probabilistic event Event $\mathsf { \Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \mathsf { d o u b l e } } ( w _ { 0 } )$ and $\operatorname* { P r } [ \mathsf { E v e n t } _ { T _ { 1 } , T _ { 0 } } ^ { \mathsf { d o u b l e } } ( w _ { 0 } ) ] \geq 1 - p / 2$ . + +The proof is a direct corollary of Lemma 3.2, by combining events Even $\mathbf { \cdot } _ { T _ { 1 } } ^ { \mathsf { s i n g l e } } ( w _ { 0 } )$ , Event $\mathbf { \cdot } _ { T _ { 0 } } ^ { \mathsf { s i n g l e } } ( w _ { 0 } )$ + +EvenLemm $\mathbf { \cdot } _ { T _ { 0 } } ^ { \mathsf { s i n g l e } } ( w _ { T _ { 0 } } )$ , ro $\mathsf { E v e n t } _ { T _ { 0 } } ^ { \mathsf { s i n g l e } } ( w _ { 2 T _ { 0 } } )$ and so on. The next lemma is a simple corollary by repeatedly applyinga. + +LemmaSuppose e, by a $C _ { 1 } = \log ( T _ { 1 } / p )$ and start $\begin{array} { r } { C _ { 2 } = \alpha ^ { 2 } \log \frac { m T _ { 1 } } { p } + \frac { \log ( T _ { 1 } / p ) } { m } } \end{array}$ .. $\eta \leq 0 . 0 1 \operatorname* { m i n } \{ 1 , \frac { 1 } { C _ { 2 } } \}$ $\begin{array} { r } { T _ { 0 } = \frac { 1 } { 1 0 0 \eta ( 1 + \sqrt { C _ { 2 } } ) } } \end{array}$ $T _ { 1 } \geq T _ { 0 }$ $w _ { 0 }$ Under event Even $\mathsf { \Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \mathsf { d o u b l e } } ( w _ { 0 } )$ , it satisfies + +$$ +f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \ge 0 . 7 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( \| \nabla f ( w _ { t } ) \| ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) +$$ + +The next lemma can be easily derived from Lemma 3.5. + +Lemma B.4 (modified from Lemma 3.5). Suppose + +$$ +R = \Theta ( \frac { \delta } { \sqrt { C _ { 1 } } \log ( \delta ^ { 3 } / \eta C _ { 2 } ) } ) a n d \nu ^ { 2 } = \Theta ( C _ { 2 } \log \frac { \delta ^ { 3 } } { \eta C _ { 2 } } ) +$$ + +Suppose $\begin{array} { r } { \eta \leq 0 . 0 1 \operatorname* { m i n } \lbrace 1 , \frac { \delta ^ { 3 } } { C _ { 2 } } \rbrace } \end{array}$ , $\begin{array} { r } { T _ { 0 } = \frac { 1 } { 1 0 0 \eta ( 1 + \sqrt { C _ { 2 } } ) } } \end{array}$ and $\begin{array} { r } { T _ { 1 } = \Theta ( \frac { 1 } { \eta \delta } \log \frac { \delta ^ { 3 } } { \eta C _ { 2 } } ) \geq T _ { 0 } } \end{array}$ . Let $\boldsymbol { w _ { 0 } } \in \mathbb { R } ^ { d }$ be any point in the space and suppose $\lambda _ { \operatorname* { m i n } } \bigl ( \nabla ^ { 2 } f ( w _ { 0 } ) \bigr ) \leq - \delta$ for some $\delta \geq 0$ . Given two coupled sequences defined as before, under events Event $\mathsf { \Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \mathsf { d o u b l e } } ( w _ { 0 } ^ { \mathsf { a } } )$ and Event $\mathsf { \Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \mathsf { d o u b l e } } ( w _ { 0 } ^ { \mathsf { b } } )$ , we have with probability at least 0.98 + +$$ +\begin{array} { l } { { \displaystyle \operatorname* { m a x } \left\{ f ( w _ { 0 } ^ { \mathsf { a } } ) - f ( w _ { T _ { 1 } } ^ { \mathsf { a } } ) , f ( w _ { 0 } ^ { \mathsf { b } } ) - f ( w _ { T _ { 1 } } ^ { \mathsf { b } } ) \right\} } \ ~ } \\ { \displaystyle \geq 0 . 5 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) + \Omega ( \frac { \delta ^ { 3 } } { C _ { 1 } \log ^ { 3 } \frac { \delta ^ { 3 } } { \eta C _ { 2 } } } ) } \end{array} +$$ + +Lemma B.4 directly proves the second half of Theorem B.1c, because given two coupled sequences with the same marginal distribution, we have + +$$ +\operatorname* { P r } [ f ( w _ { 0 } ^ { \mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \mathrm { a } } ) \ge X ] \ge \frac { 1 } { 2 } \operatorname* { P r } [ \operatorname* { m a x } \left\{ f ( w _ { 0 } ^ { \mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \mathrm { a } } ) , f ( w _ { 0 } ^ { \mathrm { b } } ) - f ( w _ { T _ { 1 } } ^ { \mathrm { b } } ) \right\} \ge X ] +$$ + +Proof of Lemma B.4. Our choice on $r$ and $R$ satisfy the requirement of Lemma 3.5. Suppose without loss of generality that the $w _ { t } ^ { \mathsf { a } }$ sequence leaves $w _ { 0 }$ by more than $R$ . Let $T _ { 1 } ^ { \mathsf { a } }$ be the first iteration $t \leq T _ { 1 }$ in which $\| w _ { t } ^ { \mathsf { a } } - w _ { 0 } ^ { \mathsf { a } } \| \geq R$ . + +$$ +\begin{array} { c } { { \displaystyle \| w _ { T _ { 1 } ^ { \mathtt { a } } } ^ { \mathtt { a } } - w _ { 0 } ^ { \mathtt { a } } \| ^ { 2 } = \eta ^ { 2 } \| \nabla f ( w _ { 0 } ^ { \mathtt { a } } ) + \cdot \cdot \cdot + \nabla f ( w _ { T _ { 1 } ^ { \mathtt { a } } - 1 } ^ { \mathtt { a } } ) + \Xi _ { 0 } ^ { \mathtt { a } } + \cdot \cdot \cdot + \Xi _ { T _ { 1 } ^ { \mathtt { a } } - 1 } ^ { \mathtt { a } } + \xi _ { 0 } ^ { \mathtt { a } } + \cdot \cdot \cdot + \xi _ { T _ { 1 } ^ { \mathtt { a } } - 1 } ^ { \mathtt { a } } \| ^ { 2 } } } \\ { { \displaystyle \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } } \\ { { \displaystyle \le O ( \eta ^ { 2 } T _ { 1 } ) \sum _ { t = 0 } ^ { T _ { 1 } ^ { \mathtt { a } } - 1 } \| \nabla f ( w _ { t } ^ { \mathtt { a } } ) \| ^ { 2 } + O ( C _ { 2 } \eta ^ { 2 } T _ { 1 } ) + O ( C _ { 1 } \eta ^ { 2 } \nu ^ { 2 } T _ { 1 } d ) } } \end{array} +$$ + +Combining this with Lemma B.3, we have + +$$ +\begin{array} { l } { \displaystyle \lvert \left( w _ { 0 } ^ { 3 } \right) - f \left( w _ { T _ { 1 } ^ { 3 } } ^ { \mathfrak { a } } \right) \geq 0 . 5 \eta \displaystyle \sum _ { t = 0 } ^ { T _ { 1 } ^ { \sharp } - 1 } \left( \lVert \nabla f ( w _ { t } ^ { \mathfrak { a } } ) \rVert ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) + \frac { \lVert w _ { T _ { 1 } ^ { \flat } } ^ { \mathfrak { a } } - w _ { 0 } ^ { \mathfrak { a } } \rVert } { 1 0 0 \eta T _ { 1 } } } \\ { \displaystyle \qquad \geq 0 . 5 \eta \displaystyle \sum _ { t = 0 } ^ { T _ { 1 } ^ { \sharp } - 1 } \left( \lVert \nabla f ( w _ { t } ^ { \mathfrak { a } } ) \rVert ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) + \frac { R ^ { 2 } } { 1 0 0 \eta T _ { 1 } } } \end{array} +$$ + +Combining this with Lemma B.3 again but for the remainder iterations, we have + +$$ +f ( w _ { 0 } ^ { 3 } ) - f ( w _ { T _ { 1 } } ^ { 3 } ) \geq 0 . 5 \eta \sum _ { t = 0 } ^ { T _ { 1 } - 1 } \left( \| \nabla f ( w _ { t } ^ { 3 } ) \| ^ { 2 } - \eta \cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \nu ^ { 2 } \eta ( d + \sqrt { C _ { 2 } } ) ) \right) + \frac { R ^ { 2 } } { 1 0 0 \eta T _ { 1 } } +$$ + +In fact, the above same proof of Lemma B.4 also implies Theorem B.1b. These together finish the proof of Theorem B.1.  + +# C MORE ON EXPERIMENTS + +We conduct experiments on training a residual network ResNet-20 He et al. (2016) on the CIFAR-10/100 image classification tasks Krizhevsky et al. (2014). + +# C.1 SETTING AND IMPLEMENTED METHODS + +In all of our experiments, we use 10 workers and mini-batch size 10 per worker. Given any attacker and any defender algorithm, we run SGD three times for 140 epochs, each time with a different initial learning rate $\eta \in \{ 0 . 1 , 0 . 2 , 0 . 4 \}$ .7 We let the learning rate decrease by a factor of 10 on epochs 80 and 110, and present present the best testing accuracies in the three runs (each corresponding to a different initial learning rate). + +We use standard data augmentation (random crops, random flips, and channel normalization). + +We compare against Geometric Median Chen et al. (2017), Coordinate-wise Median Yin et al. (2018; 2019), Krum Blanchard et al. (2017), and Zeno Xie et al. (2018b) with attacks. We set $\alpha = 0 . 4$ so there are 4 Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.) We formally define those prior works as follows. + +Definition C.1 (GeoMed Chen et al. (2017)). The geometric median of $\{ y _ { 1 } , . . . , y _ { m } \}$ , denoted by $g e o \_ m e d \{ y _ { 1 } , . . . , y _ { m } \}$ , is + +$$ +\begin{array} { r } { g e o \ L _ { - } m e d \{ y _ { 1 } , . . . , y _ { m } \} : = \arg \operatorname* { m i n } _ { y \in \mathbb { R } ^ { d } } \sum _ { i = 1 } ^ { m } \| y - y _ { i } \| } \end{array} +$$ + +In our experiments, we choose the geometric median from set $\{ y _ { 1 } , . . . , y _ { m } \}$ . + +Definition C.2 (coordinate-wise median Yin et al. (2018; 2019)). Coordinate-wise median $g \quad =$ $m e d \{ y _ { 1 } , . . . , y _ { m } \}$ is defined as a vector with its $k$ -th coordinate being $g [ k ] = m e d \{ y _ { 1 } [ k ] , . . . , y _ { m } [ k ] \}$ for each $k \in [ d ]$ , where med is the usual (one-dimensional) median. + +Definition C.3 (Krum Blanchard et al. (2017)). + +$$ +K R \{ y _ { 1 } , . . . , y _ { m } \} : = y _ { k } \quad w h e r e \quad k = \underset { i \in [ m ] } { \arg \operatorname* { m i n } } \sum _ { i \to j } \| y _ { i } - y _ { j } \| ^ { 2 } +$$ + +and $i j$ is the indices of the $m - b - 2$ nearest neighbours of $y _ { i }$ in $\{ y 1 , . . . , y _ { m } \} \backslash \{ y _ { i } \}$ by Euclidean distances. + +Note that Krum requires $2 b + 2 < m$ . So, we have also repeated the experiments for Krum with 3 Byzantine workers (out of 10 workers) to be more fair. + +Definition C.4 (Zeno Xie et al. (2018b)). + +$$ +Z e n o _ { b } \{ y _ { 1 } , . . . , y _ { m } \} = \frac { 1 } { m - b } \sum _ { i = 1 } ^ { m - b } \widetilde { y } ( i ) +$$ + +where $\{ \widetilde { y } ( i ) : i \in [ m ] \}$ are the gradient estimators with the $m - b$ highest “scores”, and the so-called estochastic descendant score for any gradient estimator $u$ , based on the current parameter $x$ , learning rate $\eta _ { : }$ , and a constant weight $\rho > 0$ , is defined as: + +$$ +S c o r e _ { \eta , \rho } ( u , x ) = f _ { r } ( x ) - f _ { r } ( x - \eta u ) - \rho \| u \| ^ { 2 } +$$ + +$f _ { r } ( x ) - f _ { r } ( x - \eta u )$ is the estimated descendant of the loss function and $\rho \| u \| ^ { 2 }$ is the magnitude of the update. + +In our experiments, we let $f _ { r } ( x )$ be the estimated objective over a mini-batch of size $n _ { r } = 1 0$ (so the time to perform this estimation is on the same magnitude as the gradient evaluation for each individual worker). We also chose $\rho = 0 . 0 0 0 5$ (and this value does not affect our experimental results by much). + +Safeguard SGD. Our Algorithm 1 is stated in a way to make our theoretical proofs as clean as possible. Here, we discuss how we actually implement it in practice. + +First of all, as common in the literature, we omit the Gaussian noise $\xi _ { t }$ that is added for theoretical purpose, and instead rely on the natural noise in the training process to escape saddle points. + +Also, we make universal choices for our safeguard window sizes (across all attackers): for our algorithm with a single safeguard we have used a universal window size $T = 3$ epochs, and for our algorithm with double safeguards we have used window sizes $T _ { 0 } = 1$ epoch and $T _ { 1 } = 6$ epochs. + +We also provide an automatic empirical process to select safeguard thresholds and eliminate bad workers.8 The process to determine $A _ { \mathrm { m e d } }$ (and likewise for $B _ { \mathrm { m e d . } }$ ) is described as follows. In each iteration, for every worker $\hat { i } \in [ m ]$ , we sort $\left\{ \Vert A _ { i } - A _ { j } \Vert \right\} _ { j \in [ m ] }$ and pick the smallest $\lceil m / 2 + 1 \rceil$ -th entry, and let this number be the “score” for worker $_ { i }$ . We select the worker with the smallest “score” as $A _ { \mathrm { m e d } }$ and call its “score” $S$ . Then, we use $1 . 5 \mathrm { m i n } \{ S , 5 \}$ as the safeguard threshold for this iteration. Namely, we declare any worker $j$ satisfying + +$\| A _ { j } - A _ { \sf m e d } \| \geq 1 . 5 \operatorname* { m a x } \{ S , 5 \}$ as a bad worker.9 + +# C.2 EXPERIMENT RESULTS BY ATTACKS + +The ideal accuracies are $9 1 . 7 \% / 6 8 . 0 \%$ for CIFAR-10/100, which correspond to applying SGD using only the stochastic gradients from the honest workers. Below we discuss about the experimental results one attack at a time. + +# C.2.1 VARIANCE ATTACK + +We begin by looking at the hardest proposed attack from prior works. The Variance attack follows the strategy prescribed by Baruch et al. (2019), by which Byzantine workers collude in order to shift the mean among all gradients by a factor $\beta$ times the standard deviation of the gradient, while staying within population variance. More precisely, the maximal change to the mean that can be applied by an attacker without the fear of being detected, by using the properties of the Normal distribution, specifically the cumulative standard normal function, and compute the maximal possible shift so that the attackers’s values stay within population variance. (See (Baruch et al., 2019, Algorithm 3) for a precise description. Our $\beta$ is $z _ { \mathrm { m a x } }$ in their notation.) We implement this strategy coordinate-wise, the same way as they did. Their work observes that the shift $\beta$ can be non-trivial in practice, since stochastic gradients tend to have large variance in neural network training (which we also observed in our setup). Critically, the attack cannot be defended against by historyless algorithms, as the attacker’s values are statistically indistinguishable from a regular execution in a single iteration. + +In our setting, for 10 total nodes and $\alpha = 0 . 4$ , $\beta$ is upper bounded by 0.3 (following the classic tables for the cumulative normal). We also ran the same attack in the setup from their paper (50 nodes total, of which 24 are Byzantine, which allows $\beta \sim 1 . 5 )$ and observed a similar outcome. Results for this experiment are given in Figure 3. + +![](images/64aff80ab494c66dd3b961892259a7e29da2e3c1faa1ace4ae36ced5717be94e.jpg) +Figure 3: Performance comparison under the variance attack. + +As shown by the results, our algorithm provably circumvents this attack, and recovers full accuracy. This is explained by the fact that the algorithm has memory: in particular, Byzantine nodes following this strategy will progressively diverge from the (honest) “median” $A _ { \mathrm { m e d } }$ (at a “linear” rate, recall Figure 2(a)), and therefore will eventually exceed the threshold and be marked as malicious by the algorithm. + +Specifically, both variants of the algorithm successfully catch all the bad nodes after at most 150 iterations. Indeed, at the 100-th iteration, the pair-wise distances $\| A _ { i } - A _ { j } \|$ among good workers $i , j \in$ good are between 5.3 and 6.3, but the pair-wise distance between a good and a bad worker is at least 12.5. + +# C.2.2 SIGN-FLIPPING ATTACK + +We next move onto the sign-flipping attack. Recall that, in a sign-flipping attack, each Byzantine worker sends the negative gradient to the master. This is still a strong attack since if one does not avoid any bad workers, the test accuracy will suffer from a significant drop. The results are in Figure 4. + +From the plots, one can see that again our single and double safe-guard algorithms both outperform prior works. They also successfully catch all the bad workers within 150 iterations. (For instance, at iteration 150 for CIFAR-10 training, the distance $\| A _ { \sf m e d } - A _ { j } \|$ for a good worker $j \in \mathtt { g o o d }$ is at most 6.9, but for a bad + +![](images/0830d7ad719c3952f22957fe4c980b78a111cbf63eee160f2e9a694efee17c31.jpg) +Figure 4: Performance comparison under the sign-flipping attack. + +worker $j \not \in$ good it can be more than 11.) + +In contrast, prior work Zeno completely fails because locally at a training step, using merely $n _ { r } = 1 0$ samples to evaluate the objective, it is statistically not possible to even distinguish if the sign of the stochastic gradient is flipped or not. For prior works Krum and GeoMedian, although they appear to have some non-negligible performances, but they are actually no better than simply applying SGD with the naive mean of gradients from all the workers (including those from bad workers).10 Therefore, we conclude that prior works all fail to be Byzantine fault tolerant under this attack. + +# C.2.3 DELAYED-GRADIENT ATTACK + +Recall that, in a delayed-gradient attack, each Byzantine worker sends an old gradient to the master. In our experiments, the delay is of $D = 1 0 0 0$ iterations ${ \it \Delta \phi } = 2 { \it \Delta \Psi }$ epochs). We believe this is not a very strong attack, because delayed gradients are not sufficiently malicious: they are still “correct” to certain extent albeit being delayed. The results are shown in Figure 5. + +![](images/f098289e873c796a5efe146a5bd454d7949f3336d83227ba94172db45a7fec07.jpg) +Figure 5: Performance comparison under the delayed-gradient attack. + +From the plots, one can see that our single and double safe-guard algorithms again both match the ideal accuracies. All the prior works suffer from a significant performance loss under this attack. + +It is worth noting that our single and double safe-guard algorithms do not catch any bad worker under this attack, so they simply use the “naive mean” of gradients from all the workers (including those delayed gradients from bad workers). However, there is no performance loss even if we use those delayed gradients. That is why we believe the delayed-gradient attack is not very strong, as the gradients are not sufficiently malicious. + +Prior work Zeno suffers from some performance loss, because it only uses 6 workers out of 10, in which statistically only $6 \times 0 . 6 \approx 3 \sim 4$ gradients are correct.11 Other prior works suffer from performance loss, because they only pick one single stochastic gradient from the 10 workers, and it is sometimes even from the bad worker. + +# C.2.4 LABEL-FLIPPING ATTACK + +Recall that, in the label-flipping attack, each Byzantine worker computes its gradient based on the cross-entropy loss with flipped labels: for CIFAR-10, label $\ell \in \{ 0 , . . . , 9 \}$ is flipped to $9 - \ell$ , and for CIFAR-100, label $\ell$ is flipped to $9 9 - \ell$ . The results are shown in Figure 6. + +![](images/7db635d9c4932b3527dd2bd95505447e076401213199725b8306f27dfeae79d7.jpg) +Figure 6: Performance comparison under the label-flipping attack. + +From the plots, one can see that our single and double safe-guard algorithms even outperform the “ideal accuracies.” $9 2 . 4 \%$ accuracy vs “ideal accuracy” $9 1 . 7 \%$ under CIFAR-10; $6 9 . 4 \%$ accuracy vs “ideal accuracy” 68.0 under CIFAR-100.) In addition, we have found out that the safeguard algorithms did not catch any bad worker. This should not be surprising, since label-flipping (a.k.a. label smoothing) is known to be a regularization technique to actually improve test accuracy, as opposed to hurt performance. + +Zeno also performs well under this attack (but it does not outperform the ideal accuracy). We have investigated into Zeno, and found out that it cannot distinguish good workers from bad workers under label-flipping attack; and therefore Zeno effectively always runs under 6 random workers as opposed to using the full power of the $m = 1 0$ workers (recall Zeno picks 6 workers with the topmost scores, see Definition C.4). This explains its (minor) under-performance comparing to safeguard. + +Other prior works perform significantly worse, and this should be alarming since label-flipping is one type of smoothing technique to improve test accuracy, as opposed to an actual “attack” to hurt performance. + +# C.2.5 SAFEGUARD ATTACKS + +Finally, in the safeguard attack that we design, Byzantine workers send negative but re-scaled gradient to the master. We choose the re-scale factor so that it hardly triggers the safe-guard conditions at the master. From our experiment, choosing the re-scale factor as 0.6 in all the cases do not trigger the safe-guard conditions, while choosing a re-scale factor as 0.7 enables the algorithm to catch Byzantine workers occasionally. Our results are shown in Figure 7 (for re-scale factor 0.6) and Figure 8 (for re-scale factor 0.7). + +![](images/a4d787756f71e5c67aa9104e5787baf0489820e638d83faedff07126f19b5336.jpg) +Figure 7: Performance comparison under the safeguard attack with re-scale factor 0.6. (Recall this attack is designed to maximally impact the performance of our algorithm.) + +Re-scale factor 0.6. In Figure 7, the performance of our (single and double) safeguard algorithms indeed get hurt a bit. Recall in Figure 7 the re-scale factor 0.6 is chosen to maximally impact our algorithm. The test accuracy drops from $9 1 . 7 \%$ to $8 9 . 3 \%$ under CIFAR-10; and drops from $6 8 . 0 \%$ to $6 \mathrm { { 0 . 0 \% } }$ under CIFAR-100 (for both single and double safeguards). In these cases, we confirm that both versions of the safeguard algorithms did not catch any bad worker. However, this still significantly outperforms all prior works. + +![](images/49b2f75c69622e5f6fd3ef4971d15626ae1482d4b1c61087e1ae4910caf31307.jpg) +Figure 8: Performance comparison under the safeguard attack with re-scale factor 0.7. (In this case, our our algorithm can catch some bad workers, and thus perform nearly optimally.) + +Re-scale factor 0.7. In Figure 8, we present the scenario when the re-scale factor is 0.7, so that the safeguard algorithms can occasionally catch some bad workers (depending on the randomness and learning rate). We confirm that in the three runs of single safeguard, it catches 1, 2, 3 bad workers for CIFAR-10, and $1 , 0 , 0$ bad workers for CIFAR-100 respectively; in the three runs of double safeguard, it catches $1 , 2 , 4$ bad workers for CIFAR-10, and 2, 2, 2 bad workers for CIFAR-100 respectively. + +Since there is a significant performance gain when our safeguard algorithms catch bad workers, this explains why safeguard algorithms in Figure 8 outperform their counterparts in Figure 7 with rescale factor 0.6. At the same time, we notice that the double safeguard algorithm has the ability to catch bad workers more easily. This is why double safeguard significantly outperforms single sageguard in Figure 8. + +In contrast, all other prior algorithms perform extremely bad under this attack. To some extent, safeguard attack is even stronger than the previously proposed variance attack, since it can drag the 100-class test accuracy on CIFAR-100 for all prior defense algorithms to nearly $1 \%$ , while variance attack can only drag them down to around $10 \%$ . + +# C.3 FULL COMPARISON TABLE + +We also include the full test accuracy comparison table in Table 1. + +
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack92.0291.7521.4322.0121.4722.433.42
sign-flipping attack91.9392.082259.6557.0370.8722.4
label-flipping attack92.3392.4431.9383.0783.1883.5291.66
delayed-gradient attack91.5891.4229.4374.3661.8179.2977.27
safeguard(x0.6) attack89.0189.2621.4423.1212.4828.6674.46
safeguard(x0.7) attack91.2492.0821.6119.9515.1724.5233.36
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack68.2767.956.65.815.056.110.87
sign-flipping attack68.0268.082.1310.1910.9328.342.59
label-flipping attack69.4368.85.3451.8552.1351.6667.86
delayed-gradient attack67.0366.424.0436.3431.4344.8536.6
safeguard(x0.6) attack59.87602.011.91.251.725.02
safeguard(x0.7) attack59.8464.912.071.971.321.553.31
+ +Table 1: Table of test accuracy performances for CIFAR-10 (above) and CIFAR-100 (below). + +# C.4 ATTACK AGAINST THE CONVEX ALGORITHM OF ALISTARH ET AL. (2018) + +We now briefly describe an attack against this algorithm. The attack specifically leverages the fact that the algorithm does not use sliding windows. + +One can first run the vanilla SGD to compute “the maximum deviation per good worker” for the accumulation vector used by the algorithm $\textstyle \sum _ { t = 0 } ^ { T } \nabla _ { t }$ . This maximum deviation is therefore a lower bound for the threshold used in their algorithm. Next, we design an attacker who evenly distributes this total allowed deviation to e.g. 5 consecutive epochs, and behaves honestly for the remaining epochs. Such an attacker cannot be identified by this algorithm, because its total deviation across all the iterations is identical to that of a good worker. However, this leads the algorithm to divergence. + +Specifically, suppose 4 Byzantine workers all maliciously report their stochastic gradients multiplied by the scalar $^ { - 5 }$ , and the remaining 6 good workers report their true stochastic gradients. One can verify numerically that this attacker can run for 5 consecutive epochs (say, epochs $a , a + 1 , a + 2 , a + 3 , a + 4 )$ without being caught by the algorithm. Now, + +• if $a \leq 7 5$ , within just 1 epoch of attack, the neural net weights diverge (value NAN). +• if $8 0 \leq a \leq 1 1 5$ , this attack is applied after the first learning rate decay. Within just 1 epoch of the attack, the objective explodes and accuracy becomes $1 0 \%$ (random), and within 3 epochs the algorithm diverges completely. +• if $1 2 0 \leq a \leq 1 5 5$ , this attack is after the second learning rate decay. Within just 2 epochs of attack, the accuracy drops to $1 1 \%$ . Later, the accuracy never recovers above $4 0 \%$ . \ No newline at end of file diff --git a/parse/train/PbEHqvFtcS/PbEHqvFtcS_content_list.json b/parse/train/PbEHqvFtcS/PbEHqvFtcS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..0798cf8418abcc1d1a31051dfd268faa57b462e7 --- /dev/null +++ b/parse/train/PbEHqvFtcS/PbEHqvFtcS_content_list.json @@ -0,0 +1,3558 @@ +[ + { + "type": "text", + "text": "BYZANTINE-RESILIENT NON-CONVEX STOCHASTIC GRADIENT DESCENT∗ ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zeyuan Allen-Zhu†, Faeze Ebrahimian‡, Jerry Li§, Dan Alistarh¶ ", + "bbox": [ + 184, + 165, + 656, + 183 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 218, + 544, + 233 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We study adversary-resilient stochastic distributed optimization, in which $m$ machines can independently compute stochastic gradients, and cooperate to jointly optimize over their local objective functions. However, an $\\alpha$ -fraction of the machines are Byzantine, in that they may behave in arbitrary, adversarial ways. We consider a variant of this procedure in the challenging non-convex case. Our main result is a new algorithm SafeguardSGD which can provably escape saddle points and find approximate local minima of the non-convex objective. The algorithm is based on a new concentration filtering technique, and its sample and time complexity bounds match the best known theoretical bounds in the stochastic, distributed setting when no Byzantine machines are present. ", + "bbox": [ + 232, + 247, + 764, + 386 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our algorithm is very practical: it improves upon the performance of all prior methods when training deep neural networks, it is relatively lightweight, and it is the first method to withstand two recently-proposed Byzantine attacks. ", + "bbox": [ + 232, + 387, + 766, + 429 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 445, + 336, + 460 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Motivated by the pervasiveness of large-scale distributed machine learning, there has recently been significant interest in providing distributed optimization algorithms with strong fault-tolerance guarantees. In this context, the strongest, most stringent fault model is that of Byzantine faults (Lamport et al., 1982): given $m$ machines, each having access to private data, at most an $\\alpha$ fraction of the machines can behave in arbitrary, possibly adversarial ways, with the goal of breaking or slowing down the algorithm. Although extremely harsh, this fault model is the “gold standard” in distributed computing (Lynch, 1996; Lamport et al., 1982; Castro et al., 1999), as algorithms proven to be correct in this setting are guaranteed to converge under arbitrary system behaviour. ", + "bbox": [ + 173, + 472, + 825, + 583 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A setting of particular interest in this context has been that of distributed stochastic optimization. Here, the task is to minimize some stochastic function $f ( x ) = \\mathbb { E } _ { s \\sim \\mathcal { D } } [ f _ { s } ( x ) ]$ over a distribution $\\mathcal { D }$ , where $f _ { s } ( \\cdot )$ can be viewed as the loss function for sample $s \\sim \\mathcal { D }$ . We assume there are $m$ machines (workers) and an honest master, and $\\alpha < 1 / 2$ fraction of the workers may be Byzantine. In each iteration $t$ , each worker has access to a version of the global iterate $x _ { t }$ , which is maintained by the master. The worker can independently sample $s \\sim \\mathcal { D }$ , compute $\\nabla f _ { s } ( x _ { t } )$ , and then synchronously send this stochastic gradient to the master. The master aggregates the workers’ messages, and sends an updated iterate $x _ { t + 1 }$ to all the workers. Eventually, the master has to output an approximate minimizer of $f$ . Clearly, the above description only applies to honest workers; Byzantine workers may deviate arbitrarily and return adversarial “gradient” vectors to the master in every iteration. ", + "bbox": [ + 174, + 587, + 825, + 726 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This distributed framework is quite general and well studied. One of the first references in this setting studied distributed PCA and regression (Feng et al., 2014). Other early approaches (Blanchard et al., 2017; Chen et al., 2017; Su & Vaidya, 2016a;b; Xie et al., 2018a) relied on defining generalizations of the geometric median. These approaches can withstand up to half of the nodes being malicious, but can have relatively high local computational cost $\\Omega ( m ^ { 2 } { \\dot { d } } )$ (Blanchard et al., 2017; Chen et al., 2017), where $m$ is the number of nodes and $d$ is the problem dimension, and usually have suboptimal sample and iteration complexities. ", + "bbox": [ + 174, + 729, + 825, + 827 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Follow-up work resolved this last issue when the objective $f ( \\cdot )$ is convex, leading to tight sample complexity bounds. Specifically, Yin et al. (2018) provided bounds for gradient descent-type algorithms, and showed that the bounds are tight when the dimension is constant. Alistarh et al. (2018) provided a stochastic gradient descent (SGD) type algorithm and showed that its sample and time complexities are asymptotically optimal even when the dimension is large. ", + "bbox": [ + 178, + 832, + 823, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Non-convex Byzantine-resilient stochastic optimization. In this paper, we focus on the more challenging non-convex setting, and shoot for the strong goal of finding approximate local minima (a.k.a. second-order critical points). In a nutshell, our main result is the following. Fix $d$ to denote the dimension, and let the objective $f : \\mathbb { R } ^ { d } \\mathbb { R }$ be Lipschitz smooth and second-order smooth. We have $m$ worker machines, each having access to unbiased, bounded estimators of the gradient of $f$ . Given an initial point $x _ { 0 }$ , the SafeguardSGD algorithm ensures that, even if at most $\\bar { \\alpha } < 1 / 2$ fraction of the machines are Byzantine, after ", + "bbox": [ + 173, + 164, + 825, + 261 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/e378316cad9f2fda93181dca2e62c878e00a67398c3427f7916b7a79e9edafee.jpg", + "text": "$$\n\\begin{array} { r } { T = \\widetilde { O } \\left( \\left( \\alpha ^ { 2 } + \\frac { 1 } { m } \\right) \\frac { d \\left( f \\left( x _ { 0 } \\right) - \\operatorname* { m i n } f \\left( x \\right) \\right) } { \\varepsilon ^ { 4 } } \\right) \\qquad \\mathrm { p a r a l l e l i t e r a t i o n s } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 263, + 697, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "for at least a constant fraction of the indices $t \\in [ T ]$ , the following hold: ", + "bbox": [ + 174, + 294, + 645, + 309 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/63efb86c1b4d22f974ade23c83eb2d8879c3eadd25398bcb716b723c9bd487a5.jpg", + "text": "$$\n\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon \\quad { \\mathrm { a n d } } \\quad \\nabla ^ { 2 } f ( x _ { t } ) \\succeq - { \\sqrt { \\varepsilon } } \\mathbf { I } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 313, + 637, + 330 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "If the goal is simply $\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon$ , then $\\begin{array} { r } { T = \\widetilde { O } \\big ( \\left( \\alpha ^ { 2 } + \\frac { 1 } { m } \\right) \\frac { \\left( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) \\right) } { \\varepsilon ^ { 4 } } \\big ) } \\end{array}$ iterations suffice. Here, the $\\widetilde O$ notation serves to hide logarithmic factors for readability. We spell out these factors in the detailed analysis. ", + "bbox": [ + 174, + 333, + 825, + 383 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• When $\\alpha < 1 / \\sqrt { m }$ , our sample complexity $( = m T ,$ ) matches the best known result in the nonByzantine case (Jin et al., 2019) without additional assumptions, and enjoys linear parallel speedup: with $m$ workers of which $< \\sqrt { m }$ are Byzantine, the parallel speedup is $\\widetilde \\Omega ( m )$ . 1 • For $\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )$ , our parallel time complexity is ${ \\widetilde { \\cal O } } ( \\alpha ^ { 2 } )$ times that needed when no parallelism is used. This still gives parallel speedup. This $\\alpha ^ { 2 }$ factor appears in convex Byzantine distributed optimization, where it is tight (Yin et al., 2018; Alistarh et al., 2018). • The Lipschitz and second-order smoothness assumptions are the minimal assumptions needed to derive convergence rates for finding second-order critical points (Jin et al., 2019). ", + "bbox": [ + 181, + 387, + 825, + 511 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Comparison with prior bounds. The closest known bounds are by Yin et al. (2019), who derived three gradient descent-type of algorithms (based on median, mean, and iterative filtering) to find a weaker type of approximate local minima. Since it relies on full gradients, their algorithm is arguably less practical, and their time complexities are generally higher than ours (see Section 2.1). ", + "bbox": [ + 173, + 516, + 825, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Other prior works consider a weaker goal: to find approximate stationary points $\\| \\nabla f ( x ) \\| \\leq \\varepsilon$ only: Bulusu et al. (2020) additionally assumed there is a guaranteed good (i.e. non-Byzantine) worker known by the master, Xie et al. (2018b) gave a practical algorithm when the Byzantine attackers have no information about the loss function or its gradient, Yang et al. (2019); Xie et al. (2018a); Blanchard et al. (2017) derived eventual convergence without an explicit complexity bound, and the non-convex result obtained in Yin et al. (2018) is subsumed by Yin et al. (2019), discussed above. ", + "bbox": [ + 174, + 575, + 825, + 660 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our algorithm and techniques. The structure of our algorithm is deceptively simple. The master node keeps track of the sum of gradients produced by each worker across time. It labels (allegedly) good workers as those whose sum of gradients “concentrate” well with respect to a surrogate of the median vector, and labels bad workers otherwise. Once a worker is labelled bad, it is removed from consideration forever. The master then performs the vanilla $S G D$ , by moving in the negative direction of the average gradients produced by those workers currently labelled as good. ", + "bbox": [ + 173, + 662, + 825, + 747 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We call our algorithm SafeguardSGD, since it behaves like having a safe guard to filter away bad workers. Its processing overhead at the master is $O ( m d )$ , negligible compared to standard SGD. ", + "bbox": [ + 173, + 751, + 823, + 780 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As the astute reader may have guessed, the key non-trivial technical ingredient is to identify the right quantity to check for concentration, and make it compatible with the task of non-convex optimization. In particular, we manage to construct such quantities so that (1) good non-Byzantine workers never get mislabelled as bad ones; (2) Byzantine workers may be labelled as good ones (which is inevitable) but when they do, the convergence rates are not impacted significantly; and (3) the notion does not require additional assumptions or running time overhead. ", + "bbox": [ + 174, + 782, + 825, + 867 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The idea of using concentration (for each worker across time) to filter out Byzantine machines traces back to the convex setting (Alistarh et al., 2018). However, the quantities used in (Alistarh et al., 2018) to check for concentration are necessarily different from this paper, and our analysis is completely new, as deriving non-convex rates is known to be much more delicate and challenging. Recently, Bulusu et al. (2020) used similar concentration filters to Alistarh et al. (2018) in the nonconvex setting, but under stronger assumptions, and for the simpler task of finding stationary points. ", + "bbox": [ + 178, + 871, + 821, + 886 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Many other algorithms do not rely on concentration filters. In each iteration, they ask each worker to compute a batch of stochastic gradients, and then use coordinate-wise median or mean over the batch average (e.g. Yin et al. (2018; 2019); Yang et al. (2019)) or iterative filtering (e.g. Su & Xu (2018); Yin et al. (2019)) by the master to derive a “robust mean.” These works fundamentally rely on each iteration to calculate an almost precise full gradient, so that they can apply a surrogate of full gradient descent. Such algorithms can introduce higher sample and time complexities (see Section 2), are less practical than stochastic gradient schemes, require additional restrictions on the resilience factor $\\alpha$ , e.g. $\\alpha < 1 / 4$ (Su & Xu, 2018), and, critically, have been shown to be vulnerable to recent attacks (Baruch et al., 2019; Xie et al., 2020). ", + "bbox": [ + 174, + 176, + 825, + 303 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Attack resilience and experimental validation. There is a growing literature on customized attacks against Byzantine-resilient algorithms, showing that many defenses can be entirely circumvented in real-world scenarios (Baruch et al., 2019; Xie et al., 2020). Our algorithm is provably correct against these attacks, a fact we also validate experimentally. We implemented SafeguardSGD to examine its practical performance against a range of prior works (Xie et al., 2018b; Blanchard et al., 2017; Chen et al., 2017; Yin et al., 2018; 2019), and against recent attacks on the distributed task of training deep neural networks. Our experiments show that SafeguardSGD generally outperforms previous methods in convergence speed and final accuracy, sometimes by a wide accuracy margin. This is true not only against known Byzantine attacks, but also against attack variants we fine-crafted to specifically slow down our algorithm, and against transient node failures. ", + "bbox": [ + 174, + 306, + 825, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 STATEMENT OF OUR THEORETICAL RESULT ", + "text_level": 1, + "bbox": [ + 174, + 462, + 573, + 478 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We denote by $\\| \\cdot \\|$ the Euclidean norm and $[ n ] : = \\{ 1 , 2 , \\dots , n \\}$ . Given symmetric matrices $\\mathbf { A } , \\mathbf { B }$ , we let $\\| \\mathbf { A } \\| _ { 2 }$ denote the spectral norm of A. We use $\\succeq$ to denote Loewner ordering, i.e. $\\mathbf { A } \\succeq \\mathbf { B }$ if $\\mathbf { A } - \\mathbf { B }$ is positive semi-definite. We denote by $\\lambda _ { \\mathrm { m i n } } ( \\mathbf { A } )$ the minimum eigenvalue of matrix $\\mathbf { A }$ . ", + "bbox": [ + 174, + 488, + 825, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider arbitrary $d$ -dimensional non-convex functions $f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ satisfying the following: ", + "bbox": [ + 173, + 534, + 805, + 549 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• $f ( x )$ is $L$ -Lipschitz smooth: meaning $\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq L \\| x - y \\|$ for any $x , y \\in \\mathbb { R } ^ { d }$ ; \n• $f ( x )$ is $L _ { 2 }$ -second-order smooth: $\\| \\nabla ^ { 2 } f ( x ) - \\nabla ^ { 2 } f ( y ) \\| _ { 2 } \\leq L _ { 2 } \\cdot \\| x - y \\|$ for any $x , y \\in \\mathbb { R } ^ { d }$ ; ", + "bbox": [ + 181, + 551, + 799, + 585 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For notational simplicity of the proofs, we assume $L = L _ { 2 } = \\mathcal { V } = 1$ .2 Note that we have also assumed the domain of $f$ is the entire space $\\mathbb { R } ^ { d }$ . If instead there is a compact domain $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ , then one can use projected SGD and re-derive similar results of this paper. We choose to present our result in the simplest setting to convey our main ideas. ", + "bbox": [ + 176, + 588, + 825, + 643 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Byzantine non-convex stochastic distributed optimization. We let $m$ be the number of worker machines and assume at most an $\\alpha$ fraction of them are Byzantine for $\\alpha \\in [ 0 , \\frac { 1 } { 2 } )$ . We denote by good $\\subseteq [ m ]$ the set of good (i.e. non-Byzantine) machines, and the algorithm does not know good. ", + "bbox": [ + 174, + 647, + 825, + 691 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Assumption 2.1. In each iteration $t _ { : }$ , the algorithm (on the master) is allowed to specify a point $x _ { t }$ and query m machines. Each machine $i \\in [ m ]$ gives back a vector $\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }$ satisfying ", + "bbox": [ + 174, + 696, + 825, + 726 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• If $i \\in \\mathtt { g o o d }$ , the stochastic gradient $\\nabla _ { t , i }$ satisfies $\\mathbb { E } [ \\nabla _ { t , i } ] = \\nabla f ( x _ { t } )$ and $\\lVert \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\rVert \\leq \\mathcal { V }$ • If $i \\in [ m ]$ \\ good, then $\\nabla _ { t , i }$ can be arbitrary (w.l.o.g. we assume $\\lVert \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\rVert \\leq \\mathcal { V } )$ .4 ", + "bbox": [ + 179, + 729, + 818, + 766 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Remark 2.2. For each $t$ and $i \\not \\in { \\mathfrak { g o o d } }$ , the vector $\\nabla _ { t , i }$ can be adversarially chosen and may depend ", + "bbox": [ + 176, + 768, + 823, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Input: point $x _ { 0 } \\in \\mathbb { R } ^ { d }$ , rate $\\eta > 0$ , lengths $T \\geq T _ { 1 } \\geq T _ { 0 } \\geq 1$ , threshold $\\mathfrak { T } _ { 1 } > \\mathfrak { T } _ { 0 } > 0$ ; \n1: $\\mathbf { g } _ { 0 0 } \\mathsf { d } _ { 0 } \\gets [ m ]$ ; \n2: for $t \\gets 0$ to $T - 1$ do \n3: $l a s t _ { 1 } \\gets \\operatorname* { m a x } \\{ t _ { 1 } \\in [ t ] : t _ { 1 }$ is a multiple of $T _ { 1 } \\}$ ; \n4: $l a s t _ { 0 } \\gets \\operatorname* { m a x } \\{ t _ { 0 } \\in [ t ] : t _ { 0 }$ is a multiple of $T _ { 0 } \\}$ \n5: for each $i \\in \\mathtt { g o o d } _ { t }$ do \n6: 7: $\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }$ macand $i$ \nAi ← Ptk=last1 ∇k,i|good | $\\begin{array} { r } { B _ { i } \\gets \\sum _ { k = l a s t _ { 0 } } ^ { t } \\frac { \\nabla _ { k , i } } { | \\mathbf { g } \\circ \\mathbf { o } \\mathbf { d } _ { k } | } } \\end{array}$ \n8: $A _ { \\sf m e d } A _ { i }$ where $i \\in { \\mathfrak { g o o d } } _ { t }$ is any machine s.t. $\\left| \\left\\{ j \\in \\operatorname { g o o d } _ { t } \\colon \\left\\| A _ { j } - A _ { i } \\right\\| \\leq \\mathfrak { T } _ { 1 } \\right\\} \\right| > m / 2$ . \n9: $B _ { \\sf m e d } B _ { i }$ where $i \\in \\mathtt { g o o d } _ { t }$ is any machine s.t. $\\left| \\left\\{ j \\in \\mathbf { g 0 0 d } _ { t } \\colon \\left\\| B _ { j } - B _ { i } \\right\\| \\le \\mathfrak { T } _ { 0 } \\right\\} \\right| > m / 2$ . \n10: $\\begin{array} { r l r } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } : \\| A _ { i } - A _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 1 } \\bigwedge \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 0 } \\big \\} } & \\\\ & { x _ { t + 1 } = x _ { t } - \\eta \\Big ( \\xi _ { t } + \\frac { 1 } { \\vert \\mathsf { g o o d } _ { t } \\vert } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\Big ) ; } & { \\diamondsuit a u s s i a n } \\end{array}$ \n11: noise $\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )$ ", + "bbox": [ + 178, + 125, + 826, + 323 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "on $\\{ \\nabla _ { t ^ { \\prime } , i } \\} _ { t ^ { \\prime } \\leq t , i \\in [ m ] }$ . In particular, the Byzantine machines can even collude during an iteration. ", + "bbox": [ + 171, + 358, + 799, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.1 OUR ALGORITHM AND THEOREM ", + "text_level": 1, + "bbox": [ + 176, + 386, + 447, + 401 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our algorithm is based on arguably the simplest possible method for achieving this goal, (perturbed) stochastic gradient descent $( S G D )$ (Ge et al., 2015). Our techniques more broadly apply to more complicated methods (e.g. at least to Allen-Zhu (2018a;b)), but we choose to analyze the simplest variant of SGD, since it is the most widely applied method in modern non-convex machine learning. ", + "bbox": [ + 173, + 406, + 825, + 463 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As illustrated in Algorithm 1, in each iteration $t = 0 , 1 , \\dots , T - 1$ , we maintain a set of (allegedly) good machines $\\mathsf { g o o d } _ { t } \\subseteq [ m ]$ . We begin with $\\mathtt { g o o d } _ { 0 } = [ m ]$ and start to detect malicious machines and remove them from the set. We choose a learning rate $\\eta > 0$ , and perform the SGD update ", + "bbox": [ + 174, + 467, + 825, + 510 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fef879fcba28b5759684029a259322eb9f2ace16065ea7970bf4277bf0697f8f.jpg", + "text": "$$\n\\begin{array} { r } { x _ { t + 1 } = x _ { t } + \\xi _ { t } - \\eta \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } } \\nabla _ { t , i } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 511, + 633, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )$ is a random Gaussian perturbation that is added for theoretical purpose. ", + "bbox": [ + 169, + 536, + 789, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For each machine $i \\in [ m ]$ , we keep track of the history of its stochastic gradients up to two windows. Namely, Ai ← Ptk=last1 k,i|good | and Bi ← P k=last0 k,i|good | for windows sizes $T _ { 0 } \\leq T _ { 1 } \\leq T$ . We compare among remaining machines in $\\mathtt { g o o d } _ { t }$ , and kick out those ones whose $A _ { i }$ or $B _ { i }$ deviate “more than usual” to construct $\\mathtt { g o o d } _ { t + 1 }$ . Conceptually, we view these two as safe guards. ", + "bbox": [ + 174, + 554, + 825, + 617 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our theory makes sure that, when the “window sizes” and the thresholds for “more than usual” are defined properly, then $\\mathtt { g o o d } _ { t }$ shall always include good, and the algorithm shall proceed to find approximate local minima. Formally, we have (letting the $\\widetilde O$ notion to hide polylogarithmic factors) ", + "bbox": [ + 173, + 621, + 826, + 666 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2.3. Let $\\begin{array} { r } { C _ { 3 } = \\alpha ^ { 2 } + \\frac { 1 } { m } } \\end{array}$ . Suppose we choose $\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )$ , $\\begin{array} { r } { \\eta = \\widetilde { \\Theta } \\big ( \\frac { \\varepsilon ^ { 2 } } { d C _ { 3 } } \\big ) } \\end{array}$ , $T _ { 0 } = \\widetilde { \\Theta } ( \\textstyle { \\frac { 1 } { \\eta } } )$ $\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } ( \\frac { 1 } { \\eta \\sqrt { \\varepsilon } } ) } \\end{array}$ , ${ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )$ , and $\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )$ , then after ", + "bbox": [ + 173, + 671, + 820, + 709 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2748fee727c58ddaf44a1063d330a1f4039a7906174f66156826a917636e2952.jpg", + "text": "$$\n\\begin{array} { r } { T = \\widetilde { O } \\left( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } ( \\alpha ^ { 2 } + \\frac { 1 } { m } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 715, + 622, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "iterations, with high probability, for at least constant fraction of the indices $t \\in [ T ]$ , they satisfy ", + "bbox": [ + 171, + 744, + 799, + 760 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b4d6cf83efa994c492e447ee88cfe8f23a120d1c42deec52db29bb31712342b4.jpg", + "text": "$$\n\\| \\nabla f ( x _ { t } ) \\| \\le \\varepsilon \\quad a n d \\quad \\nabla ^ { 2 } f ( x _ { t } ) \\succeq - \\sqrt { \\varepsilon } { \\bf I } .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 762, + 643, + 781 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Remark 2.4. If one only wishes to achieve a significantly simpler goal — finding first-order critical points $\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon$ — the analysis becomes much easier (see Section 3.1). In particular, having one safe guard without perturbation (i.e. $\\nu = 0$ ) suffices, and the iteration complexity reduces to T = Oe\u0000 f(x0)−min f(x)ε4 (α2 + 1m )\u0001. Bulusu et al. (2020) achieves this easier goal but requires an additional assumption: there is one guaranteed good worker known by the master. ", + "bbox": [ + 173, + 784, + 825, + 858 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our contribution. We reiterate our theoretical contributions from three perspectives. 1) When $\\alpha < 1 / \\sqrt { m }$ , our algorithm requires $\\begin{array} { r } { m T = \\widetilde { O } \\big ( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } \\big ) } \\end{array}$ stochastic gradient computations. This matches the best known result (Jin et al., 2019) under our minimal assumptions of the nonconvex objective. (There exist other works in the stochastic setting that break the $\\varepsilon ^ { - 4 }$ barrier and get rid of the dimension dependence $d$ under stronger assumptions.)5. 2) When $\\alpha < 1 / \\sqrt { m }$ , our algorithm enjoys linear parallel speed-up: the parallel time complexity reduces by a factor of $\\Theta ( m )$ . When $\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )$ , our parallel time complexity is $\\widetilde { O } ( \\alpha ^ { 2 } )$ times that needed when no parallelism is used, still giving noticeable speedup. The $\\textstyle { \\hat { \\alpha } } ^ { 2 }$ factor also appeared in convex Byzantine distributed optimization (and is known to be tight there) (Yin et al., 2018; Alistarh et al., 2018). ", + "bbox": [ + 173, + 864, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 116, + 825, + 189 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Comparison to (Yin et al., 2019). Yin et al. (2019) derived three gradient descent-type algorithms to find points with a weaker (and less standard) guarantee: $\\| \\nabla f ( x ) \\| ~ \\leq ~ \\varepsilon$ and $\\bar { \\nabla ^ { 2 } } f ( x ) ~ \\succeq ~ - ( \\varepsilon ^ { 2 } d ) ^ { \\bar { 1 } / 5 } \\mathbf { I }$ . Despite practical differences (namely, gradient descent may be less favorable comparing to stochastic gradient descent especially in deep learning applications), the parallel time complexities derived from their result are also generally larger than ours. ", + "bbox": [ + 174, + 193, + 825, + 263 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Their paper focuses on bounding the number of sampled stochastic functions, as opposed to the number of stochastic gradient evaluations like we do. When translated to our language, each of the workers in their setting needs to evaluate T stochastic gradients, where (1) T = Oe\u0000 α2dε4 $\\begin{array} { r } { 1 ) T = \\widetilde { O } \\big ( \\frac { \\alpha ^ { 2 } \\overline { { d } } } { \\varepsilon ^ { 4 } } + \\frac { d ^ { 2 } } { \\varepsilon ^ { 4 } m } + \\frac { \\sqrt { d } } { \\varepsilon ^ { 3 } } \\big ) } \\end{array}$ if using coordinate-wise median, (2) T = Oe\u0000 α2d2ε4 $\\begin{array} { r } { T = \\widetilde { O } \\big ( \\frac { \\alpha } { \\varepsilon ^ { 4 } } + \\frac { d } { \\varepsilon ^ { 4 } m } \\big ) } \\end{array}$ if using iterative filt if using trimmed mean, and (3)(1) and (2) are larger than ours (also with a weaker guarantee); the complexity (3) seems incomparable to ours, but when translating to the more standard (ε, √ε) guarantee, becomes T = Oe\u0000 αd2ε5 $\\begin{array} { r } { T = \\widetilde { \\cal O } \\big ( \\frac { \\alpha d ^ { 2 } } { \\varepsilon ^ { 5 } } + \\frac { d ^ { 3 } } { \\varepsilon ^ { 5 } m } \\big ) } \\end{array}$ so is also larger than ours. It is worth noting that (3) requires $\\alpha < 1 / 4$ so cannot withstand half of the machines being Byzantine. ", + "bbox": [ + 173, + 267, + 825, + 396 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Resilience against practical attacks. Our algorithm’s filtering is based upon tracking $B _ { i }$ (resp. $A _ { i , { \\bf \\Pi } }$ ), the stochastic gradients of each machine $i$ averaged over a window of $T _ { 0 }$ (resp. $T _ { 1 . }$ ) iterations. This is a departure from previous defenses, most of which are history-less, and enables us to be provably Byzantine-resilient against state-of-the-art attacks (Baruch et al., 2019; Xie et al., 2020). ", + "bbox": [ + 174, + 400, + 825, + 455 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Baruch et al. (2019), Byzantine workers collude to shift the gradient mean by a factor $\\beta$ times the standard deviation of the (true stochastic) gradient, while staying within population variance. They noticed $\\beta$ can be quite large especially in neural network training. Their attack circumvent existing defenses because those defense algorithms are “historyless”, while their attack is statistically indistinguishable from an honest execution in any single iteration. However, our algorithm can provably defend against this attack since it has memory: Byzantine workers following their strategy will progressively diverge from the (honest) “median” √ $B _ { \\mathrm { m e d } }$ (by an amount proportional to $\\Omega ( T )$ in $T$ iterations as opposed to $\\sqrt { T }$ ), and be marked as malicious by our algorithm. (See Figure 2(a).) In Xie et al. (2020), Byzantine workers deviate in the negative direction of the gradient. However, to avoid being caught by our algorithm, the maximum “magnitude” of this attack has to stay within our thresholds. We implemented both attacks and showed our algorithm’s robustness experimentally. ", + "bbox": [ + 173, + 458, + 825, + 614 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, we note that prior “historyless” schemes, such as Krum or median-based schemes, could be thought of as providing stronger guarantees, as they in theory allow Byzantine nodes to change IDs during the computation: such schemes only require an upper bound on the number of Byzantine agents in each round. However, the attack of Baruch et al. (2019) essentially shows that all such schemes are vulnerable to variance attacks, and that such attacks are eminently plausible in practice. Thus, this suggests that the use of historical information, which requires that Byzantine nodes cannot change their IDs during the execution, may be necessary for Byzantine resilience. ", + "bbox": [ + 173, + 618, + 825, + 717 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Tolerating transient failures and node ID relabeling. Our algorithm can also withstand transient node failures and some degrees of ID relabeling, by resetting the set of good nodes $\\mathsf { g o o d } _ { t }$ to include all nodes every $T _ { 1 }$ steps. The algorithm then proceeds as usual. The key observation behind this relaxation is the fact that our analysis only requires that the attack conditions hold inside the current window. (Please see the Theorem B.1 for details.) We validate this experimentally in Section 5. ", + "bbox": [ + 174, + 719, + 825, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 WARMUP: SINGLE SAFE GUARD", + "text_level": 1, + "bbox": [ + 176, + 805, + 475, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As a warmup, let us first analyze the behavior of perturbed SGD with a single safe guard. Consider Algorithm 2, where we start with a point $w _ { 0 }$ , a set $\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }$ , and perform $T$ steps of perturbed SGD. (We use the $w _ { t }$ sequence instead of the $x _ { t }$ sequence to emphasize that we are in Algorithm 2.) ", + "bbox": [ + 174, + 832, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input: point $w _ { 0 } \\in \\mathbb { R } ^ { d }$ , set $\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }$ , rate $\\eta > 0$ , length $T \\geq 1$ , threshold $\\mathfrak T > 0$ ; \n1: for $t \\gets 0$ to $T - 1$ do \n2: for each $i \\in \\mathtt { g o o d } _ { t }$ do \n3: receive $\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }$ from machine $i$ ; \n4: Bi ← Ptk=0 ∇k,i|goodk| ; \n5: $B _ { \\sf m e d } B _ { i }$ where $i \\in \\mathtt { g o o d } _ { t }$ is any machine s.t. $\\left| \\left\\{ j \\in \\mathbf { g 0 0 d } _ { t } \\colon \\left\\| B _ { j } - B _ { i } \\right\\| \\le \\mathfrak { T } \\right\\} \\right| > m / 2 .$ . \n6: $\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } \\colon \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\Big ( \\xi _ { t } + \\frac { 1 } { | \\mathsf { g o o d } _ { t } | } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\Big ) ; } \\end{array}$ ; \n7: \u0005 Gaussian noise $\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )$ ", + "bbox": [ + 176, + 123, + 821, + 263 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 3.1. We make the following definition to simplify notations: let $\\Xi _ { t } : = \\sigma _ { t } + \\Delta _ { t }$ where ", + "bbox": [ + 171, + 284, + 802, + 299 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2c08d6eb1052e237c59c211849be5ecaecb5e743c1ca8ee7ebfb5b633f1102c8.jpg", + "text": "$$\n\\begin{array} { r l } & { \\sigma _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\\\ & { \\Delta _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } \\backslash \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 299, + 503, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Therefore, we can re-write the SGD update as $w _ { t + 1 } = w _ { t } - \\eta ( \\nabla f ( w _ { t } ) + \\xi _ { t } + \\Xi _ { t } )$ ", + "bbox": [ + 174, + 347, + 727, + 363 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The following lemma is fairly immediate to prove: ", + "bbox": [ + 173, + 369, + 506, + 386 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lemma 3.2 (single safe guard). In Algorithm 2, suppose we choose $\\mathfrak { T } = 8 \\sqrt { T \\log ( 1 6 m T / p ) }$ . Then, with probability at least $1 - p / 4$ , for every $t = 0 , \\ldots , T - 1$ , ", + "bbox": [ + 173, + 391, + 821, + 421 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6b0a758c05f6a45046d4b3f6f38f2b4438c3506da58e940bacc88a861b71fce4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\bullet \\Vert \\sigma _ { t } \\Vert ^ { 2 } \\leq O ( \\frac { \\log ( T / p ) } { m } ) a n d \\Vert \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } ) } \\\\ & { \\bullet \\Vert \\Delta _ { t } \\Vert ^ { 2 } \\leq \\alpha ^ { 2 } a n d \\Vert \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\alpha ^ { 2 } T \\log ( m T / p ) ) } \\\\ & { \\bullet \\vert \\langle \\nabla f ( w _ { t } ) , \\xi _ { t } \\rangle \\vert \\leq \\Vert \\nabla f ( w _ { t } ) \\Vert \\cdot O ( \\nu \\sqrt { \\log ( T / p ) } ) , } \\\\ & { \\bullet \\Vert \\xi _ { t } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T / p ) ) , \\Vert \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d T \\log ( T / p ) ) } \\\\ & { \\textit { e c a l l t h i s p r o b a b i l i s t i c e n t \\ 5 \\mathrm { v e n t } } \\mathrm { t v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) a n d \\operatorname* { P r } [ \\mathrm { E v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) ] \\geq 1 - p / 4 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 424, + 718, + 546 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(The third property above is ensured by our choice of $\\mathfrak { T }$ and the use of safe guard, and the rest of the properties follow from simple martingale concentration arguments. Details are in Appendix A.1.) ", + "bbox": [ + 173, + 550, + 823, + 579 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 CORE TECHNICAL LEMMA 1: OBJECTIVE DECREASE ", + "text_level": 1, + "bbox": [ + 171, + 589, + 586, + 606 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our first main technical lemma is the following: ", + "bbox": [ + 174, + 611, + 488, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lemma 3.3. Suppose we choose $\\mathfrak { T }$ as in Lemma 3.2. Denote by $C _ { 1 } = \\log ( T / p )$ and $C _ { 2 } \\ =$ $\\begin{array} { r } { \\alpha ^ { 2 } \\log { \\frac { m T } { p } } + { \\frac { \\log ( T / p ) } { m } } } \\end{array}$ . Suppose $\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}$ , 1100η(1+√C2) and we start from w0 and apply Algorithm 2. Under event Even $\\mathbf { \\Pi } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )$ , it satisfies ", + "bbox": [ + 173, + 631, + 826, + 684 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/cf041560eae4b456c1cd25849fc5d274a7a6d973b5b21586d1f8aa27510fc58b.jpg", + "text": "$$\n\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 688, + 807, + 709 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lemma 3.3 says after $\\begin{array} { r } { T \\approx \\frac { 1 } { \\eta } } \\end{array}$ steps of perturbed SGD, the objective value decreases by, up to some small additive error and up to logarithmic factors, $\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } ( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - } \\end{array}$ $\\eta C _ { 2 } )$ . This immediately implies, if we choose $\\begin{array} { r } { \\eta \\approx \\frac { \\varepsilon ^ { 2 } } { C _ { 2 } } } \\end{array}$ , then by repeating this analysis for $\\begin{array} { r } { O ( \\frac { C _ { 2 } } { \\varepsilon ^ { 4 } } ) = O ( \\frac { \\alpha ^ { 2 } + 1 / m } { \\varepsilon ^ { 4 } } ) } \\end{array}$ ( α2+1/mε4 ) iterations, we can find approximate critical point x with k∇f (x)k ≤ ε. ", + "bbox": [ + 173, + 715, + 826, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof sketch of Lemma 3.3. The full proof is in Appendix A.2 but we illustrate the main idea and difficulties below. After simple manipulations, it is not hard to derive that ", + "bbox": [ + 171, + 794, + 823, + 823 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/89c3e604404cfa7fafaf5f0d084d6fa58cc7871c9fe8fdd33578cf56cbb3775b.jpg", + "text": "$$\n\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\gtrapprox 0 . 9 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - \\eta \\right) + \\underbrace { \\eta \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle } _ { \\mathrm { r e m a i n d e r ~ t e r m s } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 824, + 750, + 864 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where recall that $\\Xi _ { t } = \\sigma _ { t } + \\Delta _ { t }$ . When there are no Byzantine machines, we have $\\mathbb { E } [ \\Xi _ { t } ] = \\mathbb { E } [ \\sigma _ { t } ] = 0$ so the remainder terms must be small by martingale concentration. Therefore, the main technical difficulty arises to deal with those Byzantine machines, who can adversarially design their $\\nabla _ { t }$ (even by collusion) so as to negatively correlate with $\\nabla f ( w _ { t } )$ to “maximally destroy” the above inequality. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our main idea is to use second-order smoothness to write $\\nabla f ( w _ { t } ) \\approx \\nabla f ( w _ { 0 } ) + \\nabla ^ { 2 } f ( w _ { 0 } ) \\cdot ( w _ { t } - w _ { 0 } )$ . To illustrate our idea, let us ignore the constant vector and assume that the Hessian is the identity: that is, imagine as if $\\nabla f ( w _ { t } ) \\mathbf { \\bar { \\approx } } w _ { t } - w _ { 0 }$ . Using $\\begin{array} { r } { w _ { t } - w _ { 0 } = - \\sum _ { k < t } \\Xi _ { t } + \\xi _ { t } } \\end{array}$ , we immediately have ", + "bbox": [ + 173, + 103, + 825, + 147 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3315623a3cd648ed53891363b61039c36b9bf574c6b4ad5106975dd21198c861.jpg", + "text": "$$\n\\begin{array} { r } { - \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle \\approx - \\langle w _ { t } - w _ { 0 } , \\Xi _ { t } \\rangle = \\sum _ { k < t } \\langle \\Xi _ { k } , \\Xi _ { t } \\rangle + \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 150, + 735, + 169 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the first partial sum $\\textstyle \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle$ in (3.1), it is easy to bound its magnitude using our safeguard. Indeed, we have $\\begin{array} { r } { \\big | \\sum _ { t } \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle \\big | \\leq \\| \\sum _ { t } \\Xi _ { t } \\| ^ { 2 } + \\sum _ { t } \\| \\Xi _ { t } \\| ^ { 2 } } \\end{array}$ so we can apply Lemma 3.2. For the second partial sum $\\begin{array} { r } { \\sum _ { t } \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}$ , we can apply the concentration Proposition 3.4 below. \u0003 ", + "bbox": [ + 173, + 174, + 826, + 226 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 3.4. Fix the dimension parameter $d \\ge 1$ . Suppose $\\xi _ { 0 \\underline { { : } } } \\dots , \\xi _ { T - 1 } \\ \\in \\ \\mathbb { R } ^ { d }$ are i.i.d. drawn from $\\mathcal { N } ( 0 , \\bf { I } )$ , and that $\\Delta _ { 1 } , \\ldots , \\Delta _ { T - 1 }$ are arbitrary vectors in $\\mathbb { R } ^ { d }$ . Here, each vector $\\Delta _ { t }$ with $t = 1 , \\dots , T - 1$ can depend on $\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 }$ but not on $\\xi _ { t } , \\dots , \\xi _ { T - 1 }$ . Suppose that these vectors satisfy $\\| \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\| ^ { 2 } \\leq \\mathfrak { T }$ for every $t = 1 , \\dots , T - 1$ . Then, with probability at least $1 - p$ , ", + "bbox": [ + 173, + 233, + 825, + 290 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/4490f7c290ac0e971389f87ab29e6f26d4459fb36fe16ae481c193d32c11963b.jpg", + "text": "$$\n\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 292, + 684, + 320 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 CORE TECHNICAL LEMMA 2: RANDOMNESS COUPLING ", + "bbox": [ + 174, + 325, + 604, + 340 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our next technical lemma studies that, if run Algorithm 2 from a point $w _ { 0 }$ so that the Hessian $\\nabla ^ { 2 } f ( w _ { 0 } )$ has a eigenvalue which is less than $- \\delta$ (think of $w _ { 0 }$ as a saddle point), then with good probability, after sufficiently many iterations, the sequence $w _ { 1 } , w _ { 2 } , \\dots , w _ { T }$ shall escape from $w _ { 0 }$ to distance at least $R$ for some parameter $R \\approx \\delta$ . To prove this, motivated by Jin et al. (2017), we study two executions of Algorithm 2 where their randomness are coupled. We then argue that at least one of them has to escape from $w _ { 0 }$ . For any vector $v$ , let $[ v ] _ { i }$ denote the $i$ -th coordinate of $v$ . ", + "bbox": [ + 173, + 345, + 825, + 430 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 3.5. Suppose we choose $\\mathfrak { T }$ as in Lemma 3.2 and $C _ { 1 } , C _ { 2 }$ as in Lemma 3.3. Suppose $\\boldsymbol { w } _ { 0 } \\in \\mathbb { R } ^ { d }$ satisfies $\\dot { \\lambda } _ { \\mathrm { m i n } } ( \\nabla ^ { 2 } f ( w _ { 0 } ) ) = - \\delta$ for some $\\delta \\geq 0$ . Without loss of generality let $\\mathbf { e } _ { 1 }$ be the eigenvector of $\\nabla ^ { 2 } f ( w _ { 0 } )$ with smallest eigenvalue. Consider now two executions of Algorithm 2, both starting from $w _ { 0 } ^ { \\mathsf { a } } = w _ { 0 } ^ { \\mathsf { b } } = w _ { 0 }$ , and suppose their randomness $\\{ \\xi _ { t } ^ { \\mathsf { a } } \\} _ { t }$ and $\\{ \\xi _ { t } ^ { \\mathrm { b } } \\} _ { t }$ are coupled so that $[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } = - [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 }$ but $[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { i } = [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { i }$ for $i > 1$ . In words, the randomness is the same orthogonal to $\\mathbf { e } _ { 1 }$ , but along $\\mathbf { e } _ { 1 }$ , the two have opposite signs. Now, suppose we perform $\\begin{array} { r } { T = \\Theta \\big ( \\frac { 1 } { \\eta \\delta } \\log \\frac { R ^ { 2 } \\delta } { \\eta \\nu ^ { 2 } } \\big ) } \\end{array}$ steps of perturbed SGD from $w _ { 0 } ^ { \\mathsf { a } } , w _ { 0 } ^ { \\mathsf { b } }$ respectively using Algorithm 2. Suppose ", + "bbox": [ + 173, + 434, + 825, + 541 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/6dc82af2029d61380e159b33aa634e018e7b1f60fefef4261df05b4c31f7d919.jpg", + "text": "$$\n\\begin{array} { r } { R \\leq { \\cal O } \\big ( { \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( R ^ { 2 } \\delta / \\eta \\nu ^ { 2 } ) } } \\big ) \\quad a n d \\quad \\nu ^ { 2 } \\geq \\Omega \\big ( C _ { 2 } \\log { \\frac { R ^ { 2 } \\delta } { \\eta \\nu } } \\big ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 544, + 687, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Then, under events Even $\\mathbf { t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { a } } )$ and $\\mathsf {bar { c } v e n t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { b } } )$ , with probability at least 0.98, either $\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } \\| > R$ or $\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } \\| > R$ for some $t \\in [ T ]$ . ", + "bbox": [ + 174, + 570, + 825, + 604 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof details in Appendix A.4. The main proof difficulty is to analyze a noisy version of the power method, where the noise comes from (1) Gaussian perturbation (which is the good noise), (2) stochastic gradients (which has zero mean), and (3) Byzantine workers (which can be adversarial). ", + "bbox": [ + 174, + 609, + 825, + 654 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 FROM WARMUP TO FINAL THEOREM WITH DOUBLE SAFE GUARDS ", + "text_level": 1, + "bbox": [ + 171, + 669, + 766, + 685 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "At a high level, Lemma 3.3 ensures that if we keep encountering points with large gradient $\\| \\nabla f ( w _ { t } ) \\|$ , then the objective should sufficiently decrease; in contrast, Lemma 3.5 says that if we keep encountering points with negative Hessian directions (i.e., $\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( w _ { t } ) ) < \\dot { - } \\delta )$ , then the points must move a lot (i.e., by more than $R$ in $T$ iterations, which can also lead to sufficient objective decrease, see Lemma B.4). Therefore, at a high level, when the two lemmas are combined, they tell that we must not encounter points with $\\| \\nabla f ( x ) \\|$ being large, or $\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( x ) )$ being very negative, for too many iterations. Therefore, the algorithm can find approximate local minima. ", + "bbox": [ + 173, + 694, + 825, + 794 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The reason we need two safe guards, is because the number of rounds $T$ for Lemma 3.3 and Lemma 3.5 differ by a factor. We need two safe guards with different window sizes to ensure the two lemmas simultaneously hold. We encourage the reader to examine the full analysis in Appendix B. ", + "bbox": [ + 174, + 796, + 825, + 839 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL VALIDATION ", + "text_level": 1, + "bbox": [ + 176, + 854, + 446, + 871 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate the convergence of SafeguardSGD to examine its practical performance against prior works. We perform the non-convex task of training a residual network ResNet-20 (He et al., 2016) on the CIFAR-10/100 datasets (Krizhevsky et al., 2014). More details are given in Appendix C. ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/fd2edb2655662c1193ffe6a690559d8147cb61fcf659521ef7f5a1fb281c36fa.jpg", + "image_caption": [ + "Figure 1: Convergence comparison (CIFAR-10 test accuracy) under different attacks. (In Appendix C.2, one can find additional CIFAR-100 experiments, more discussions, and bigger plots.) " + ], + "image_footnote": [], + "bbox": [ + 171, + 93, + 826, + 337 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We instantiate $m = 1 0$ workers and one master node executing data-parallel SGD for 140 passes (i.e. epochs) over the training dataset. The results for higher number of workers and epochs are similar, and therefore omitted. We compare against Geometric Median (Chen et al., 2017), Coordinatewise Median (Yin et al., 2018; 2019), Krum (Blanchard et al., 2017), and Zeno (Xie et al., 2018b). Overall, our experimental setup is very similar to Zeno (Xie et al., 2018b) but with additional attacks. ", + "bbox": [ + 173, + 402, + 825, + 473 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We implemented the approach of Yang et al. (2019), but found it very sensitive to hyper-parameter values and were unable to make it converge across all attacks even after significant tuning of its $\\gamma$ parameter. We also implemented the convex algorithm of Alistarh et al. (2018), and executed it in our non-convex setting. We found their algorithm can be easily attacked on our ResNet training tasks. There exists a simple attack, described in Appendix C.4 which causes their algorithm to either mislabel most good workers as Byzantine, or diverge, or converge to very poor solutions. This is not surprising, since their algorithm is designed for, and only guaranteed to work in, the convex setting. ", + "bbox": [ + 174, + 476, + 825, + 574 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To make the comparison stronger, when implementing SafeguardSGD, we have chosen fixed window sizes $T _ { 0 } = 1$ epoch and $T _ { 1 } = 6$ epochs across all experiments, and adopted an automated process to select $\\mathfrak { T } _ { 0 } , \\mathfrak { T } _ { 1 }$ . Determining these thresholds requires being able to pre-run the task on an honest worker. We have also implemented a single safeguard variant of SafeguardSGD, with window size $T = 3$ epochs. ", + "bbox": [ + 174, + 578, + 825, + 647 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Attacks. We set $\\alpha = 0 . 4$ , which means that there are 4 Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.) ", + "bbox": [ + 173, + 651, + 823, + 680 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "• LABEL-FLIPPING ATTACK: each Byzantine worker computes its gradient based on the crossentropy loss with flipped labels: for CIFAR-10, label $\\ell \\in \\{ 0 , . . . , 9 \\}$ is flipped to $9 - \\ell$ . • DELAYED-GRADIENT ATTACK: each Byzantine worker sends an old gradient to master. In our experiments, the delay is of $D = 1 0 0 0$ iterations. • VARIANCE ATTACK (Baruch et al., 2019): Byzantine workers measure the mean and the standard-deviation of gradients at each round, and collude to move the mean by the largest value which still operates within population variance. (For our parameter settings, this is 0.3 times the standard deviation. We discuss results for additional parameter values in the Appendix.) • SIGN-FLIPPING ATTACK: each Byzantine worker sends the negative gradient to the master. • SAFEGUARD ATTACK: each Byzantine workers sends a negative but re-scaled gradient to the master. We use re-scale factors 0.6 and 0.7 in our experiments. The re-scale factor 0.6 avoids triggering the safe-guard conditions at the master, and the re-scale factor 0.7 occasionally triggers the safe-guard conditions. This attack is an instantiation of the inner-product attack (Xie et al., 2020), customized specifically to maximally affect our SafeguardSGD algorithm. ", + "bbox": [ + 181, + 685, + 826, + 890 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Main experimental results. The ideal test accuracy is $9 1 . 7 \\%$ , which corresponds to applying SGD using only the stochastic gradients from the honest workers. Figure 1 compares the performances in test accuracy. Below we summarize our main findings for the experiments, and we defer detailed discussions (and additional experiments for CIFAR-100) to Appendix C. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/383ad3190abfe8f2cfb2caf94b401335009e80d8bed41ff9a0c8c89b12115762.jpg", + "image_caption": [ + "(a) $\\| B _ { i } - B _ { \\mathrm { m e d } } \\|$ between a good node (blue), and a bad node (red) which pretends to be honest and then starts to apply the variance attack. " + ], + "image_footnote": [], + "bbox": [ + 205, + 113, + 459, + 222 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/cae7626417996e121b4e9ed1583ac1e9c08ef0c2c60549476e6ad5df3383dbb6.jpg", + "image_caption": [ + "(b) Convergence for our safeguard algorithms under the variance attack, after periodically resetting the set of good nodes. ", + "Figure 2 " + ], + "image_footnote": [], + "bbox": [ + 531, + 107, + 787, + 222 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 291, + 823, + 320 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• SafeguardSGD generally outperforms all the previous methods in test accuracy. The test accuracy difference can be ${ } ^ { 6 6 } 9 0 \\%$ vs. $< 4 0 \\%$ ” between our algorithm and the best prior work. • The variance attack is indeed very strong, in that it severely affects the accuracy of all prior works (test accuracy $< 3 5 \\%$ ). This is because thesese defenses are “historyless.” By contrast, our algorithm not only provably but also empirically defends against it. • Our safeguard attack (especially with re-scale factor 0.7) is as strong as the variance attack, and even stronger on the CIFAR-100 dataset; please see the results in Appendix C.2.5. • The label-flipping attack is rather weak: although some defenses, such as Zeno, did not determine which of the workers are malicious, they still converge well under this attack. • The sign-flipping and delayed-gradient attacks are moderate: the best prior works can achieve accuracy $6 0 \\% \\sim 7 0 \\%$ . It is worth noting that the sign-flipping attack can already nullify the Zeno defence (test accuracy $2 0 \\%$ ). The issue seems to be that it can be very hard for Zeno to use relatively few samples to determine if the gradient direction is flipped to negative. • SafeguardSGD can easily catch all the bad workers under sign-flipping and variance attacks, and thus leads to gives ideal performance. It cannot catch any bad worker for label-flipping and delayed-gradient attacks, but there is no performance loss anyways if we use such bad gradients. • The safeguard attacks, designed to maximally impact the performance of our SafeguardSGD, can indeed affect our performance. Specifically, under re-scale factor 0.6, the test accuracy drops from $9 1 . 7 \\%$ to $8 9 . 3 \\%$ because SafeguardSGD cannot catch any bad worker; however, under rescale factor 0.7, the test accuracy no longer drops because SafeguardSGD can begin to catch some bad workers (it can catch between 0 and 4 bad workers depending on the randomness.) • In most cases, the single-safeguard algorithm is close to double-safeguard, except for the safeguard $( \\mathrm { x 0 . 7 } )$ attack, in which using double-safeguard one can more easily catch bad workers. (This is more apparent in the CIFAR-100 experiment, see Appendix C.2.5.) ", + "bbox": [ + 181, + 324, + 826, + 674 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We conclude that SafeguardSGD can be practical, and outperforms previous approaches. ", + "bbox": [ + 174, + 676, + 758, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A deeper dive: how the algorithm works. Let us explain the inner workings of our algorithm in the context of a “delayed” attack, where the Byzantine nodes collude to execute an attack only after a specific, given point in the execution (in this case, the first half-epoch). Figure 2(a) presents the results from the perspective of the value of $\\| B _ { i } - B _ { \\mathrm { m e d } } \\|$ registered at the master server, for two nodes, an honest one, and a Byzantine one. The value of √ $\\| B _ { i } - B _ { \\sf m e d } \\|$ increases for all the nodes (at a rate of roughly $\\sqrt { t }$ at step $t$ ); but, once the attack starts, the statistic for the Byzantine node grows linearly in $t$ , leading to fast detection. ", + "bbox": [ + 173, + 695, + 825, + 795 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Transient attacks and node ID relabeling. Finally, in Figure 2(b) we analyze the behaviour of our algorithm when it periodically (every 3 epochs for single safeguard and 6 epochs for double safeguard) resets the set of good nodes to include all nodes, restarting the detection process from scratch. Our theoretical result still applies after this relaxation. This relaxation has two benefits. First, it benefits from bad workers that under transient failures (e.g., the node fails for 10 epochs but resumes to work correctly after a while), and thus benefits from the data stored on this worker. Second, it can defend against certain degree of node ID relabeling: it supports the case when good and bad workers exchange their IDs every 6 epochs. In Figure 2(b), we see even under the (very strong) variance attack, relaxed safeguard maintains good performance. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "F. E. and D. A. were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 805223 ScaleML). ", + "bbox": [ + 176, + 125, + 823, + 154 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES \nDan Alistarh, Zeyuan Allen-Zhu, and Jerry Li. Byzantine stochastic gradient descent. In Advances in Neural Information Processing Systems, pp. 4613–4623, 2018. \nZeyuan Allen-Zhu. Natasha 2: Faster Non-Convex Optimization Than SGD. In NeurIPS, 2018a. Full version available at http://arxiv.org/abs/1708.08694. \nZeyuan Allen-Zhu. How To Make the Gradients Small Stochastically. In NeurIPS, 2018b. Full version available at http://arxiv.org/abs/1801.02982. \nZeyuan Allen-Zhu and Yuanzhi Li. Feature purification: How adversarial training performs robust deep learning. arXiv preprint arXiv:2005.10190, 2020. \nGilad Baruch, Moran Baruch, and Yoav Goldberg. A little is enough: Circumventing defenses for distributed learning. In Advances in Neural Information Processing Systems, pp. 8635–8645, 2019. \nPeva Blanchard, El Mahdi El Mhamdi, Rachid Guerraoui, and Julien Stainer. Machine learning with adversaries: Byzantine tolerant gradient descent. In NIPS, pp. 118–128, 2017. \nSaikiran Bulusu, Prashant Khanduri, Pranay Sharma, and Pramod K Varshney. On distributed stochastic gradient descent for nonconvex functions in the presence of byzantines. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3137–3141. IEEE, 2020. \nMiguel Castro, Barbara Liskov, et al. Practical byzantine fault tolerance. 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Defending against saddle point attack in byzantine-robust distributed learning. In International Conference on Machine Learning, pp. 7074–7084, 2019. ", + "bbox": [ + 171, + 145, + 828, + 921 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 826, + 449 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 408, + 101, + 589, + 127 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A MISSING PROOFS FOR SECTION 3 ", + "text_level": 1, + "bbox": [ + 174, + 142, + 491, + 160 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 PROOF OF LEMMA 3.2 ", + "bbox": [ + 174, + 170, + 375, + 185 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Recall the following, useful inequality. ", + "bbox": [ + 174, + 190, + 405, + 204 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lemma A.1 (Pinelis’ 1994 inequality (Pinelis, 1994)). Let $X _ { 1 } , \\dots , X _ { T } \\in \\mathbb { R } ^ { d }$ be a random process satisfying $\\mathbb { E } [ X _ { t } | X _ { 1 } , \\dots , X _ { t - 1 } ] = 0$ and $\\| X _ { t } \\| \\leq M$ . Then, $\\operatorname* { P r } \\left[ \\| X _ { 1 } + \\cdot \\cdot \\cdot + X _ { T } \\| ^ { 2 } > 2 \\log ( 2 / \\delta ) M ^ { \\hat { 2 } } T \\right] \\leq \\delta .$ . ", + "bbox": [ + 174, + 208, + 823, + 237 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lemma 3.2 is in fact a direct corollary of the following claim, whose proof is quite classical. Denote by $C = \\log ( 1 6 m T / p )$ . Denote by ", + "bbox": [ + 171, + 242, + 823, + 270 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/b9561823442ca9dc63974e4c38f59f46c4ad4c4846fd7b870b52044459c1b8bf.jpg", + "text": "$$\nB _ { i } ^ { ( t ) } : = \\frac { \\nabla _ { 0 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla _ { t - 1 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } \\quad \\mathrm { a n d } \\quad B _ { \\star } ^ { ( t ) } : = \\frac { \\nabla f ( w _ { 0 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla f ( w _ { t - 1 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 273, + 751, + 304 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Recall at iteration $t - 1$ , Algorithm 2 computes $\\{ B _ { 1 } ^ { ( t ) } , \\ldots , B _ { m } ^ { ( t ) } \\}$ as well as some $B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ where $i$ is any machine in $\\tt g o o d _ { t - 1 }$ such that at least half of $j \\in [ m ]$ satisfies $\\lVert B _ { j } ^ { ( t ) } - B _ { i } ^ { ( t ) } \\rVert \\leq 8 \\sqrt { t C } / m$ . ", + "bbox": [ + 171, + 308, + 821, + 342 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Claim A.2. Let $C = \\log ( 1 6 m T / p )$ . Then, with probability at least $1 - p / 4$ , we have ", + "bbox": [ + 171, + 345, + 686, + 361 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) for all $i \\in$ good and $\\in [ T ] , \\| B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\| \\leq 4 \\sqrt { t C } / m .$ . \n(b) for all $t \\in [ T ]$ , each $i \\in$ good is a valid choice for $B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ = B ( t )i . \n(c) for all $i \\in$ good and $t \\in [ T ] .$ , $\\lVert B _ { i } ^ { ( t ) } - B _ { \\mathrm { m e d } } ^ { ( t ) } \\rVert \\leq 1 6 \\sqrt { t C } / m$ and $\\| B _ { \\star } ^ { ( t ) } - B _ { \\sf m e d } ^ { ( t ) } \\| \\le 1 2 \\sqrt { t C } / m$ (d) for all $i \\in$ good and $t \\in [ T ]$ , we also have $i \\in \\mathbf { g } \\circ \\circ \\mathsf { d } _ { t + 1 }$ . \n(e) $\\begin{array} { r } { \\big \\| \\sum _ { i \\in \\mathrm { g o o d } } \\big ( B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) . } \\end{array}$ ", + "bbox": [ + 169, + 364, + 759, + 463 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof of Claim A.2. We prove by induction. Suppose the statements hold for $t - 1$ and we now move to $t$ . ", + "bbox": [ + 171, + 484, + 802, + 500 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) $\\textit { i } \\in$ $\\mathbb { E } [ \\nabla _ { t , i } ] ~ = ~ \\nabla _ { t }$ and n thu $\\| \\nabla _ { t , i } - \\nabla _ { t } \\| \\leq 1$ . Let to the $\\begin{array} { r } { { \\boldsymbol { X } } _ { t } ~ = ~ \\frac { \\nabla _ { t , i } - \\nabla _ { t } } { | { \\bf g } \\circ { \\bf o d } _ { t } | } } \\end{array}$ , so that a union $\\begin{array} { r } { \\Vert X _ { t } \\Vert ~ \\leq ~ \\frac { 1 } { \\vert \\mathbf { g } \\circ \\mathbf { o } \\mathbf { d } _ { t } \\vert } ~ \\leq ~ \\frac { 1 } { ( 1 - \\alpha ) m } ~ \\leq ~ \\frac { 2 } { m } } \\end{array}$ $X _ { t }$ bound over all $i \\in \\mathtt { g o o d }$ . Thus, with probability at least $1 - { \\frac { p } { 8 T } }$ we have $\\lVert B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\rVert \\leq 4 \\sqrt { t C } / m$ for all $i \\in { \\bf g } { \\sf o o d }$ . The result follows from a further union bound over $t \\in [ T ]$ . ", + "bbox": [ + 171, + 503, + 826, + 566 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(b) Claim A.2a implies for every $i , j \\in \\mathtt { g o o d }$ we have $\\| B _ { i } ^ { ( t ) } - B _ { j } ^ { ( t ) } \\| \\leq 8 \\sqrt { t C } / m$ . Therefore each $i \\in$ good is a valid choice for setting $B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }$ . ", + "bbox": [ + 169, + 570, + 823, + 606 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(c) This is a consequence of the previous items and the definition of $B _ { \\mathrm { m e d } } ^ { ( t ) }$ . ", + "bbox": [ + 171, + 609, + 614, + 626 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(d) This is a consequence of the previous item. ", + "bbox": [ + 173, + 627, + 452, + 642 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(e) We can apply Lemma A.1 with $\\begin{array} { r } { \\{ X _ { 1 } , X _ { 2 } , \\dots , X _ { t \\mid \\operatorname { g o o d } \\mid } \\} ~ = ~ \\{ { \\frac { \\nabla _ { k , i } - \\nabla f ( w _ { k } ) } { | \\operatorname { g o o d } _ { k } | } } \\} _ { k \\in [ t ] , i \\in \\operatorname { g o o d } } } \\end{array}$ . It holds with probability at least $\\textstyle 1 - { \\frac { p } { 8 T } }$ that $\\begin{array} { r } { \\big \\| \\sum _ { i \\in \\mathrm { g o o d } } \\big ( B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) } \\end{array}$ . ", + "bbox": [ + 174, + 643, + 825, + 684 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof of Lemma 3.2. The property $\\mathbf { g } _ { 0 0 } \\mathsf { d } _ { t } \\supseteq \\mathsf { g } _ { 0 0 } \\mathsf { d }$ is from Claim A.2d. ", + "bbox": [ + 173, + 727, + 593, + 742 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The property $\\begin{array} { r } { \\| \\sigma _ { t } \\| ^ { 2 } \\le { \\cal O } ( \\frac { \\log ( T / p ) } { m } ) } \\end{array}$ is by standard concentration inequalities for sums of bounded random vectors. ", + "bbox": [ + 178, + 746, + 825, + 773 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The property $\\| \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } )$ is from Claim A.2e. ", + "bbox": [ + 173, + 773, + 604, + 792 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The property $\\| \\Delta _ { t } \\| \\le \\alpha$ is obvious as we have at most $\\alpha$ fraction of the bad machines. ", + "bbox": [ + 174, + 794, + 687, + 808 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The bound on $\\| \\Delta _ { 0 } + \\cdots + \\Delta _ { t - 1 } \\| ^ { 2 }$ can be derived as follows. For every $i \\in [ m ] \\ \\backslash \\ \\mathtt { g o o d }$ , let $t$ be the last iteration $_ { i }$ satisfies $i \\in \\mathtt { g o o d } _ { t }$ . Then, by the triangle inequality, ", + "bbox": [ + 178, + 810, + 825, + 838 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/013520b1e13b7e76ca8158fd6e5233c80abd87d4d98f2f52184539260dd6e591.jpg", + "text": "$$\n\\| B _ { i } ^ { ( t + 1 ) } - B _ { \\star } ^ { ( t + 1 ) } \\| \\le \\frac { 2 } { m } + \\| B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\|\n$$", + "text_format": "latex", + "bbox": [ + 366, + 840, + 630, + 867 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "On the other hand, $t \\in { \\mathsf { g o o d } } _ { t }$ implies $\\lVert B _ { i } ^ { ( t ) } - B _ { \\mathrm { m e d } } ^ { ( t ) } \\rVert \\leq 1 6 \\sqrt { t C } / m$ by the algorithm; combining this with $\\| B _ { \\star } ^ { ( t ) } - B _ { \\sf m e d } ^ { ( t ) } \\| \\le 1 2 \\sqrt { t C } / m$ , and summing up over all such bad machines $_ { i }$ finishes the proof. ", + "bbox": [ + 171, + 871, + 826, + 904 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The final two properties follow from standard facts about Gaussian random vectors. ", + "bbox": [ + 174, + 910, + 668, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 PROOF OF LEMMA 3.3 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 374, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof of Lemma 3.3. Using the Lipschitz smoothness of $f ( \\cdot )$ , we have ", + "bbox": [ + 174, + 125, + 593, + 140 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/26ea8a67357ffa26caaf183b76f3442b144d4d4fac078a4126a1f768e432fd46.jpg", + "text": "$$\n\\begin{array} { l } { f ( w _ { t } ) - f ( w _ { t + 1 } ) \\geq \\langle \\nabla f ( w _ { t } ) , w _ { t } - w _ { t + 1 } \\rangle - \\displaystyle \\frac { 1 } { 2 } \\| w _ { t } - w _ { t + 1 } \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad = \\eta \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } + \\eta \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle - \\displaystyle \\frac { 1 } { 2 } \\| w _ { t } - w _ { t + 1 } \\| ^ { 2 } + \\eta \\langle \\nabla f ( w _ { t } ) , \\xi _ { t } \\rangle } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 141, + 771, + 198 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We first show: ", + "bbox": [ + 173, + 198, + 259, + 210 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c96de59f9d7c28314fab252768f915ea27ae5dff08e28f7ca9050d9fb0d4ffd8.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\| w _ { t } - w _ { t + 1 } \\| ^ { 2 } = \\eta ^ { 2 } \\| \\nabla f ( w _ { t } ) + \\Xi _ { t } - \\xi _ { t } \\| ^ { 2 } \\le 3 \\eta ^ { 2 } ( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } + \\| \\Xi _ { t } \\| ^ { 2 } + \\| \\xi _ { t } \\| ^ { 2 } ) } } \\\\ { { \\displaystyle \\sum _ { t = 0 } ^ { T - 1 } \\eta \\langle \\nabla f ( w _ { t } ) , \\xi _ { t } \\rangle | \\le \\eta \\sqrt { \\displaystyle \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } \\cdot O ( \\nu \\sqrt { C _ { 1 } } ) } \\le \\left( 0 . 0 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } \\right) + O ( C _ { 1 } \\nu ^ { 2 } \\eta ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 210, + 803, + 277 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The first follows since $( a + b + c ) ^ { 2 } \\leq 3 ( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } )$ for any $a , b , c \\in { \\mathsf { R } }$ , and the second follows from Lemma 3.2. Combining them, and also using that $\\left. \\Xi _ { t } \\right. ^ { 2 } \\leq O ( \\dot { C } _ { 2 } )$ , $\\| \\dot { \\xi } _ { t } \\| ^ { 2 } \\le { \\cal O } ( d \\nu ^ { 2 } C _ { 1 } )$ , and $\\eta \\leq 0 . 0 1$ , we have ", + "bbox": [ + 174, + 279, + 820, + 319 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/04f41c8d21f6fa4c00eb3ed0d34ba6a5683d066914ff2e5deb61a2c0a0821e6b.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T } ) \\ge 0 . 9 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - O ( \\eta C _ { 2 } ) \\right) + \\eta \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle - O ( \\eta T \\nu ^ { 2 } C _ { 1 } ( \\eta d + \\frac { 1 } { T } ) )\n$$", + "text_format": "latex", + "bbox": [ + 187, + 319, + 808, + 358 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For the inner product on the right hand of (A.1), we have that ", + "bbox": [ + 173, + 373, + 537, + 387 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6ceef057c198b30cdb82d665f8737db14b09482d5c747c45ec92849c3f47be67.jpg", + "text": "$$\n\\eta \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle = \\underbrace { \\frac { \\eta } { T } \\sum _ { q = 0 } ^ { T - 1 } \\left. \\nabla f ( w _ { q } ) , \\sum _ { t = 0 } ^ { T - 1 } \\Xi _ { t } \\right. } _ { \\bullet } + \\underbrace { \\frac { \\eta } { T } \\sum _ { q = 0 } ^ { T - 1 } \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) - \\nabla f ( w _ { q } ) , \\Xi _ { t } \\rangle } _ { \\bullet }\n$$", + "text_format": "latex", + "bbox": [ + 210, + 390, + 754, + 450 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For the first term $\\spadesuit$ , we have ", + "bbox": [ + 173, + 450, + 346, + 464 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/8b50c2dfbb6f91ca2eac4bcc23f75e3f9e5237e16211f6d4741eb925b5118e4c.jpg", + "text": "$$\n\\begin{array} { l } { | \\bullet | \\leq \\displaystyle \\frac { \\eta } { T } \\sum _ { q = 0 } ^ { T - 1 } \\left| \\left. \\nabla f ( w _ { q } ) , \\sum _ { t = 0 } ^ { T - 1 } \\Xi _ { t } \\right. \\right| \\leq \\displaystyle \\frac { \\eta } { T } \\sum _ { q = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { q } ) \\| \\cdot \\left\\| \\sum _ { t = 0 } ^ { T - 1 } \\Xi _ { t } \\right\\| } \\\\ { \\leq 0 . 1 \\eta \\displaystyle \\sum _ { q = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { q } ) \\| ^ { 2 } + \\displaystyle \\frac { O ( \\eta ) } { T ^ { 2 } } \\sum _ { q = 0 } ^ { T - 1 } \\left\\| \\sum _ { t = 0 } ^ { T - 1 } \\Xi _ { t } \\right\\| ^ { 2 } } \\\\ { \\leq 0 . 1 \\eta \\displaystyle \\sum _ { q = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { q } ) \\| ^ { 2 } + O \\left( \\eta C _ { 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 465, + 756, + 590 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where the last inequality follows from Lemma 3.2. ", + "bbox": [ + 176, + 592, + 472, + 604 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For the second term $\\clubsuit$ , we have ", + "bbox": [ + 173, + 608, + 364, + 622 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d7fbc329282d3f7b7720a7a8a1e0eaf11064b5cc06a798d5d7743e75eaf5dfcd.jpg", + "text": "$$\n\\begin{array} { r } { | \\bullet | \\le \\underbrace { \\eta } _ { T } \\displaystyle \\sum _ { q = 0 } ^ { T - 1 } \\Big | \\displaystyle \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) - \\nabla f ( w _ { q } ) , \\Xi _ { t } \\rangle \\Big | \\le \\underbrace { \\frac { \\eta } { T } \\displaystyle \\sum _ { q = 0 } ^ { T - 1 } \\Big | \\displaystyle \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \\Xi _ { t } \\rangle \\Big | } _ { \\diamond } } \\\\ { + \\underbrace { \\eta } _ { T } \\displaystyle \\sum _ { q = 0 } ^ { T - 1 } \\sum _ { t = 0 } ^ { T - 1 } ( \\| w _ { t } - w _ { 0 } \\| + \\| w _ { q } - w _ { 0 } \\| ) \\| w _ { t } - w _ { q } \\| \\| \\Xi _ { t } \\| } _ { \\displaystyle \\int _ { 0 } ^ { T } \\eta = 0 } , \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 622, + 767, + 726 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Using $\\left\\| w _ { t } - w _ { q } \\right\\| \\leq \\left\\| w _ { t } - w _ { 0 } \\right\\| + \\left\\| w _ { q } - w _ { 0 } \\right\\|$ , one can derive ", + "bbox": [ + 173, + 744, + 550, + 760 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1b7d418d156c7072b8d9151c6778d3339da920fb38f7ec346ec7ec25ecbf5cef.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\bigtriangledown \\le \\frac { \\eta } { T } \\sum _ { q = 0 } ^ { T - 1 } \\sum _ { t = 0 } ^ { T - 1 } ( \\| w _ { t } - w _ { 0 } \\| + \\| w _ { q } - w _ { 0 } \\| ) ^ { 2 } \\cdot O ( \\sqrt { C _ { 2 } } ) } \\\\ & { \\displaystyle \\le \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| w _ { t } - w _ { 0 } \\| ^ { 2 } \\cdot O ( \\sqrt { C _ { 2 } } ) } \\\\ & { \\displaystyle \\le \\eta ^ { 3 } \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { 0 } ) + \\cdot \\cdot \\cdot + \\nabla f ( w _ { t - 1 } ) + \\Xi _ { 0 } + \\cdot \\cdot \\cdot + \\Xi _ { t - 1 } + \\xi _ { 0 } + \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\cdot O ( \\sqrt { C _ { 2 } } ) } \\\\ & { \\displaystyle \\le O ( \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } + O ( \\sqrt { C _ { 2 } } C \\eta ^ { 3 } T ^ { 2 } ) + O ( \\eta ^ { 3 } \\nu ^ { 2 } T ^ { 2 } d C _ { 1 } \\sqrt { C _ { 2 } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 765, + 785, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As for $\\diamondsuit$ ", + "bbox": [ + 173, + 104, + 233, + 117 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8162b2fce8d52fb9d26e3b6ef39e4362814fbdcd16e15acfe57f25afd775b4a2.jpg", + "text": "$$\n\\Big | \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \\Xi _ { t } \\rangle \\Big | \\leq \\Big | \\sum _ { t = q + 1 } ^ { T - 1 } \\langle \\nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \\Xi _ { t } \\rangle \\Big | + \\Big | \\sum _ { t = 0 } ^ { q - 1 } \\langle \\nabla ^ { 2 } f ( w _ { 0 } ) ( w _ { t } - w _ { q } ) , \\Xi _ { t } \\rangle \\Big |\n$$", + "text_format": "latex", + "bbox": [ + 183, + 119, + 812, + 160 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For the first term (and the second term is analogous), we have ", + "bbox": [ + 174, + 162, + 539, + 176 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e1934493f9808ebf33cf20bd26ff2ad7d52c453c7a8074adb6a87561bacb12b7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\begin{array} { r l } & { \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( j ) ( x , \\theta ) - x _ { j } \\right| , \\mathcal { E } _ { k } \\Biggr | } \\\\ & { = - \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) \\right| , } \\end{array} } \\\\ & { \\begin{array} { r l } & { \\le - \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) - x _ { j } - \\mathbb { F } ^ { j } ( x ( x , x _ { j } ) ) + \\mathbb { E } _ { \\eta _ { k } + 1 } - \\mathbb { E } _ { \\eta _ { k - 1 } + 1 } + ( \\epsilon + \\epsilon - \\epsilon + \\epsilon - \\epsilon ) \\mathbb { E } _ { k } \\right| , \\xi _ { 1 } } \\\\ & { \\le \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) \\right| , } \\end{array} } \\\\ & \\begin{array} { r l } & { \\ge \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) \\right| , \\mathcal { E } _ { k } - \\eta - \\frac { 1 } { \\gamma } \\mathcal { F } _ { k } \\Big | , } \\\\ & { \\le \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) - x _ { j } \\right| , \\mathcal { E } _ { k } \\Big | - \\eta \\left| \\displaystyle \\sum _ { j = 1 } ^ { N } \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) \\right| , \\mathcal { E } _ { k } + \\cdots - \\mathbb { E } _ { \\eta _ { k - 1 } - 1 } ) \\mathbb { E } _ { \\eta _ { k - 1 } + 1 } ^ { j } } \\\\ & { \\le \\eta \\left( \\frac { \\gamma ^ { j } } { \\gamma } \\mathcal { F } ^ { j } ( x ) ( x , \\theta ) - x _ { j } \\right) , } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 176, + 181, + 816, + 494 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Above, inequality $\\textcircled{1}$ uses $\\| \\Xi _ { 0 } + \\cdot \\cdot \\cdot \\Xi _ { t } \\| \\le { \\cal O } ( \\sqrt { T C _ { 2 } } )$ for $\\begin{array} { r } { C _ { 2 } = \\alpha ^ { 2 } \\log \\frac { m T } { p } + \\frac { \\log ( T / p ) } { m } } \\end{array}$ (see Lemma 3.2) and a delicate application of Azuma’s inequality that we state at the end of this subsection (see Proposition 3.4); Inequality $\\textcircled{2}$ uses Young’s inequality and Lemma 3.2. ", + "bbox": [ + 174, + 496, + 826, + 539 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Putting this back to the formula of $\\diamondsuit$ , we have ", + "bbox": [ + 176, + 541, + 449, + 555 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6249c968eaaee3d902f31851cfe349e656b0670972d1ffaf8630b40d27800f23.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\diamond \\lambda \\leq O ( \\eta ^ { 2 } \\sqrt { T C _ { 2 } } ) \\cdot \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| + O ( T \\eta ^ { 2 } C _ { 2 } + T \\eta ^ { 2 } \\nu ^ { 2 } d C _ { 1 } ) } \\\\ { \\displaystyle \\qquad \\leq 0 . 1 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } + O ( \\eta ^ { 3 } T ^ { 2 } C _ { 2 } + T \\eta ^ { 2 } C _ { 2 } + T \\eta ^ { 2 } \\nu ^ { 2 } d C _ { 1 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 558, + 700, + 638 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Finally, putting $\\diamondsuit$ and $\\heartsuit$ back to $\\clubsuit$ , and putting $\\clubsuit$ and $\\spadesuit$ back to (A.2) and (A.1), we have ", + "bbox": [ + 171, + 643, + 710, + 659 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/73d20642e74334221ecebb186b6e94d4f02ee4cdcefe6880ff755824495c98e2.jpg", + "text": "$$\n\\begin{array} { l } \\displaystyle { \\mathrm { ~ \\gamma ~ } _ { \\mathrm { { \\Theta } } } ) - f ( w _ { T } ) \\geq 0 . 8 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - O ( \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad - C _ { 2 } \\cdot O ( \\eta + \\eta ^ { 2 } T + \\eta ^ { 3 } T ^ { 2 } + \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \\cdot O ( T \\eta ^ { 2 } \\nu ^ { 2 } d + T ^ { 2 } \\eta ^ { 3 } \\nu ^ { 2 } \\sqrt { C _ { 2 } } + \\eta T \\nu ^ { 2 } ( \\eta d + \\frac { 1 } { T } ) ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 660, + 861, + 727 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { T = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}$ and $\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}$ ", + "bbox": [ + 173, + 729, + 583, + 748 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b02c5875e9361055b04e9aec5db6929204e34db32506261376def8049a9823c1.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\dot { \\tau } } ( w _ { 0 } ) - f ( w _ { T } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - C _ { 2 } \\cdot O ( \\eta + \\eta ^ { 2 } T + \\eta ^ { 3 } T ^ { 2 } + \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \\cdot O ( T \\eta \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ^ { 2 } ) } } \\\\ { { \\displaystyle \\qquad = 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - C _ { 2 } \\cdot O ( \\frac { 1 } { T } + \\eta + \\eta ^ { 2 } T + \\sqrt { C _ { 2 } } \\eta ^ { 2 } T ) - C _ { 1 } \\cdot O ( \\eta T \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) } } \\\\ { { \\displaystyle \\qquad \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) \\ . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 751, + 839, + 872 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 PROOF OF PROPOSITION 3.4 ", + "text_level": 1, + "bbox": [ + 176, + 103, + 415, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proposition 3.4. Fix the dimension parameter $d \\geq 1$ . Suppose $\\xi _ { 0 } , \\ldots , \\xi _ { T - 1 } \\in \\mathbb { R } ^ { d }$ are i.i.d. drawn from $\\mathcal { N } ( 0 , \\bf { I } )$ , and that $\\Delta _ { 1 } , \\ldots , \\Delta _ { T - 1 }$ are arbitrary vectors in $\\mathbb { R } ^ { d }$ . Here, each vector $\\Delta _ { t }$ with $t = 1 , \\dots , T - 1$ can depend on $\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 }$ but not on $\\xi _ { t } , \\dots , \\xi _ { T - 1 }$ . Suppose that these vectors satisfy $\\| \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\| ^ { 2 } \\leq \\mathfrak { T } _ { j }$ or every $t = 1 , \\dots , T - 1$ . Then, with probability at least $1 - p ,$ , ", + "bbox": [ + 173, + 126, + 826, + 179 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/906cf685822d4e99239c9192719e45f6f34f058ec802f69f6d93ad2cd650ce5e.jpg", + "text": "$$\n\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 181, + 669, + 204 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof of Proposition 3.4. Using the identity formula6 ", + "bbox": [ + 176, + 215, + 490, + 229 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/033543a9e2d0e7c570956a509a551f90a74b038f4d51ed3baf87b4812a32529a.jpg", + "text": "$$\n\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle = \\left( \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right) \\left( \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right) - \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle = \\Delta _ { 0 } ^ { 2 } \\Delta _ { t } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 232, + 766, + 256 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "we have ", + "bbox": [ + 173, + 257, + 225, + 270 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9307e5d36c5cb0f6c1fa33fe3f6999cf01db532521918c99303003456b056e31.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta t \\rangle \\right| \\leq \\left\\| \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right\\| \\cdot \\left\\| \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right\\| + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| } \\\\ & { \\qquad \\leq O ( \\sqrt { d T ^ { \\sum } \\log ( T / p ) } ) + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\ . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 268, + 787, + 320 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the last inequality uses $\\| \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { T - 2 } \\| \\le O ( \\sqrt { d T \\log ( 1 / p ) } )$ with probability at least $1 - p / 2$ . Furthermore, we note that $\\xi _ { t }$ is independent of $\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 } , \\Delta _ { 1 } , \\ldots , \\Delta _ { t }$ and $\\mathbb { E } [ \\xi _ { t } ] = 0$ . Therefore, letting $S _ { t } = \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle$ , we have $\\mathbb { E } [ S _ { t } | \\xi _ { 0 } , \\dots , \\xi _ { t - 1 } ] = 0$ ; furthermore, with probability at least $1 - p / 2$ , it satisfies $| S _ { t } | \\leq O ( \\sqrt { d \\mathfrak { T } \\log ( T / p ) } )$ for every $t$ . Finally, by Azuma’s inequality, we have ", + "bbox": [ + 173, + 323, + 826, + 378 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/db940c92d791db75fe64be42e15a5df5fd004067d7b79dd9f36aa34ed1facaa6.jpg", + "text": "$$\n\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 381, + 666, + 406 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.4 PROOF OF LEMMA 3.5 ", + "text_level": 1, + "bbox": [ + 174, + 416, + 374, + 430 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof of Lemma 3.5. Let us denote by $\\begin{array} { r } { r _ { t } = \\frac { [ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } } { 2 } = - \\frac { [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 } } { 2 } } \\end{array}$ and we know $\\begin{array} { r } { r _ { t } \\sim \\mathcal { N } ( 0 , \\frac { \\nu ^ { 2 } } { 4 } ) } \\end{array}$ . We can write ", + "bbox": [ + 176, + 439, + 787, + 458 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a1f34a11af70377a67f341f5253d4a55947b8704dad796c1881cf38f5c3e6f83.jpg", + "text": "$$\nw _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } - \\eta ( \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } )\n$$", + "text_format": "latex", + "bbox": [ + 269, + 460, + 727, + 478 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Using the second-order smoothness, we have ", + "bbox": [ + 176, + 479, + 442, + 492 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/e6b952ac6a13dc351897ac31a436c7f5dffe8bfb39ca83bc1cb258ae0560d2b9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) = \\displaystyle \\int _ { \\tau = 0 } ^ { 1 } \\nabla ^ { 2 } f \\big ( { w } _ { t } ^ { \\mathsf { a } } + \\tau ( w _ { t } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { a } } ) \\big ) \\big ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\big ) d \\tau } \\\\ & { \\qquad = \\nabla ^ { 2 } f \\big ( { w } _ { 0 } \\big ) \\cdot \\big ( { w } _ { t } ^ { \\mathsf { a } } - { w } _ { t } ^ { \\mathsf { b } } \\big ) + \\theta _ { t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 494, + 702, + 545 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "for some vector $\\| \\theta _ { t } \\| \\leq \\operatorname* { m a x } \\{ \\| w _ { 0 } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { a } } \\| , \\| w _ { 0 } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { b } } \\| \\} \\cdot \\| w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\|$ . Therefore, we have ", + "bbox": [ + 176, + 547, + 709, + 564 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/30ff4dbae63bb63312ce28242825b7f508e71385711c567d855da5d5d81e5a41.jpg", + "text": "$$\nw _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 274, + 565, + 720, + 583 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now, giving $\\psi _ { 0 } = \\overline { { \\psi } } _ { 0 } = 0$ , imagine two sequences ", + "bbox": [ + 173, + 585, + 475, + 601 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• $\\psi _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) \\psi _ { t }$ and • $\\overline { { \\psi } } _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) \\overline { { \\psi } } _ { t } - \\eta \\big ( \\Xi _ { t } ^ { 3 } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } \\big ) = w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } }$ ", + "bbox": [ + 183, + 603, + 647, + 640 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We will inductively prove $\\lVert \\psi _ { t } - \\overline { { \\psi } } _ { t } \\rVert \\leq \\frac 1 2 \\lVert \\psi _ { t } \\rVert$ . On one hand, it is easy to see that $\\psi _ { t }$ is zero except in the first coordinate, in which it behaves as a Gaussian with zero mean and variance Pt−1k=0(1 + ηδ)2k · η2ν24 Θ \u0010 (1+ηδ)2tηδ · η2ν2\u0011. By Gaussian tail bounds, we know that ", + "bbox": [ + 174, + 642, + 825, + 696 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• with probability at least 0.99, it satisfies $\\begin{array} { r } { \\| \\psi _ { t } \\| \\leq O ( \\frac { \\sqrt { \\eta C _ { 1 } } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } ) } \\end{array}$ for every $t$ • with probability at least 0.99, it satisfies $\\begin{array} { r } { \\| \\psi _ { T } \\| \\ge \\frac { 1 } { 1 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { T } } { \\sqrt { \\delta } } \\big ) } \\end{array}$ ", + "bbox": [ + 181, + 700, + 661, + 746 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In the rest of the proof, we condition on this event happens. We prove towards contradiction by assuming $\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } ^ { \\mathsf { a } } \\| \\leq R$ and $\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } ^ { \\mathsf { b } } \\| \\leq R$ for all $t \\in [ T ]$ . ", + "bbox": [ + 171, + 750, + 823, + 776 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We will inductively prove that $\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}$ . We calculate the difference ", + "bbox": [ + 174, + 780, + 720, + 800 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/83d83c17d711d93c9eb740d9b432eb48f9f512cd2553ebe5e21bcc0a79f92e19.jpg", + "text": "$$\n\\psi _ { t } - { \\overline { { \\psi } } } _ { t } = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\left( \\mathbf { I } - \\eta { \\boldsymbol \\nabla } ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } ( { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { a } } - { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { b } } + { \\boldsymbol \\theta } _ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 331, + 803, + 671, + 840 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let g = $\\begin{array} { r } { g = \\frac { \\psi _ { t } - \\overline { { \\psi } } _ { t } } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| } } \\end{array}$ , then we can inner-product the above equation by vector $g$ , which gives ", + "bbox": [ + 173, + 844, + 702, + 869 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a650dcb471bce83dc72faedab74503e39191a292b0a2ab01a7b7dfb6ea9fa774.jpg", + "text": "$$\n\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\Big \\langle \\Xi _ { i } ^ { \\mathrm { a } } - \\Xi _ { i } ^ { \\mathrm { b } } + \\theta _ { i } , \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) ^ { t - 1 - i } g \\Big \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 178, + 866, + 557, + 905 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "6We thank an anonymous reviewer on openreview who pointed out this simpler proof to us. ", + "bbox": [ + 191, + 911, + 733, + 926 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cdf21f4c57748ce7ff780d02f57dd2438b5f85ee7d62154dde9928a0e85504ef.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overset { \\circledast } { \\leq } \\eta \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\left( \\left. \\Xi _ { i } ^ { \\mathsf { a } } - \\Xi _ { i } ^ { \\mathsf { b } } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. + R \\cdot O ( \\frac { \\sqrt { \\eta C _ { 1 } } \\nu \\left( 1 + \\eta \\delta \\right) ^ { i } } { \\sqrt { \\delta } } ) \\cdot ( 1 + \\eta \\delta ) ^ { t - 1 - i } \\right) } \\\\ & { \\leq \\eta \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\left. \\Xi _ { i } ^ { \\mathsf { a } } - \\Xi _ { i } ^ { \\mathsf { b } } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. + O ( R \\eta T \\frac { \\sqrt { \\eta C _ { 1 } } } { \\sqrt { \\delta } } \\nu ( 1 + \\eta \\delta ) ^ { t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 113, + 820, + 189 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the inequality $\\textcircled{1}$ uses $\\| \\theta _ { i } \\| \\leq R \\cdot \\| \\overline { { \\psi } } _ { i } \\| \\leq R \\cdot ( \\| \\psi _ { i } \\| + \\| \\psi _ { i } - \\overline { { \\psi } } _ { i } \\| )$ , $\\begin{array} { r l } { \\Vert \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\Vert } & { { } \\leq } \\end{array}$ $( 1 + \\eta \\delta ) ^ { t - 1 - i }$ , and the inductive assumption. Let us call $M = \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right)$ , and focus on ", + "bbox": [ + 173, + 194, + 825, + 227 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/09b1d9101bd28e7dee275dbbc230f22bc627008d052872758af55e104207bd2b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left| \\displaystyle \\sum _ { i = 0 } ^ { \\ell - 1 } \\left. \\Xi _ { i } ^ { 3 } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. \\right| } \\\\ & { = \\left| \\displaystyle \\langle \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } , g \\rangle + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\left. \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } , M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\right. \\right| } \\\\ & { \\leq \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } \\| \\cdot \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } \\| \\cdot \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\left( \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| \\right) } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\cdot ( 1 + \\eta \\delta ) ^ { t - 1 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 231, + 738, + 409 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Together, we have ", + "bbox": [ + 173, + 409, + 284, + 422 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3b7d722923f62180d19941de86c94aa7a00331c3eec18295687b3457afce6f05.jpg", + "text": "$$\n\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq O ( \\eta \\sqrt { T C _ { 2 } } ) \\cdot ( 1 + \\eta \\delta ) ^ { t } + O ( R \\eta T \\frac { \\sqrt { \\eta C _ { 1 } } } { \\sqrt { \\delta } } \\nu ( 1 + \\eta \\delta ) ^ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 292, + 426, + 704, + 457 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Under our assumption, we have $\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| < \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}$ and therefore $\\lVert \\overline { { \\psi } } _ { T } \\rVert \\ge \\lVert \\psi _ { T } \\rVert - \\lVert \\psi _ { T } - \\overline { { \\psi } } _ { T } \\rVert \\ge$ 0 ( √ην(1+ηδ)t√ ). Thus, within T iterations, we have kψtk > R and this gives a contradiction. \u0003 ", + "bbox": [ + 174, + 460, + 825, + 501 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B FINAL: DOUBLE SAFE GUARD ", + "text_level": 1, + "bbox": [ + 176, + 515, + 464, + 532 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We now come to our final Algorithm 3 which is our perturbed SGD algorithm with two safeguards. The two safeguard algorithm naturally divides itself into epochs, each consisting of $T _ { 1 }$ iterations. We will demonstrate that within most epochs, we make good progress. Thus, consider some iterate $x _ { m T _ { 1 } }$ , for some $m < T / T _ { 1 }$ . Our goal will be to argue that we make good function value progress by iterate $x _ { ( m + 1 ) T _ { 1 } }$ , and that we do not settle into any saddle points. To slightly simplify notation, let $w _ { 0 } = x _ { m T _ { 1 } }$ , and let the sequence of iterates be $w _ { 0 } , \\dots , w _ { T _ { 1 } - 1 }$ , so that $w _ { T _ { 1 } - 1 } = x _ { ( m + 1 ) T _ { 1 } - 1 }$ . For completeness’ sake we rewrite this as Algorithm 1. ", + "bbox": [ + 173, + 541, + 826, + 621 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Algorithm 3 Perturbed SGD with double safe guard (for analysis purpose) ", + "text_level": 1, + "bbox": [ + 176, + 647, + 665, + 662 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Input: $w _ { 0 } \\in \\mathbb { R } ^ { d }$ , set $\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }$ , rate $\\eta > 0$ , lengths $T _ { 1 } \\ge T _ { 0 } \\ge 1$ , threshold $\\mathfrak { T } _ { 1 } > \\mathfrak { T } _ { 0 } > 0$ ; \n1: for $t \\gets 0$ to $T _ { 1 } - 1$ do \n2: $l a s t \\gets \\operatorname* { m a x } \\{ t _ { 0 } \\in [ t ] \\colon t _ { 0 }$ is a multiple of $T _ { 0 } \\}$ \n3: for each $i \\in \\mathtt { g o o d } _ { t }$ do \n4: receive $\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }$ from machine $i$ ; \n5: Ai ← Ptk=0 ∇k,i|good | and Bi ← Ptk=last ∇k,i|good | ; \n6: $A _ { \\sf m e d } A _ { i }$ where $i \\in { \\mathfrak { g o o d } } _ { t }$ is any machine s.t. $\\left| \\left\\{ j \\in \\operatorname { g o o d } _ { t } \\colon \\left\\| A _ { j } - A _ { i } \\right\\| \\leq \\mathfrak { T } _ { 1 } \\right\\} \\right| > m / 2$ . \n7: $B _ { \\sf m e d } B _ { i }$ where $i \\in \\mathtt { g o o d } _ { t }$ is any machine s.t. $\\left| \\left\\{ j \\in \\mathbf { g 0 0 d } _ { t } \\colon \\left\\| B _ { j } - B _ { i } \\right\\| \\le \\mathfrak { T } _ { 0 } \\right\\} \\right| > m / 2$ . \n8: $\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } : \\| A _ { i } - A _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 1 } \\bigwedge \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 0 } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\left( \\xi _ { t } + \\frac { 1 } { \\vert \\mathsf { g o o d } _ { t } \\vert } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\right) ; } \\end{array}$ ; \n9: ", + "bbox": [ + 176, + 667, + 825, + 837 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our main result is the following theorem. ", + "bbox": [ + 174, + 864, + 418, + 878 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Theorem B.1. Let $C _ { 3 } = \\alpha ^ { 2 } + { \\textstyle { \\frac { 1 } { m } } }$ . Suppose we pick parameters $p , \\delta \\in ( 0 , 1 )$ , $\\begin{array} { r } { \\eta \\le \\widetilde { O } ( \\frac { \\delta ^ { 3 } } { C _ { 3 } } ) } \\end{array}$ , $\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )$ $\\begin{array} { r } { T _ { 0 } = \\widetilde { \\Theta } \\left( \\frac { 1 } { \\eta } \\right) } \\end{array}$ , $\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } \\big ( \\frac { 1 } { \\eta \\delta } \\big ) \\geq T _ { 0 } } \\end{array}$ , $\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )$ , and ${ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )$ . Then, starting from $w _ { 0 }$ , ", + "bbox": [ + 173, + 883, + 821, + 921 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) with probability at least $1 - p$ we have ", + "bbox": [ + 171, + 104, + 429, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/f5ce478307ac4462e58ec1c9150577d414608ca2b228c375ee251c03bc716873.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - \\widetilde { O } ( \\eta C _ { 3 } d ) \\right) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 121, + 694, + 160 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(b) As long as $\\| w _ { t } - w _ { 0 } \\| \\ge R$ for some $t \\in \\{ 1 , 2 , \\dots , T _ { 1 } \\}$ and for $\\begin{array} { r } { R = \\widetilde { \\Theta } ( \\delta ) \\leq \\frac { \\delta } { 2 } } \\end{array}$ , then with probability at least $1 - p$ we have then ", + "bbox": [ + 173, + 166, + 823, + 196 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/675f751a4d41cac9c171475e47e413370d537b4f7b8feb6cab903b0bd7b475ae.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\Big ( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\Big ) + \\widetilde { \\Omega } ( \\delta ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 198, + 676, + 238 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(c) i $f \\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( w _ { 0 } ) ) \\leq - \\delta$ , we also have with probability at least 0.45, ", + "bbox": [ + 173, + 244, + 607, + 261 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5c4a89bd641d5f843ca361f5283db612d80b5f7444d48b02dcfbd3d86ffd0e3f.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\Big ( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\Big ) + \\widetilde { \\Omega } ( \\delta ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 262, + 674, + 303 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.1 WHY THEOREM B.1 IMPLIES THEOREM 2.3 ", + "text_level": 1, + "bbox": [ + 173, + 311, + 522, + 327 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Using the parameter choice $\\begin{array} { r } { \\eta = \\widetilde { \\Theta } ( \\frac { \\varepsilon ^ { 2 } } { C _ { 3 } d } ) } \\end{array}$ from Theorem 2.3, we know $\\widetilde { \\cal O } ( \\eta C _ { 3 } d ) \\le 0 . 1 \\varepsilon ^ { 2 }$ . We claim two things: ", + "bbox": [ + 176, + 334, + 823, + 366 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• For at least $9 0 \\%$ of the epochs, they must satisfy (denoting by $w _ { 0 }$ and $w _ { T _ { 1 } }$ the beginning and ending points of this epoch) ", + "bbox": [ + 181, + 368, + 821, + 395 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/552df0d7326fc1432351a7449bcc1c89b32ce7dcbd3a5278a619a999fedd3633.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\leq 2 0 \\frac { f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } \\leq \\varepsilon ^ { 1 . 5 }\n$$", + "text_format": "latex", + "bbox": [ + 361, + 395, + 660, + 425 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The last inequality uses our choice of $T$ and $\\delta = \\widetilde { \\Theta } ( \\sqrt { \\varepsilon } )$ . ", + "bbox": [ + 197, + 429, + 535, + 443 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The reason for this is by way of contradiction. Suppose for at least $1 0 \\%$ of the epochs it satisfies $\\begin{array} { r } { f ( w _ { 0 } ) \\ : - \\ : f ( w _ { T _ { 1 } } ) > 2 0 \\frac { \\dot { f } ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } } \\end{array}$ , then, for the remainder of the epochs, they must at least satisfy $f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq - 0 . 7 \\eta T _ { 1 } \\cdot 0 . 1 \\varepsilon ^ { 2 }$ . Summing over all the epochs, we shall obtain $f ( x _ { 0 } ) - f ( x _ { T } ) >$ $f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x )$ but this gives a contradiction. ", + "bbox": [ + 196, + 444, + 825, + 502 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• For at least $4 0 \\%$ of the epochs, they must satisfy the three properties from Theorem B.1. ", + "bbox": [ + 184, + 502, + 717, + 517 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In particular, for at least $3 0 \\%$ of the epochs, they must satisfy both. Since $\\varepsilon ^ { 1 . 5 }$ is so small that ", + "bbox": [ + 171, + 518, + 725, + 534 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/4582c309c5cb7a8c589c4f9111e26a8cd1c1496ae52ed095ec5a0cbce50b38b9.jpg", + "text": "$$\n\\varepsilon ^ { 1 . 5 } \\ge f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\right) + \\widetilde { \\Omega } ( \\delta ^ { 3 } ) \\ge \\widetilde { \\Omega } ( \\delta ^ { 3 } ) - 0 . 0 5 \\eta T _ { 1 } \\varepsilon ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 241, + 536, + 754, + 575 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "would give a contradiction (for instance, one can choose $\\delta$ to be slightly larger than $\\sqrt { \\varepsilon }$ by some log factors), this means, for those $3 0 \\%$ of the epochs, they must satisfy: ", + "bbox": [ + 173, + 578, + 823, + 606 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• $\\begin{array} { r } { \\varepsilon ^ { 1 . 5 } \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - 0 . 1 \\varepsilon ^ { 2 } \\right) \\ , } \\end{array}$ , • $\\begin{array} { r } { \\| w _ { t } - w _ { 0 } \\| \\leq \\frac { \\delta } { 2 } } \\end{array}$ for every $t = 1 , 2 , \\dots , T _ { 1 }$ , and • $\\nabla ^ { 2 } f ( w _ { 0 } ) \\succeq - \\delta \\mathbf { I } .$ . ", + "bbox": [ + 183, + 607, + 483, + 659 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The latter two properties together implies $\\nabla ^ { 2 } f ( w _ { t } ) \\succeq - \\frac { \\delta } { 2 } \\mathbf { I }$ for every $t = 1 , 2 , \\ldots , T _ { 1 }$ (by the second-order smoothness). The first property implies for at least $9 0 \\%$ of the iterations $t$ in this epoch, they must satisfy $\\| \\nabla f ( x ) \\| \\leq \\varepsilon$ . This finishes the proof of Theorem 2.3. \u0004 ", + "bbox": [ + 174, + 662, + 825, + 705 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.2 PROOF OF THEOREM B.1 ", + "text_level": 1, + "bbox": [ + 174, + 717, + 392, + 731 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We first have the following lemma ", + "bbox": [ + 174, + 737, + 379, + 751 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma B.2 (double safe guard). In Algorithm 3, suppose $\\mathfrak { T } _ { 1 } ~ = ~ 8 \\sqrt { T _ { 1 } \\log ( 1 6 m T _ { 1 } / p ) }$ and $\\mathfrak { T } _ { 0 } ~ =$ $8 \\sqrt { T _ { 0 } \\log ( 1 6 m T _ { 1 } / p ) }$ . Then, with probability at least $1 - p / 2 ,$ , for every $t = 0 , \\ldots , T _ { 1 } - 1$ , ", + "bbox": [ + 174, + 756, + 823, + 787 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "• $\\mathbf { g } _ { 0 0 } \\mathbf { d } _ { t } \\supseteq \\mathbf { g } _ { 0 0 } \\mathbf { d }$ . \n• • $\\begin{array} { r l } & { | \\sigma _ { t } | | ^ { 2 } \\leq O ( \\frac { \\log ( T _ { 1 } / p ) } { m } ) , \\| \\Delta _ { t } \\| ^ { 2 } \\leq \\alpha ^ { 2 } , \\| \\xi _ { t } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T _ { 1 } / p ) ) , } \\\\ & { | \\sigma _ { 0 } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 1 } \\log ( T _ { 1 } / p ) } { m } ) , \\| \\sigma _ { t a s t } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 0 } \\log ( T _ { 1 } / p ) } { m } ) } \\\\ & { | \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 1 } \\log ( m T _ { 1 } / p ) ) a n d \\| \\Delta _ { l a s t } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 0 } \\log ( m T _ { 1 } / p ) ) } \\\\ & { | \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 1 } \\log ( T _ { 1 } / p ) ) a n d \\| \\xi _ { l a s t } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 0 } \\log ( T _ { 1 } / p ) ) . } \\\\ & { \\quad \\cdot \\quad \\cdot \\quad \\cdot \\quad \\cdot } \\end{array}$ • \n• ", + "bbox": [ + 181, + 790, + 813, + 890 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We call this probabilistic event Event $\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )$ and $\\operatorname* { P r } [ \\mathsf { E v e n t } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ) ] \\geq 1 - p / 2$ . ", + "bbox": [ + 174, + 883, + 681, + 901 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The proof is a direct corollary of Lemma 3.2, by combining events Even $\\mathbf { \\cdot } _ { T _ { 1 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )$ , Event $\\mathbf { \\cdot } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )$ ", + "bbox": [ + 171, + 909, + 820, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "EvenLemm $\\mathbf { \\cdot } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { T _ { 0 } } )$ , ro $\\mathsf { E v e n t } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { 2 T _ { 0 } } )$ and so on. The next lemma is a simple corollary by repeatedly applyinga. ", + "bbox": [ + 173, + 103, + 823, + 131 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "LemmaSuppose e, by a $C _ { 1 } = \\log ( T _ { 1 } / p )$ and start $\\begin{array} { r } { C _ { 2 } = \\alpha ^ { 2 } \\log \\frac { m T _ { 1 } } { p } + \\frac { \\log ( T _ { 1 } / p ) } { m } } \\end{array}$ .. $\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}$ $\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}$ $T _ { 1 } \\geq T _ { 0 }$ $w _ { 0 }$ Under event Even $\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )$ , it satisfies ", + "bbox": [ + 174, + 136, + 825, + 185 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d60869e06ea14288f79f8940a2cbea5df5826aa6f03ccfb49c175ded6b3f1f0f.jpg", + "text": "$$\nf ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 214, + 189, + 782, + 229 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The next lemma can be easily derived from Lemma 3.5. ", + "bbox": [ + 174, + 236, + 503, + 250 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma B.4 (modified from Lemma 3.5). Suppose ", + "bbox": [ + 173, + 253, + 478, + 267 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/31b7245e824074578fbf95553f57a5df19d563db350646d4ccc23cab8452b24a.jpg", + "text": "$$\nR = \\Theta ( \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( \\delta ^ { 3 } / \\eta C _ { 2 } ) } ) a n d \\nu ^ { 2 } = \\Theta ( C _ { 2 } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } )\n$$", + "text_format": "latex", + "bbox": [ + 320, + 271, + 678, + 303 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Suppose $\\begin{array} { r } { \\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\lbrace 1 , \\frac { \\delta ^ { 3 } } { C _ { 2 } } \\rbrace } \\end{array}$ , $\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}$ and $\\begin{array} { r } { T _ { 1 } = \\Theta ( \\frac { 1 } { \\eta \\delta } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } ) \\geq T _ { 0 } } \\end{array}$ . Let $\\boldsymbol { w _ { 0 } } \\in \\mathbb { R } ^ { d }$ be any point in the space and suppose $\\lambda _ { \\operatorname* { m i n } } \\bigl ( \\nabla ^ { 2 } f ( w _ { 0 } ) \\bigr ) \\leq - \\delta$ for some $\\delta \\geq 0$ . Given two coupled sequences defined as before, under events Event $\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ^ { \\mathsf { a } } )$ and Event $\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ^ { \\mathsf { b } } )$ , we have with probability at least 0.98 ", + "bbox": [ + 174, + 306, + 825, + 356 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/9f1aab65008332c4ba73404ea5369c81fb947e771b72169aa7dbadb4c75b3616.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\operatorname* { m a x } \\left\\{ f ( w _ { 0 } ^ { \\mathsf { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathsf { a } } ) , f ( w _ { 0 } ^ { \\mathsf { b } } ) - f ( w _ { T _ { 1 } } ^ { \\mathsf { b } } ) \\right\\} } \\ ~ } \\\\ { \\displaystyle \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\Omega ( \\frac { \\delta ^ { 3 } } { C _ { 1 } \\log ^ { 3 } \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 358, + 754, + 419 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma B.4 directly proves the second half of Theorem B.1c, because given two coupled sequences with the same marginal distribution, we have ", + "bbox": [ + 171, + 424, + 826, + 450 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ac1ca7536fd15a004e1bffc1afa24097042bc9aad4fcf8cc39bdaaec7d19a0cf.jpg", + "text": "$$\n\\operatorname* { P r } [ f ( w _ { 0 } ^ { \\mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { a } } ) \\ge X ] \\ge \\frac { 1 } { 2 } \\operatorname* { P r } [ \\operatorname* { m a x } \\left\\{ f ( w _ { 0 } ^ { \\mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { a } } ) , f ( w _ { 0 } ^ { \\mathrm { b } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { b } } ) \\right\\} \\ge X ]\n$$", + "text_format": "latex", + "bbox": [ + 240, + 454, + 758, + 481 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof of Lemma B.4. Our choice on $r$ and $R$ satisfy the requirement of Lemma 3.5. Suppose without loss of generality that the $w _ { t } ^ { \\mathsf { a } }$ sequence leaves $w _ { 0 }$ by more than $R$ . Let $T _ { 1 } ^ { \\mathsf { a } }$ be the first iteration $t \\leq T _ { 1 }$ in which $\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } ^ { \\mathsf { a } } \\| \\geq R$ . ", + "bbox": [ + 176, + 508, + 825, + 550 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/01a014c680c9853ae0eb278f04660b7558cc3bfa95b5d84e0c92d220132950aa.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\| w _ { T _ { 1 } ^ { \\mathtt { a } } } ^ { \\mathtt { a } } - w _ { 0 } ^ { \\mathtt { a } } \\| ^ { 2 } = \\eta ^ { 2 } \\| \\nabla f ( w _ { 0 } ^ { \\mathtt { a } } ) + \\cdot \\cdot \\cdot + \\nabla f ( w _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } ) + \\Xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\Xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } + \\xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } \\| ^ { 2 } } } \\\\ { { \\displaystyle \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } } \\\\ { { \\displaystyle \\le O ( \\eta ^ { 2 } T _ { 1 } ) \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\mathtt { a } } - 1 } \\| \\nabla f ( w _ { t } ^ { \\mathtt { a } } ) \\| ^ { 2 } + O ( C _ { 2 } \\eta ^ { 2 } T _ { 1 } ) + O ( C _ { 1 } \\eta ^ { 2 } \\nu ^ { 2 } T _ { 1 } d ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 550, + 787, + 614 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Combining this with Lemma B.3, we have ", + "bbox": [ + 174, + 616, + 426, + 630 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/9a3c48a8ab46594dc021223bd4ac9c34aa958f9ae2c84d06fd0eb4f6694271c4.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\lvert \\left( w _ { 0 } ^ { 3 } \\right) - f \\left( w _ { T _ { 1 } ^ { 3 } } ^ { \\mathfrak { a } } \\right) \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { \\lVert w _ { T _ { 1 } ^ { \\flat } } ^ { \\mathfrak { a } } - w _ { 0 } ^ { \\mathfrak { a } } \\rVert } { 1 0 0 \\eta T _ { 1 } } } \\\\ { \\displaystyle \\qquad \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 632, + 839, + 717 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Combining this with Lemma B.3 again but for the remainder iterations, we have ", + "bbox": [ + 173, + 718, + 648, + 732 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/deb8ffcd9383de8a3eda38953948f33f4d103572a2b900e77ba78203442ccbfa.jpg", + "text": "$$\nf ( w _ { 0 } ^ { 3 } ) - f ( w _ { T _ { 1 } } ^ { 3 } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ^ { 3 } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 733, + 823, + 775 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In fact, the above same proof of Lemma B.4 also implies Theorem B.1b. These together finish the proof of Theorem B.1. \u0004 ", + "bbox": [ + 173, + 824, + 828, + 854 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C MORE ON EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 871, + 416, + 887 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We conduct experiments on training a residual network ResNet-20 He et al. (2016) on the CIFAR-10/100 image classification tasks Krizhevsky et al. (2014). ", + "bbox": [ + 173, + 897, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.1 SETTING AND IMPLEMENTED METHODS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 496, + 117 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In all of our experiments, we use 10 workers and mini-batch size 10 per worker. Given any attacker and any defender algorithm, we run SGD three times for 140 epochs, each time with a different initial learning rate $\\eta \\in \\{ 0 . 1 , 0 . 2 , 0 . 4 \\}$ .7 We let the learning rate decrease by a factor of 10 on epochs 80 and 110, and present present the best testing accuracies in the three runs (each corresponding to a different initial learning rate). ", + "bbox": [ + 173, + 123, + 825, + 175 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We use standard data augmentation (random crops, random flips, and channel normalization). ", + "bbox": [ + 178, + 179, + 723, + 193 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We compare against Geometric Median Chen et al. (2017), Coordinate-wise Median Yin et al. (2018; 2019), Krum Blanchard et al. (2017), and Zeno Xie et al. (2018b) with attacks. We set $\\alpha = 0 . 4$ so there are 4 Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.) We formally define those prior works as follows. ", + "bbox": [ + 174, + 195, + 825, + 247 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.1 (GeoMed Chen et al. (2017)). The geometric median of $\\{ y _ { 1 } , . . . , y _ { m } \\}$ , denoted by $g e o \\_ m e d \\{ y _ { 1 } , . . . , y _ { m } \\}$ , is ", + "bbox": [ + 173, + 252, + 823, + 280 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b7976376997ce922fcbac75cb86e0a9ae225dbb6a492f3ccd9fb8747facd20cb.jpg", + "text": "$$\n\\begin{array} { r } { g e o \\ L _ { - } m e d \\{ y _ { 1 } , . . . , y _ { m } \\} : = \\arg \\operatorname* { m i n } _ { y \\in \\mathbb { R } ^ { d } } \\sum _ { i = 1 } ^ { m } \\| y - y _ { i } \\| } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 281, + 665, + 299 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In our experiments, we choose the geometric median from set $\\{ y _ { 1 } , . . . , y _ { m } \\}$ . ", + "bbox": [ + 169, + 301, + 619, + 315 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.2 (coordinate-wise median Yin et al. (2018; 2019)). Coordinate-wise median $g \\quad =$ $m e d \\{ y _ { 1 } , . . . , y _ { m } \\}$ is defined as a vector with its $k$ -th coordinate being $g [ k ] = m e d \\{ y _ { 1 } [ k ] , . . . , y _ { m } [ k ] \\}$ for each $k \\in [ d ]$ , where med is the usual (one-dimensional) median. ", + "bbox": [ + 173, + 320, + 826, + 359 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.3 (Krum Blanchard et al. (2017)). ", + "bbox": [ + 174, + 363, + 454, + 377 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b7da2aecba71433a34e3055cc290be30ac33049b0d8f490fd7678b21c6419f4b.jpg", + "text": "$$\nK R \\{ y _ { 1 } , . . . , y _ { m } \\} : = y _ { k } \\quad w h e r e \\quad k = \\underset { i \\in [ m ] } { \\arg \\operatorname* { m i n } } \\sum _ { i \\to j } \\| y _ { i } - y _ { j } \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 308, + 381, + 687, + 411 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "and $i j$ is the indices of the $m - b - 2$ nearest neighbours of $y _ { i }$ in $\\{ y 1 , . . . , y _ { m } \\} \\backslash \\{ y _ { i } \\}$ by Euclidean distances. ", + "bbox": [ + 166, + 415, + 825, + 430 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Note that Krum requires $2 b + 2 < m$ . So, we have also repeated the experiments for Krum with 3 Byzantine workers (out of 10 workers) to be more fair. ", + "bbox": [ + 173, + 436, + 825, + 462 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.4 (Zeno Xie et al. (2018b)). ", + "bbox": [ + 174, + 467, + 418, + 481 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/188567243fdce1e8046be6ee3d4ad7a0725d581e6074ebf14fa63a857740b3bf.jpg", + "text": "$$\nZ e n o _ { b } \\{ y _ { 1 } , . . . , y _ { m } \\} = \\frac { 1 } { m - b } \\sum _ { i = 1 } ^ { m - b } \\widetilde { y } ( i )\n$$", + "text_format": "latex", + "bbox": [ + 380, + 484, + 617, + 522 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\{ \\widetilde { y } ( i ) : i \\in [ m ] \\}$ are the gradient estimators with the $m - b$ highest “scores”, and the so-called estochastic descendant score for any gradient estimator $u$ , based on the current parameter $x$ , learning rate $\\eta _ { : }$ , and a constant weight $\\rho > 0$ , is defined as: ", + "bbox": [ + 174, + 525, + 825, + 564 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/f2957468c08b0590e3d0fce50ba76472b832024fc6f3e7083ef8697ba2b43195.jpg", + "text": "$$\nS c o r e _ { \\eta , \\rho } ( u , x ) = f _ { r } ( x ) - f _ { r } ( x - \\eta u ) - \\rho \\| u \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 352, + 568, + 645, + 585 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$f _ { r } ( x ) - f _ { r } ( x - \\eta u )$ is the estimated descendant of the loss function and $\\rho \\| u \\| ^ { 2 }$ is the magnitude of the update. ", + "bbox": [ + 174, + 587, + 820, + 602 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In our experiments, we let $f _ { r } ( x )$ be the estimated objective over a mini-batch of size $n _ { r } = 1 0$ (so the time to perform this estimation is on the same magnitude as the gradient evaluation for each individual worker). We also chose $\\rho = 0 . 0 0 0 5$ (and this value does not affect our experimental results by much). ", + "bbox": [ + 174, + 608, + 821, + 648 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Safeguard SGD. Our Algorithm 1 is stated in a way to make our theoretical proofs as clean as possible. Here, we discuss how we actually implement it in practice. ", + "bbox": [ + 171, + 651, + 823, + 678 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "First of all, as common in the literature, we omit the Gaussian noise $\\xi _ { t }$ that is added for theoretical purpose, and instead rely on the natural noise in the training process to escape saddle points. ", + "bbox": [ + 171, + 681, + 821, + 707 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Also, we make universal choices for our safeguard window sizes (across all attackers): for our algorithm with a single safeguard we have used a universal window size $T = 3$ epochs, and for our algorithm with double safeguards we have used window sizes $T _ { 0 } = 1$ epoch and $T _ { 1 } = 6$ epochs. ", + "bbox": [ + 174, + 710, + 825, + 750 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We also provide an automatic empirical process to select safeguard thresholds and eliminate bad workers.8 The process to determine $A _ { \\mathrm { m e d } }$ (and likewise for $B _ { \\mathrm { m e d . } }$ ) is described as follows. In each iteration, for every worker $\\hat { i } \\in [ m ]$ , we sort $\\left\\{ \\Vert A _ { i } - A _ { j } \\Vert \\right\\} _ { j \\in [ m ] }$ and pick the smallest $\\lceil m / 2 + 1 \\rceil$ -th entry, and let this number be the “score” for worker $_ { i }$ . We select the worker with the smallest “score” as $A _ { \\mathrm { m e d } }$ and call its “score” $S$ . Then, we use $1 . 5 \\mathrm { m i n } \\{ S , 5 \\}$ as the safeguard threshold for this iteration. Namely, we declare any worker $j$ satisfying ", + "bbox": [ + 173, + 753, + 825, + 820 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$\\| A _ { j } - A _ { \\sf m e d } \\| \\geq 1 . 5 \\operatorname* { m a x } \\{ S , 5 \\}$ as a bad worker.9 ", + "bbox": [ + 176, + 128, + 468, + 143 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2 EXPERIMENT RESULTS BY ATTACKS ", + "text_level": 1, + "bbox": [ + 176, + 155, + 470, + 169 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The ideal accuracies are $9 1 . 7 \\% / 6 8 . 0 \\%$ for CIFAR-10/100, which correspond to applying SGD using only the stochastic gradients from the honest workers. Below we discuss about the experimental results one attack at a time. ", + "bbox": [ + 174, + 175, + 825, + 213 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2.1 VARIANCE ATTACK ", + "text_level": 1, + "bbox": [ + 174, + 226, + 366, + 239 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We begin by looking at the hardest proposed attack from prior works. The Variance attack follows the strategy prescribed by Baruch et al. (2019), by which Byzantine workers collude in order to shift the mean among all gradients by a factor $\\beta$ times the standard deviation of the gradient, while staying within population variance. More precisely, the maximal change to the mean that can be applied by an attacker without the fear of being detected, by using the properties of the Normal distribution, specifically the cumulative standard normal function, and compute the maximal possible shift so that the attackers’s values stay within population variance. (See (Baruch et al., 2019, Algorithm 3) for a precise description. Our $\\beta$ is $z _ { \\mathrm { m a x } }$ in their notation.) We implement this strategy coordinate-wise, the same way as they did. Their work observes that the shift $\\beta$ can be non-trivial in practice, since stochastic gradients tend to have large variance in neural network training (which we also observed in our setup). Critically, the attack cannot be defended against by historyless algorithms, as the attacker’s values are statistically indistinguishable from a regular execution in a single iteration. ", + "bbox": [ + 174, + 246, + 825, + 386 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In our setting, for 10 total nodes and $\\alpha = 0 . 4$ , $\\beta$ is upper bounded by 0.3 (following the classic tables for the cumulative normal). We also ran the same attack in the setup from their paper (50 nodes total, of which 24 are Byzantine, which allows $\\beta \\sim 1 . 5 )$ and observed a similar outcome. Results for this experiment are given in Figure 3. ", + "bbox": [ + 173, + 388, + 825, + 440 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/64aff80ab494c66dd3b961892259a7e29da2e3c1faa1ace4ae36ced5717be94e.jpg", + "image_caption": [ + "Figure 3: Performance comparison under the variance attack. " + ], + "image_footnote": [], + "bbox": [ + 169, + 458, + 825, + 625 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "As shown by the results, our algorithm provably circumvents this attack, and recovers full accuracy. This is explained by the fact that the algorithm has memory: in particular, Byzantine nodes following this strategy will progressively diverge from the (honest) “median” $A _ { \\mathrm { m e d } }$ (at a “linear” rate, recall Figure 2(a)), and therefore will eventually exceed the threshold and be marked as malicious by the algorithm. ", + "bbox": [ + 173, + 665, + 825, + 715 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Specifically, both variants of the algorithm successfully catch all the bad nodes after at most 150 iterations. Indeed, at the 100-th iteration, the pair-wise distances $\\| A _ { i } - A _ { j } \\|$ among good workers $i , j \\in$ good are between 5.3 and 6.3, but the pair-wise distance between a good and a bad worker is at least 12.5. ", + "bbox": [ + 174, + 719, + 823, + 758 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2.2 SIGN-FLIPPING ATTACK ", + "text_level": 1, + "bbox": [ + 176, + 770, + 398, + 784 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We next move onto the sign-flipping attack. Recall that, in a sign-flipping attack, each Byzantine worker sends the negative gradient to the master. This is still a strong attack since if one does not avoid any bad workers, the test accuracy will suffer from a significant drop. The results are in Figure 4. ", + "bbox": [ + 176, + 790, + 821, + 829 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "From the plots, one can see that again our single and double safe-guard algorithms both outperform prior works. They also successfully catch all the bad workers within 150 iterations. (For instance, at iteration 150 for CIFAR-10 training, the distance $\\| A _ { \\sf m e d } - A _ { j } \\|$ for a good worker $j \\in \\mathtt { g o o d }$ is at most 6.9, but for a bad ", + "bbox": [ + 176, + 833, + 823, + 872 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/0830d7ad719c3952f22957fe4c980b78a111cbf63eee160f2e9a694efee17c31.jpg", + "image_caption": [ + "Figure 4: Performance comparison under the sign-flipping attack. " + ], + "image_footnote": [], + "bbox": [ + 169, + 102, + 826, + 270 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "worker $j \\not \\in$ good it can be more than 11.) ", + "bbox": [ + 176, + 325, + 421, + 338 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In contrast, prior work Zeno completely fails because locally at a training step, using merely $n _ { r } = 1 0$ samples to evaluate the objective, it is statistically not possible to even distinguish if the sign of the stochastic gradient is flipped or not. For prior works Krum and GeoMedian, although they appear to have some non-negligible performances, but they are actually no better than simply applying SGD with the naive mean of gradients from all the workers (including those from bad workers).10 Therefore, we conclude that prior works all fail to be Byzantine fault tolerant under this attack. ", + "bbox": [ + 174, + 343, + 825, + 417 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.2.3 DELAYED-GRADIENT ATTACK ", + "text_level": 1, + "bbox": [ + 176, + 430, + 441, + 444 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Recall that, in a delayed-gradient attack, each Byzantine worker sends an old gradient to the master. In our experiments, the delay is of $D = 1 0 0 0$ iterations ${ \\it \\Delta \\phi } = 2 { \\it \\Delta \\Psi }$ epochs). We believe this is not a very strong attack, because delayed gradients are not sufficiently malicious: they are still “correct” to certain extent albeit being delayed. The results are shown in Figure 5. ", + "bbox": [ + 173, + 450, + 825, + 502 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/f098289e873c796a5efe146a5bd454d7949f3336d83227ba94172db45a7fec07.jpg", + "image_caption": [ + "Figure 5: Performance comparison under the delayed-gradient attack. " + ], + "image_footnote": [], + "bbox": [ + 169, + 518, + 826, + 685 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "From the plots, one can see that our single and double safe-guard algorithms again both match the ideal accuracies. All the prior works suffer from a significant performance loss under this attack. ", + "bbox": [ + 173, + 724, + 823, + 751 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "It is worth noting that our single and double safe-guard algorithms do not catch any bad worker under this attack, so they simply use the “naive mean” of gradients from all the workers (including those delayed gradients from bad workers). However, there is no performance loss even if we use those delayed gradients. That is why we believe the delayed-gradient attack is not very strong, as the gradients are not sufficiently malicious. ", + "bbox": [ + 174, + 755, + 825, + 805 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Prior work Zeno suffers from some performance loss, because it only uses 6 workers out of 10, in which statistically only $6 \\times 0 . 6 \\approx 3 \\sim 4$ gradients are correct.11 Other prior works suffer from performance loss, because they only pick one single stochastic gradient from the 10 workers, and it is sometimes even from the bad worker. ", + "bbox": [ + 176, + 809, + 823, + 835 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 117, + 823, + 142 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.2.4 LABEL-FLIPPING ATTACK", + "text_level": 1, + "bbox": [ + 176, + 155, + 411, + 169 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Recall that, in the label-flipping attack, each Byzantine worker computes its gradient based on the cross-entropy loss with flipped labels: for CIFAR-10, label $\\ell \\in \\{ 0 , . . . , 9 \\}$ is flipped to $9 - \\ell$ , and for CIFAR-100, label $\\ell$ is flipped to $9 9 - \\ell$ . The results are shown in Figure 6. ", + "bbox": [ + 173, + 176, + 825, + 214 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/7db635d9c4932b3527dd2bd95505447e076401213199725b8306f27dfeae79d7.jpg", + "image_caption": [ + "Figure 6: Performance comparison under the label-flipping attack. " + ], + "image_footnote": [], + "bbox": [ + 169, + 233, + 826, + 398 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "From the plots, one can see that our single and double safe-guard algorithms even outperform the “ideal accuracies.” $9 2 . 4 \\%$ accuracy vs “ideal accuracy” $9 1 . 7 \\%$ under CIFAR-10; $6 9 . 4 \\%$ accuracy vs “ideal accuracy” 68.0 under CIFAR-100.) In addition, we have found out that the safeguard algorithms did not catch any bad worker. This should not be surprising, since label-flipping (a.k.a. label smoothing) is known to be a regularization technique to actually improve test accuracy, as opposed to hurt performance. ", + "bbox": [ + 173, + 440, + 825, + 505 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Zeno also performs well under this attack (but it does not outperform the ideal accuracy). We have investigated into Zeno, and found out that it cannot distinguish good workers from bad workers under label-flipping attack; and therefore Zeno effectively always runs under 6 random workers as opposed to using the full power of the $m = 1 0$ workers (recall Zeno picks 6 workers with the topmost scores, see Definition C.4). This explains its (minor) under-performance comparing to safeguard. ", + "bbox": [ + 174, + 508, + 825, + 571 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Other prior works perform significantly worse, and this should be alarming since label-flipping is one type of smoothing technique to improve test accuracy, as opposed to an actual “attack” to hurt performance. ", + "bbox": [ + 176, + 575, + 825, + 602 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.2.5 SAFEGUARD ATTACKS ", + "text_level": 1, + "bbox": [ + 174, + 613, + 387, + 627 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Finally, in the safeguard attack that we design, Byzantine workers send negative but re-scaled gradient to the master. We choose the re-scale factor so that it hardly triggers the safe-guard conditions at the master. From our experiment, choosing the re-scale factor as 0.6 in all the cases do not trigger the safe-guard conditions, while choosing a re-scale factor as 0.7 enables the algorithm to catch Byzantine workers occasionally. Our results are shown in Figure 7 (for re-scale factor 0.6) and Figure 8 (for re-scale factor 0.7). ", + "bbox": [ + 173, + 633, + 825, + 698 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/a4d787756f71e5c67aa9104e5787baf0489820e638d83faedff07126f19b5336.jpg", + "image_caption": [ + "Figure 7: Performance comparison under the safeguard attack with re-scale factor 0.6. (Recall this attack is designed to maximally impact the performance of our algorithm.) " + ], + "image_footnote": [], + "bbox": [ + 169, + 717, + 826, + 883 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Re-scale factor 0.6. In Figure 7, the performance of our (single and double) safeguard algorithms indeed get hurt a bit. Recall in Figure 7 the re-scale factor 0.6 is chosen to maximally impact our algorithm. The test accuracy drops from $9 1 . 7 \\%$ to $8 9 . 3 \\%$ under CIFAR-10; and drops from $6 8 . 0 \\%$ to $6 \\mathrm { { 0 . 0 \\% } }$ under CIFAR-100 (for both single and double safeguards). In these cases, we confirm that both versions of the safeguard algorithms did not catch any bad worker. However, this still significantly outperforms all prior works. ", + "bbox": [ + 173, + 104, + 825, + 169 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/49b2f75c69622e5f6fd3ef4971d15626ae1482d4b1c61087e1ae4910caf31307.jpg", + "image_caption": [ + "Figure 8: Performance comparison under the safeguard attack with re-scale factor 0.7. (In this case, our our algorithm can catch some bad workers, and thus perform nearly optimally.) " + ], + "image_footnote": [], + "bbox": [ + 169, + 196, + 825, + 362 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Re-scale factor 0.7. In Figure 8, we present the scenario when the re-scale factor is 0.7, so that the safeguard algorithms can occasionally catch some bad workers (depending on the randomness and learning rate). We confirm that in the three runs of single safeguard, it catches 1, 2, 3 bad workers for CIFAR-10, and $1 , 0 , 0$ bad workers for CIFAR-100 respectively; in the three runs of double safeguard, it catches $1 , 2 , 4$ bad workers for CIFAR-10, and 2, 2, 2 bad workers for CIFAR-100 respectively. ", + "bbox": [ + 174, + 431, + 825, + 494 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Since there is a significant performance gain when our safeguard algorithms catch bad workers, this explains why safeguard algorithms in Figure 8 outperform their counterparts in Figure 7 with rescale factor 0.6. At the same time, we notice that the double safeguard algorithm has the ability to catch bad workers more easily. This is why double safeguard significantly outperforms single sageguard in Figure 8. ", + "bbox": [ + 174, + 500, + 825, + 550 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "In contrast, all other prior algorithms perform extremely bad under this attack. To some extent, safeguard attack is even stronger than the previously proposed variance attack, since it can drag the 100-class test accuracy on CIFAR-100 for all prior defense algorithms to nearly $1 \\%$ , while variance attack can only drag them down to around $10 \\%$ . ", + "bbox": [ + 174, + 554, + 825, + 604 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.3 FULL COMPARISON TABLE ", + "text_level": 1, + "bbox": [ + 176, + 617, + 405, + 631 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We also include the full test accuracy comparison table in Table 1. ", + "bbox": [ + 174, + 637, + 563, + 651 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/13e28cf13d108a2ccad363e9bf3180cb61e26e5dd371738eb440bbe3697cfe15.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Table of test accuracy performances for CIFAR-10 (above) and CIFAR-100 (below). " + ], + "table_body": "
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack92.0291.7521.4322.0121.4722.433.42
sign-flipping attack91.9392.082259.6557.0370.8722.4
label-flipping attack92.3392.4431.9383.0783.1883.5291.66
delayed-gradient attack91.5891.4229.4374.3661.8179.2977.27
safeguard(x0.6) attack89.0189.2621.4423.1212.4828.6674.46
safeguard(x0.7) attack91.2492.0821.6119.9515.1724.5233.36
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack68.2767.956.65.815.056.110.87
sign-flipping attack68.0268.082.1310.1910.9328.342.59
label-flipping attack69.4368.85.3451.8552.1351.6667.86
delayed-gradient attack67.0366.424.0436.3431.4344.8536.6
safeguard(x0.6) attack59.87602.011.91.251.725.02
safeguard(x0.7) attack59.8464.912.071.971.321.553.31
", + "bbox": [ + 235, + 674, + 761, + 901 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.4 ATTACK AGAINST THE CONVEX ALGORITHM OF ALISTARH ET AL. (2018) ", + "text_level": 1, + "bbox": [ + 174, + 103, + 733, + 117 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We now briefly describe an attack against this algorithm. The attack specifically leverages the fact that the algorithm does not use sliding windows. ", + "bbox": [ + 174, + 123, + 825, + 150 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "One can first run the vanilla SGD to compute “the maximum deviation per good worker” for the accumulation vector used by the algorithm $\\textstyle \\sum _ { t = 0 } ^ { T } \\nabla _ { t }$ . This maximum deviation is therefore a lower bound for the threshold used in their algorithm. Next, we design an attacker who evenly distributes this total allowed deviation to e.g. 5 consecutive epochs, and behaves honestly for the remaining epochs. Such an attacker cannot be identified by this algorithm, because its total deviation across all the iterations is identical to that of a good worker. However, this leads the algorithm to divergence. ", + "bbox": [ + 174, + 154, + 825, + 232 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Specifically, suppose 4 Byzantine workers all maliciously report their stochastic gradients multiplied by the scalar $^ { - 5 }$ , and the remaining 6 good workers report their true stochastic gradients. One can verify numerically that this attacker can run for 5 consecutive epochs (say, epochs $a , a + 1 , a + 2 , a + 3 , a + 4 )$ without being caught by the algorithm. Now, ", + "bbox": [ + 174, + 234, + 825, + 286 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "• if $a \\leq 7 5$ , within just 1 epoch of attack, the neural net weights diverge (value NAN). \n• if $8 0 \\leq a \\leq 1 1 5$ , this attack is applied after the first learning rate decay. Within just 1 epoch of the attack, the objective explodes and accuracy becomes $1 0 \\%$ (random), and within 3 epochs the algorithm diverges completely. \n• if $1 2 0 \\leq a \\leq 1 5 5$ , this attack is after the second learning rate decay. Within just 2 epochs of attack, the accuracy drops to $1 1 \\%$ . Later, the accuracy never recovers above $4 0 \\%$ . 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The worker can independently sample", + "type": "text" + }, + { + "bbox": [ + 296, + 520, + 325, + 530 + ], + "score": 0.91, + "content": "s \\sim \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 520, + 367, + 533 + ], + "score": 1.0, + "content": ", compute", + "type": "text" + }, + { + "bbox": [ + 367, + 520, + 402, + 532 + ], + "score": 0.93, + "content": "\\nabla f _ { s } ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 520, + 505, + 533 + ], + "score": 1.0, + "content": ", and then synchronously", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "send this stochastic gradient to the master. 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We assume there are", + "type": "text" + }, + { + "bbox": [ + 454, + 489, + 464, + 497 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "machines", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 256, + 510 + ], + "score": 1.0, + "content": "(workers) and an honest master, and", + "type": "text" + }, + { + "bbox": [ + 257, + 498, + 295, + 510 + ], + "score": 0.92, + "content": "\\alpha < 1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "fraction of the workers may be Byzantine. In each", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 142, + 522 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 142, + 510, + 147, + 519 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 509, + 379, + 522 + ], + "score": 1.0, + "content": ", each worker has access to a version of the global iterate", + "type": "text" + }, + { + "bbox": [ + 379, + 511, + 389, + 520 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 509, + 505, + 522 + ], + "score": 1.0, + "content": ", which is maintained by the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 296, + 533 + ], + "score": 1.0, + "content": "master. The worker can independently sample", + "type": "text" + }, + { + "bbox": [ + 296, + 520, + 325, + 530 + ], + "score": 0.91, + "content": "s \\sim \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 520, + 367, + 533 + ], + "score": 1.0, + "content": ", compute", + "type": "text" + }, + { + "bbox": [ + 367, + 520, + 402, + 532 + ], + "score": 0.93, + "content": "\\nabla f _ { s } ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 520, + 505, + 533 + ], + "score": 1.0, + "content": ", and then synchronously", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "send this stochastic gradient to the master. The master aggregates the workers’ messages, and sends", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 183, + 555 + ], + "score": 1.0, + "content": "an updated iterate", + "type": "text" + }, + { + "bbox": [ + 183, + 544, + 204, + 554 + ], + "score": 0.9, + "content": "x _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "to all the workers. Eventually, the master has to output an approximate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 161, + 566 + ], + "score": 1.0, + "content": "minimizer of", + "type": "text" + }, + { + "bbox": [ + 162, + 553, + 169, + 565 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 552, + 505, + 566 + ], + "score": 1.0, + "content": ". Clearly, the above description only applies to honest workers; Byzantine workers", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 491, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 491, + 577 + ], + "score": 1.0, + "content": "may deviate arbitrarily and return adversarial “gradient” vectors to the master in every iteration.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 464, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "This distributed framework is quite general and well studied. One of the first references in this setting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 504, + 601 + ], + "score": 1.0, + "content": "studied distributed PCA and regression (Feng et al., 2014). Other early approaches (Blanchard et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "2017; Chen et al., 2017; Su & Vaidya, 2016a;b; Xie et al., 2018a) relied on defining generalizations", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "of the geometric median. These approaches can withstand up to half of the nodes being malicious,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 322, + 635 + ], + "score": 1.0, + "content": "but can have relatively high local computational cost", + "type": "text" + }, + { + "bbox": [ + 322, + 622, + 356, + 634 + ], + "score": 0.92, + "content": "\\Omega ( m ^ { 2 } { \\dot { d } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "(Blanchard et al., 2017; Chen et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 164, + 646 + ], + "score": 1.0, + "content": "2017), where", + "type": "text" + }, + { + "bbox": [ + 164, + 635, + 174, + 644 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 632, + 293, + 646 + ], + "score": 1.0, + "content": "is the number of nodes and", + "type": "text" + }, + { + "bbox": [ + 293, + 634, + 300, + 643 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "is the problem dimension, and usually have sub-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 278, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 278, + 657 + ], + "score": 1.0, + "content": "optimal sample and iteration complexities.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 576, + 506, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 659, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 347, + 673 + ], + "score": 1.0, + "content": "Follow-up work resolved this last issue when the objective", + "type": "text" + }, + { + "bbox": [ + 348, + 659, + 365, + 671 + ], + "score": 0.91, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 656, + 505, + 673 + ], + "score": 1.0, + "content": "is convex, leading to tight sample", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "complexity bounds. 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Alistarh et al.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "(2018) provided a stochastic gradient descent (SGD) type algorithm and showed that its sample and", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 426, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 426, + 129 + ], + "score": 1.0, + "content": "time complexities are asymptotically optimal even when the dimension is large.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 656, + 505, + 673 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "complexity bounds. 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Alistarh et al.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "(2018) provided a stochastic gradient descent (SGD) type algorithm and showed that its sample and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 426, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 426, + 129 + ], + "score": 1.0, + "content": "time complexities are asymptotically optimal even when the dimension is large.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 130, + 505, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "Non-convex Byzantine-resilient stochastic optimization. In this paper, we focus on the more", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 506, + 153 + ], + "score": 1.0, + "content": "challenging non-convex setting, and shoot for the strong goal of finding approximate local minima", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 457, + 164 + ], + "score": 1.0, + "content": "(a.k.a. second-order critical points). In a nutshell, our main result is the following. Fix", + "type": "text" + }, + { + "bbox": [ + 457, + 153, + 464, + 162 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 504, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 255, + 174 + ], + "score": 1.0, + "content": "the dimension, and let the objective", + "type": "text" + }, + { + "bbox": [ + 255, + 162, + 311, + 174 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 162, + 504, + 174 + ], + "score": 1.0, + "content": "be Lipschitz smooth and second-order smooth.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 144, + 186 + ], + "score": 1.0, + "content": "We have", + "type": "text" + }, + { + "bbox": [ + 144, + 176, + 154, + 184 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "worker machines, each having access to unbiased, bounded estimators of the gradient", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 504, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 117, + 198 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 185, + 124, + 196 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 183, + 216, + 198 + ], + "score": 1.0, + "content": ". 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We spell out these factors in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 193, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 193, + 305 + ], + "score": 1.0, + "content": "the detailed analysis.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 111, + 307, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 110, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 110, + 307, + 147, + 321 + ], + "score": 1.0, + "content": "• When", + "type": "text" + }, + { + "bbox": [ + 147, + 307, + 197, + 320 + ], + "score": 0.93, + "content": "\\alpha < 1 / \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 307, + 297, + 321 + ], + "score": 1.0, + "content": ", our sample complexity", + "type": "text" + }, + { + "bbox": [ + 298, + 308, + 330, + 319 + ], + "score": 0.88, + "content": "( = m T ,", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 307, + 505, + 321 + ], + "score": 1.0, + "content": ") matches the best known result in the non-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 120, + 319, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 120, + 319, + 504, + 331 + ], + "score": 1.0, + "content": "Byzantine case (Jin et al., 2019) without additional assumptions, and enjoys linear parallel speed-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 119, + 329, + 453, + 344 + ], + "spans": [ + { + "bbox": [ + 119, + 331, + 156, + 344 + ], + "score": 1.0, + "content": "up: with", + "type": "text" + }, + { + "bbox": [ + 157, + 334, + 167, + 342 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 331, + 240, + 344 + ], + "score": 1.0, + "content": "workers of which", + "type": "text" + }, + { + "bbox": [ + 240, + 331, + 269, + 344 + ], + "score": 0.92, + "content": "< \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 331, + 421, + 344 + ], + "score": 1.0, + "content": "are Byzantine, the parallel speedup is", + "type": "text" + }, + { + "bbox": [ + 421, + 329, + 446, + 344 + ], + "score": 0.92, + "content": "\\widetilde \\Omega ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 331, + 453, + 344 + ], + "score": 1.0, + "content": ". 1", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 111, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 111, + 345, + 138, + 360 + ], + "score": 1.0, + "content": "• For", + "type": "text" + }, + { + "bbox": [ + 139, + 346, + 218, + 359 + ], + "score": 0.92, + "content": "\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 345, + 358, + 360 + ], + "score": 1.0, + "content": ", our parallel time complexity is", + "type": "text" + }, + { + "bbox": [ + 358, + 345, + 386, + 359 + ], + "score": 0.92, + "content": "{ \\widetilde { \\cal O } } ( \\alpha ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "times that needed when no", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 119, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 119, + 358, + 350, + 371 + ], + "score": 1.0, + "content": "parallelism is used. This still gives parallel speedup. This", + "type": "text" + }, + { + "bbox": [ + 351, + 358, + 362, + 368 + ], + "score": 0.87, + "content": "\\alpha ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "factor appears in convex Byzantine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 120, + 370, + 442, + 381 + ], + "spans": [ + { + "bbox": [ + 120, + 370, + 442, + 381 + ], + "score": 1.0, + "content": "distributed optimization, where it is tight (Yin et al., 2018; Alistarh et al., 2018).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 112, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 112, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "• The Lipschitz and second-order smoothness assumptions are the minimal assumptions needed to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 120, + 394, + 447, + 406 + ], + "spans": [ + { + "bbox": [ + 120, + 394, + 447, + 406 + ], + "score": 1.0, + "content": "derive convergence rates for finding second-order critical points (Jin et al., 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "Comparison with prior bounds. The closest known bounds are by Yin et al. (2019), who derived", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "three gradient descent-type of algorithms (based on median, mean, and iterative filtering) to find", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 432, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 442 + ], + "score": 1.0, + "content": "a weaker type of approximate local minima. Since it relies on full gradients, their algorithm is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "arguably less practical, and their time complexities are generally higher than ours (see Section 2.1).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 424, + 469 + ], + "score": 1.0, + "content": "Other prior works consider a weaker goal: to find approximate stationary points", + "type": "text" + }, + { + "bbox": [ + 425, + 456, + 481, + 469 + ], + "score": 0.93, + "content": "\\| \\nabla f ( x ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "only:", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "Bulusu et al. (2020) additionally assumed there is a guaranteed good (i.e. non-Byzantine) worker", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "known by the master, Xie et al. (2018b) gave a practical algorithm when the Byzantine attackers", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "have no information about the loss function or its gradient, Yang et al. (2019); Xie et al. (2018a);", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Blanchard et al. (2017) derived eventual convergence without an explicit complexity bound, and the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 499, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 499, + 524 + ], + "score": 1.0, + "content": "non-convex result obtained in Yin et al. (2018) is subsumed by Yin et al. (2019), discussed above.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "Our algorithm and techniques. The structure of our algorithm is deceptively simple. The master", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "node keeps track of the sum of gradients produced by each worker across time. It labels (allegedly)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "good workers as those whose sum of gradients “concentrate” well with respect to a surrogate of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "the median vector, and labels bad workers otherwise. Once a worker is labelled bad, it is removed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 372, + 582 + ], + "score": 1.0, + "content": "from consideration forever. The master then performs the vanilla", + "type": "text" + }, + { + "bbox": [ + 372, + 570, + 393, + 580 + ], + "score": 0.29, + "content": "S G D", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ", by moving in the negative", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 580, + 460, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 460, + 593 + ], + "score": 1.0, + "content": "direction of the average gradients produced by those workers currently labelled as good.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "We call our algorithm SafeguardSGD, since it behaves like having a safe guard to filter away bad", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 606, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 303, + 618 + ], + "score": 1.0, + "content": "workers. Its processing overhead at the master is", + "type": "text" + }, + { + "bbox": [ + 303, + 606, + 333, + 618 + ], + "score": 0.92, + "content": "O ( m d )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 606, + 492, + 618 + ], + "score": 1.0, + "content": ", negligible compared to standard SGD.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "As the astute reader may have guessed, the key non-trivial technical ingredient is to identify the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "right quantity to check for concentration, and make it compatible with the task of non-convex", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "optimization. In particular, we manage to construct such quantities so that (1) good non-Byzantine", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "workers never get mislabelled as bad ones; (2) Byzantine workers may be labelled as good ones", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "(which is inevitable) but when they do, the convergence rates are not impacted significantly; and (3)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 675, + 415, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 415, + 687 + ], + "score": 1.0, + "content": "the notion does not require additional assumptions or running time overhead.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 109, + 690, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 107, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "The idea of using concentration (for each worker across time) to filter out Byzantine machines", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "1By parallel speedup we mean the reduction in wall-clock time due to sampling gradients in parallel among", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 495, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 119, + 732 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 120, + 723, + 129, + 730 + ], + "score": 0.63, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 721, + 308, + 732 + ], + "score": 1.0, + "content": "nodes. In each time step, the algorithm generates", + "type": "text" + }, + { + "bbox": [ + 308, + 724, + 317, + 730 + ], + "score": 0.55, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 721, + 495, + 732 + ], + "score": 1.0, + "content": "new gradients, although some may be corrupted.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 129 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 130, + 505, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "Non-convex Byzantine-resilient stochastic optimization. In this paper, we focus on the more", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 506, + 153 + ], + "score": 1.0, + "content": "challenging non-convex setting, and shoot for the strong goal of finding approximate local minima", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 457, + 164 + ], + "score": 1.0, + "content": "(a.k.a. second-order critical points). In a nutshell, our main result is the following. Fix", + "type": "text" + }, + { + "bbox": [ + 457, + 153, + 464, + 162 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "to denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 504, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 255, + 174 + ], + "score": 1.0, + "content": "the dimension, and let the objective", + "type": "text" + }, + { + "bbox": [ + 255, + 162, + 311, + 174 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 162, + 504, + 174 + ], + "score": 1.0, + "content": "be Lipschitz smooth and second-order smooth.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 144, + 186 + ], + "score": 1.0, + "content": "We have", + "type": "text" + }, + { + "bbox": [ + 144, + 176, + 154, + 184 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "worker machines, each having access to unbiased, bounded estimators of the gradient", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 504, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 117, + 198 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 185, + 124, + 196 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 183, + 216, + 198 + ], + "score": 1.0, + "content": ". Given an initial point", + "type": "text" + }, + { + "bbox": [ + 217, + 186, + 228, + 196 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 183, + 468, + 198 + ], + "score": 1.0, + "content": ", the SafeguardSGD algorithm ensures that, even if at most", + "type": "text" + }, + { + "bbox": [ + 468, + 185, + 504, + 197 + ], + "score": 0.91, + "content": "\\bar { \\alpha } < 1 / 2", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 287, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 287, + 208 + ], + "score": 1.0, + "content": "fraction of the machines are Byzantine, after", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 129, + 506, + 208 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 209, + 427, + 231 + ], + "lines": [ + { + "bbox": [ + 177, + 209, + 427, + 231 + ], + "spans": [ + { + "bbox": [ + 177, + 209, + 427, + 231 + ], + "score": 0.9, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\left( \\left( \\alpha ^ { 2 } + \\frac { 1 } { m } \\right) \\frac { d \\left( f \\left( x _ { 0 } \\right) - \\operatorname* { m i n } f \\left( x \\right) \\right) } { \\varepsilon ^ { 4 } } \\right) \\qquad \\mathrm { p a r a l l e l i t e r a t i o n s } . } \\end{array}", + "type": "interline_equation", + "image_path": "e378316cad9f2fda93181dca2e62c878e00a67398c3427f7916b7a79e9edafee.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 177, + 209, + 427, + 231 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 233, + 395, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 396, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 283, + 246 + ], + "score": 1.0, + "content": "for at least a constant fraction of the indices", + "type": "text" + }, + { + "bbox": [ + 283, + 233, + 313, + 245 + ], + "score": 0.93, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 232, + 396, + 246 + ], + "score": 1.0, + "content": ", the following hold:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 232, + 396, + 246 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 248, + 390, + 262 + ], + "lines": [ + { + "bbox": [ + 218, + 248, + 390, + 262 + ], + "spans": [ + { + "bbox": [ + 218, + 248, + 390, + 262 + ], + "score": 0.87, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon \\quad { \\mathrm { a n d } } \\quad \\nabla ^ { 2 } f ( x _ { t } ) \\succeq - { \\sqrt { \\varepsilon } } \\mathbf { I } .", + "type": "interline_equation", + "image_path": "63efb86c1b4d22f974ade23c83eb2d8879c3eadd25398bcb716b723c9bd487a5.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 218, + 248, + 390, + 262 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 102, + 259, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 102, + 259, + 192, + 284 + ], + "score": 1.0, + "content": "If the goal is simply", + "type": "text" + }, + { + "bbox": [ + 193, + 266, + 255, + 279 + ], + "score": 0.93, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 259, + 280, + 284 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 281, + 264, + 431, + 280 + ], + "score": 0.91, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\big ( \\left( \\alpha ^ { 2 } + \\frac { 1 } { m } \\right) \\frac { \\left( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) \\right) } { \\varepsilon ^ { 4 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 259, + 507, + 284 + ], + "score": 1.0, + "content": "iterations suffice.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 145, + 294 + ], + "score": 1.0, + "content": "Here, the", + "type": "text" + }, + { + "bbox": [ + 146, + 279, + 155, + 291 + ], + "score": 0.85, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "notation serves to hide logarithmic factors for readability. We spell out these factors in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 193, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 193, + 305 + ], + "score": 1.0, + "content": "the detailed analysis.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 102, + 259, + 507, + 305 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 307, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 110, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 110, + 307, + 147, + 321 + ], + "score": 1.0, + "content": "• When", + "type": "text" + }, + { + "bbox": [ + 147, + 307, + 197, + 320 + ], + "score": 0.93, + "content": "\\alpha < 1 / \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 307, + 297, + 321 + ], + "score": 1.0, + "content": ", our sample complexity", + "type": "text" + }, + { + "bbox": [ + 298, + 308, + 330, + 319 + ], + "score": 0.88, + "content": "( = m T ,", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 307, + 505, + 321 + ], + "score": 1.0, + "content": ") matches the best known result in the non-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 120, + 319, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 120, + 319, + 504, + 331 + ], + "score": 1.0, + "content": "Byzantine case (Jin et al., 2019) without additional assumptions, and enjoys linear parallel speed-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 119, + 329, + 453, + 344 + ], + "spans": [ + { + "bbox": [ + 119, + 331, + 156, + 344 + ], + "score": 1.0, + "content": "up: with", + "type": "text" + }, + { + "bbox": [ + 157, + 334, + 167, + 342 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 331, + 240, + 344 + ], + "score": 1.0, + "content": "workers of which", + "type": "text" + }, + { + "bbox": [ + 240, + 331, + 269, + 344 + ], + "score": 0.92, + "content": "< \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 331, + 421, + 344 + ], + "score": 1.0, + "content": "are Byzantine, the parallel speedup is", + "type": "text" + }, + { + "bbox": [ + 421, + 329, + 446, + 344 + ], + "score": 0.92, + "content": "\\widetilde \\Omega ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 331, + 453, + 344 + ], + "score": 1.0, + "content": ". 1", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 111, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 111, + 345, + 138, + 360 + ], + "score": 1.0, + "content": "• For", + "type": "text" + }, + { + "bbox": [ + 139, + 346, + 218, + 359 + ], + "score": 0.92, + "content": "\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 345, + 358, + 360 + ], + "score": 1.0, + "content": ", our parallel time complexity is", + "type": "text" + }, + { + "bbox": [ + 358, + 345, + 386, + 359 + ], + "score": 0.92, + "content": "{ \\widetilde { \\cal O } } ( \\alpha ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "times that needed when no", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 119, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 119, + 358, + 350, + 371 + ], + "score": 1.0, + "content": "parallelism is used. This still gives parallel speedup. This", + "type": "text" + }, + { + "bbox": [ + 351, + 358, + 362, + 368 + ], + "score": 0.87, + "content": "\\alpha ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "factor appears in convex Byzantine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 120, + 370, + 442, + 381 + ], + "spans": [ + { + "bbox": [ + 120, + 370, + 442, + 381 + ], + "score": 1.0, + "content": "distributed optimization, where it is tight (Yin et al., 2018; Alistarh et al., 2018).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 112, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 112, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "• The Lipschitz and second-order smoothness assumptions are the minimal assumptions needed to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 120, + 394, + 447, + 406 + ], + "spans": [ + { + "bbox": [ + 120, + 394, + 447, + 406 + ], + "score": 1.0, + "content": "derive convergence rates for finding second-order critical points (Jin et al., 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 110, + 307, + 506, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "Comparison with prior bounds. The closest known bounds are by Yin et al. (2019), who derived", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "three gradient descent-type of algorithms (based on median, mean, and iterative filtering) to find", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 432, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 442 + ], + "score": 1.0, + "content": "a weaker type of approximate local minima. Since it relies on full gradients, their algorithm is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "arguably less practical, and their time complexities are generally higher than ours (see Section 2.1).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 408, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 424, + 469 + ], + "score": 1.0, + "content": "Other prior works consider a weaker goal: to find approximate stationary points", + "type": "text" + }, + { + "bbox": [ + 425, + 456, + 481, + 469 + ], + "score": 0.93, + "content": "\\| \\nabla f ( x ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "only:", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "Bulusu et al. (2020) additionally assumed there is a guaranteed good (i.e. non-Byzantine) worker", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "known by the master, Xie et al. (2018b) gave a practical algorithm when the Byzantine attackers", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "have no information about the loss function or its gradient, Yang et al. (2019); Xie et al. (2018a);", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Blanchard et al. (2017) derived eventual convergence without an explicit complexity bound, and the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 499, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 499, + 524 + ], + "score": 1.0, + "content": "non-convex result obtained in Yin et al. (2018) is subsumed by Yin et al. (2019), discussed above.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 456, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "Our algorithm and techniques. The structure of our algorithm is deceptively simple. The master", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "node keeps track of the sum of gradients produced by each worker across time. It labels (allegedly)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "good workers as those whose sum of gradients “concentrate” well with respect to a surrogate of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "the median vector, and labels bad workers otherwise. Once a worker is labelled bad, it is removed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 372, + 582 + ], + "score": 1.0, + "content": "from consideration forever. The master then performs the vanilla", + "type": "text" + }, + { + "bbox": [ + 372, + 570, + 393, + 580 + ], + "score": 0.29, + "content": "S G D", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ", by moving in the negative", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 580, + 460, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 460, + 593 + ], + "score": 1.0, + "content": "direction of the average gradients produced by those workers currently labelled as good.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 526, + 506, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "We call our algorithm SafeguardSGD, since it behaves like having a safe guard to filter away bad", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 606, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 303, + 618 + ], + "score": 1.0, + "content": "workers. Its processing overhead at the master is", + "type": "text" + }, + { + "bbox": [ + 303, + 606, + 333, + 618 + ], + "score": 0.92, + "content": "O ( m d )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 606, + 492, + 618 + ], + "score": 1.0, + "content": ", negligible compared to standard SGD.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 595, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "As the astute reader may have guessed, the key non-trivial technical ingredient is to identify the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "right quantity to check for concentration, and make it compatible with the task of non-convex", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "optimization. In particular, we manage to construct such quantities so that (1) good non-Byzantine", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "workers never get mislabelled as bad ones; (2) Byzantine workers may be labelled as good ones", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "(which is inevitable) but when they do, the convergence rates are not impacted significantly; and (3)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 675, + 415, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 415, + 687 + ], + "score": 1.0, + "content": "the notion does not require additional assumptions or running time overhead.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 621, + 505, + 687 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 690, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 107, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "The idea of using concentration (for each worker across time) to filter out Byzantine machines", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "traces back to the convex setting (Alistarh et al., 2018). However, the quantities used in (Alistarh", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "et al., 2018) to check for concentration are necessarily different from this paper, and our analysis is", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "score": 1.0, + "content": "completely new, as deriving non-convex rates is known to be much more delicate and challenging.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "Recently, Bulusu et al. (2020) used similar concentration filters to Alistarh et al. (2018) in the non-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 504, + 140 + ], + "score": 1.0, + "content": "convex setting, but under stronger assumptions, and for the simpler task of finding stationary points.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 49, + "bbox_fs": [ + 107, + 689, + 505, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "traces back to the convex setting (Alistarh et al., 2018). However, the quantities used in (Alistarh", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "et al., 2018) to check for concentration are necessarily different from this paper, and our analysis is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 119 + ], + "score": 1.0, + "content": "completely new, as deriving non-convex rates is known to be much more delicate and challenging.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "Recently, Bulusu et al. (2020) used similar concentration filters to Alistarh et al. (2018) in the non-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 504, + 140 + ], + "score": 1.0, + "content": "convex setting, but under stronger assumptions, and for the simpler task of finding stationary points.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "score": 1.0, + "content": "Many other algorithms do not rely on concentration filters. In each iteration, they ask each worker", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 164 + ], + "score": 1.0, + "content": "to compute a batch of stochastic gradients, and then use coordinate-wise median or mean over the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "batch average (e.g. Yin et al. (2018; 2019); Yang et al. (2019)) or iterative filtering (e.g. Su & Xu", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "(2018); Yin et al. (2019)) by the master to derive a “robust mean.” These works fundamentally", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "rely on each iteration to calculate an almost precise full gradient, so that they can apply a surrogate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "of full gradient descent. Such algorithms can introduce higher sample and time complexities (see", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 220 + ], + "score": 1.0, + "content": "Section 2), are less practical than stochastic gradient schemes, require additional restrictions on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 172, + 230 + ], + "score": 1.0, + "content": "resilience factor", + "type": "text" + }, + { + "bbox": [ + 173, + 220, + 180, + 228 + ], + "score": 0.67, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 217, + 201, + 230 + ], + "score": 1.0, + "content": ", e.g.", + "type": "text" + }, + { + "bbox": [ + 201, + 218, + 237, + 229 + ], + "score": 0.91, + "content": "\\alpha < 1 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "(Su & Xu, 2018), and, critically, have been shown to be vulnerable", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 229, + 326, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 326, + 241 + ], + "score": 1.0, + "content": "to recent attacks (Baruch et al., 2019; Xie et al., 2020).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Attack resilience and experimental validation. There is a growing literature on customized", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "attacks against Byzantine-resilient algorithms, showing that many defenses can be entirely", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "circumvented in real-world scenarios (Baruch et al., 2019; Xie et al., 2020). Our algorithm is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "provably correct against these attacks, a fact we also validate experimentally. We implemented", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "SafeguardSGD to examine its practical performance against a range of prior works (Xie et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "2018b; Blanchard et al., 2017; Chen et al., 2017; Yin et al., 2018; 2019), and against recent attacks", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "on the distributed task of training deep neural networks. Our experiments show that SafeguardSGD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "generally outperforms previous methods in convergence speed and final accuracy, sometimes by a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "wide accuracy margin. This is true not only against known Byzantine attacks, but also against attack", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "variants we fine-crafted to specifically slow down our algorithm, and against transient node failures.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 351, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 353, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 353, + 380 + ], + "score": 1.0, + "content": "2 STATEMENT OF OUR THEORETICAL RESULT", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 164, + 400 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + }, + { + "bbox": [ + 165, + 387, + 183, + 399 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 387, + 283, + 400 + ], + "score": 1.0, + "content": "the Euclidean norm and", + "type": "text" + }, + { + "bbox": [ + 284, + 387, + 366, + 399 + ], + "score": 0.92, + "content": "[ n ] : = \\{ 1 , 2 , \\dots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 387, + 479, + 400 + ], + "score": 1.0, + "content": ". Given symmetric matrices", + "type": "text" + }, + { + "bbox": [ + 479, + 388, + 501, + 398 + ], + "score": 0.78, + "content": "\\mathbf { A } , \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 387, + 505, + 400 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 410 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 158, + 411 + ], + "score": 0.92, + "content": "\\| \\mathbf { A } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 398, + 318, + 410 + ], + "score": 1.0, + "content": "denote the spectral norm of A. We use", + "type": "text" + }, + { + "bbox": [ + 319, + 400, + 329, + 409 + ], + "score": 0.82, + "content": "\\succeq", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 398, + 462, + 410 + ], + "score": 1.0, + "content": "to denote Loewner ordering, i.e.", + "type": "text" + }, + { + "bbox": [ + 462, + 399, + 495, + 410 + ], + "score": 0.89, + "content": "\\mathbf { A } \\succeq \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 408, + 488, + 422 + ], + "spans": [ + { + "bbox": [ + 107, + 410, + 137, + 420 + ], + "score": 0.81, + "content": "\\mathbf { A } - \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 408, + 295, + 422 + ], + "score": 1.0, + "content": "is positive semi-definite. We denote by", + "type": "text" + }, + { + "bbox": [ + 295, + 410, + 333, + 421 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { m i n } } ( \\mathbf { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 408, + 474, + 422 + ], + "score": 1.0, + "content": "the minimum eigenvalue of matrix", + "type": "text" + }, + { + "bbox": [ + 474, + 410, + 484, + 419 + ], + "score": 0.46, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 408, + 488, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 493, + 435 + ], + "lines": [ + { + "bbox": [ + 104, + 419, + 494, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 195, + 439 + ], + "score": 1.0, + "content": "We consider arbitrary", + "type": "text" + }, + { + "bbox": [ + 195, + 424, + 201, + 434 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 419, + 343, + 439 + ], + "score": 1.0, + "content": "-dimensional non-convex functions", + "type": "text" + }, + { + "bbox": [ + 344, + 423, + 392, + 435 + ], + "score": 0.93, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 419, + 494, + 439 + ], + "score": 1.0, + "content": "satisfying the following:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 111, + 437, + 489, + 464 + ], + "lines": [ + { + "bbox": [ + 109, + 437, + 483, + 451 + ], + "spans": [ + { + "bbox": [ + 109, + 437, + 120, + 451 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 438, + 141, + 451 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 437, + 151, + 451 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 152, + 439, + 159, + 449 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 437, + 273, + 451 + ], + "score": 1.0, + "content": "-Lipschitz smooth: meaning", + "type": "text" + }, + { + "bbox": [ + 273, + 438, + 404, + 451 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq L \\| x - y \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 437, + 437, + 451 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 437, + 437, + 477, + 450 + ], + "score": 0.92, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 437, + 483, + 451 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 109, + 450, + 490, + 465 + ], + "spans": [ + { + "bbox": [ + 109, + 450, + 120, + 465 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 452, + 141, + 464 + ], + "score": 0.89, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 450, + 151, + 465 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 152, + 452, + 164, + 463 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 450, + 256, + 465 + ], + "score": 1.0, + "content": "-second-order smooth:", + "type": "text" + }, + { + "bbox": [ + 257, + 451, + 412, + 464 + ], + "score": 0.91, + "content": "\\| \\nabla ^ { 2 } f ( x ) - \\nabla ^ { 2 } f ( y ) \\| _ { 2 } \\leq L _ { 2 } \\cdot \\| x - y \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 450, + 444, + 465 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 445, + 451, + 485, + 463 + ], + "score": 0.91, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 450, + 490, + 465 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 108, + 466, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 316, + 478 + ], + "score": 1.0, + "content": "For notational simplicity of the proofs, we assume", + "type": "text" + }, + { + "bbox": [ + 317, + 466, + 398, + 477 + ], + "score": 0.93, + "content": "L = L _ { 2 } = \\mathcal { V } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 465, + 505, + 478 + ], + "score": 1.0, + "content": ".2 Note that we have also", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 474, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 204, + 490 + ], + "score": 1.0, + "content": "assumed the domain of", + "type": "text" + }, + { + "bbox": [ + 204, + 478, + 212, + 489 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 474, + 290, + 490 + ], + "score": 1.0, + "content": "is the entire space", + "type": "text" + }, + { + "bbox": [ + 291, + 476, + 304, + 487 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 474, + 463, + 490 + ], + "score": 1.0, + "content": ". If instead there is a compact domain", + "type": "text" + }, + { + "bbox": [ + 463, + 476, + 501, + 488 + ], + "score": 0.9, + "content": "\\mathcal { X } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 474, + 506, + 490 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "then one can use projected SGD and re-derive similar results of this paper. We choose to present our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 325, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 325, + 511 + ], + "score": 1.0, + "content": "result in the simplest setting to convey our main ideas.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 391, + 525 + ], + "score": 1.0, + "content": "Byzantine non-convex stochastic distributed optimization. We let", + "type": "text" + }, + { + "bbox": [ + 391, + 515, + 401, + 523 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "be the number of worker", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 242, + 538 + ], + "score": 1.0, + "content": "machines and assume at most an", + "type": "text" + }, + { + "bbox": [ + 243, + 527, + 250, + 534 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 523, + 394, + 538 + ], + "score": 1.0, + "content": "fraction of them are Byzantine for", + "type": "text" + }, + { + "bbox": [ + 394, + 523, + 440, + 537 + ], + "score": 0.93, + "content": "\\alpha \\in [ 0 , \\frac { 1 } { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 523, + 505, + 538 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 129, + 549 + ], + "score": 1.0, + "content": "good", + "type": "text" + }, + { + "bbox": [ + 129, + 536, + 156, + 549 + ], + "score": 0.57, + "content": "\\subseteq [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 536, + 504, + 549 + ], + "score": 1.0, + "content": "the set of good (i.e. non-Byzantine) machines, and the algorithm does not know good.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 249, + 565 + ], + "score": 1.0, + "content": "Assumption 2.1. In each iteration", + "type": "text" + }, + { + "bbox": [ + 250, + 553, + 255, + 562 + ], + "score": 0.5, + "content": "t _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 550, + 493, + 565 + ], + "score": 1.0, + "content": ", the algorithm (on the master) is allowed to specify a point", + "type": "text" + }, + { + "bbox": [ + 493, + 554, + 504, + 563 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 561, + 459, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 262, + 578 + ], + "score": 1.0, + "content": "and query m machines. Each machine", + "type": "text" + }, + { + "bbox": [ + 262, + 563, + 293, + 575 + ], + "score": 0.93, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 561, + 374, + 578 + ], + "score": 1.0, + "content": "gives back a vector", + "type": "text" + }, + { + "bbox": [ + 374, + 563, + 416, + 576 + ], + "score": 0.93, + "content": "\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 561, + 459, + 578 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 110, + 578, + 501, + 607 + ], + "lines": [ + { + "bbox": [ + 110, + 577, + 497, + 592 + ], + "spans": [ + { + "bbox": [ + 110, + 577, + 128, + 592 + ], + "score": 1.0, + "content": "• If", + "type": "text" + }, + { + "bbox": [ + 128, + 579, + 166, + 591 + ], + "score": 0.58, + "content": "i \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 577, + 262, + 592 + ], + "score": 1.0, + "content": ", the stochastic gradient", + "type": "text" + }, + { + "bbox": [ + 262, + 579, + 280, + 591 + ], + "score": 0.9, + "content": "\\nabla _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 577, + 315, + 592 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 315, + 578, + 390, + 591 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\nabla _ { t , i } ] = \\nabla f ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 577, + 408, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 408, + 578, + 497, + 592 + ], + "score": 0.9, + "content": "\\lVert \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\rVert \\leq \\mathcal { V }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 110, + 593, + 485, + 608 + ], + "spans": [ + { + "bbox": [ + 110, + 593, + 129, + 608 + ], + "score": 1.0, + "content": "• If", + "type": "text" + }, + { + "bbox": [ + 129, + 594, + 161, + 607 + ], + "score": 0.69, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 593, + 214, + 608 + ], + "score": 1.0, + "content": "\\ good, then", + "type": "text" + }, + { + "bbox": [ + 214, + 595, + 232, + 607 + ], + "score": 0.91, + "content": "\\nabla _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 593, + 382, + 608 + ], + "score": 1.0, + "content": "can be arbitrary (w.l.o.g. we assume", + "type": "text" + }, + { + "bbox": [ + 382, + 594, + 475, + 607 + ], + "score": 0.92, + "content": "\\lVert \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\rVert \\leq \\mathcal { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 593, + 485, + 608 + ], + "score": 1.0, + "content": ".4", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 108, + 609, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 197, + 623 + ], + "score": 1.0, + "content": "Remark 2.2. For each", + "type": "text" + }, + { + "bbox": [ + 197, + 610, + 203, + 619 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 606, + 220, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 221, + 609, + 258, + 621 + ], + "score": 0.75, + "content": "i \\not \\in { \\mathfrak { g o o d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 606, + 304, + 623 + ], + "score": 1.0, + "content": ", the vector", + "type": "text" + }, + { + "bbox": [ + 305, + 609, + 323, + 621 + ], + "score": 0.91, + "content": "\\nabla _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 606, + 505, + 623 + ], + "score": 1.0, + "content": "can be adversarially chosen and may depend", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 630, + 503, + 669 + ], + "lines": [ + { + "bbox": [ + 118, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 118, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "2In the literature of convergence analysis for non-convex optimization, the final complexity bounds naturally", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 639, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 286, + 651 + ], + "score": 1.0, + "content": "and polynomially depend on these parameters", + "type": "text" + }, + { + "bbox": [ + 286, + 640, + 318, + 650 + ], + "score": 0.91, + "content": "L , L _ { 2 } , \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 639, + 504, + 651 + ], + "score": 1.0, + "content": ", and the way the dependence goes is typically", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "unique (Allen-Zhu, 2018a;b; Fang et al., 2018; Jin et al., 2019). This is why it suffices to ignore their appearance", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 659, + 316, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 285, + 671 + ], + "score": 1.0, + "content": "and only compare the polynomial dependence on", + "type": "text" + }, + { + "bbox": [ + 285, + 662, + 291, + 669 + ], + "score": 0.78, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 659, + 307, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 660, + 313, + 669 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 659, + 316, + 671 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 670, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 118, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 118, + 668, + 215, + 683 + ], + "score": 1.0, + "content": "3One can instead assume", + "type": "text" + }, + { + "bbox": [ + 216, + 670, + 396, + 681 + ], + "score": 0.9, + "content": "\\operatorname* { P r } [ \\| \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\| > t ] \\le 2 \\exp ( - t ^ { 2 } / 2 \\mathcal { V } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 668, + 506, + 683 + ], + "score": 1.0, + "content": "and the results of this paper", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 680, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 504, + 691 + ], + "score": 1.0, + "content": "continue to hold up to logarithmic factors. To present the simplest theory, we do not include that version in this", + "type": "text" + } + ] + }, + { + "bbox": [ + 104, + 689, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 689, + 506, + 703 + ], + "score": 1.0, + "content": "paper. We refer interested readers to Jin et al. (2019) for how to deal with such probabilistic assumption (when", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 217, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 217, + 712 + ], + "score": 1.0, + "content": "there is no Byzantine worker).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 712, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 186, + 724 + ], + "score": 1.0, + "content": "4This requirement", + "type": "text" + }, + { + "bbox": [ + 186, + 711, + 267, + 722 + ], + "score": 0.91, + "content": "\\lVert \\nabla f ( x _ { t } ) - \\nabla _ { t , i } \\rVert \\leq \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "is “without loss of generality” because it is trivial for the algorithm", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 403, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 239, + 733 + ], + "score": 1.0, + "content": "to catch bad machines if they output", + "type": "text" + }, + { + "bbox": [ + 239, + 722, + 255, + 732 + ], + "score": 0.89, + "content": "\\nabla _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 721, + 295, + 733 + ], + "score": 1.0, + "content": "more than", + "type": "text" + }, + { + "bbox": [ + 295, + 722, + 308, + 731 + ], + "score": 0.79, + "content": "2 \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 721, + 403, + 733 + ], + "score": 1.0, + "content": "away from the majorities.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "score": 1.0, + "content": "Many other algorithms do not rely on concentration filters. In each iteration, they ask each worker", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 164 + ], + "score": 1.0, + "content": "to compute a batch of stochastic gradients, and then use coordinate-wise median or mean over the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "batch average (e.g. Yin et al. (2018; 2019); Yang et al. (2019)) or iterative filtering (e.g. Su & Xu", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "(2018); Yin et al. 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There is a growing literature on customized", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "attacks against Byzantine-resilient algorithms, showing that many defenses can be entirely", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "circumvented in real-world scenarios (Baruch et al., 2019; Xie et al., 2020). Our algorithm is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "provably correct against these attacks, a fact we also validate experimentally. We implemented", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "SafeguardSGD to examine its practical performance against a range of prior works (Xie et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "2018b; Blanchard et al., 2017; Chen et al., 2017; Yin et al., 2018; 2019), and against recent attacks", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "on the distributed task of training deep neural networks. Our experiments show that SafeguardSGD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "generally outperforms previous methods in convergence speed and final accuracy, sometimes by a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "wide accuracy margin. This is true not only against known Byzantine attacks, but also against attack", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "variants we fine-crafted to specifically slow down our algorithm, and against transient node failures.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 243, + 506, + 354 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 351, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 353, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 353, + 380 + ], + "score": 1.0, + "content": "2 STATEMENT OF OUR THEORETICAL RESULT", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 164, + 400 + ], + "score": 1.0, + "content": "We denote by", + "type": "text" + }, + { + "bbox": [ + 165, + 387, + 183, + 399 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 387, + 283, + 400 + ], + "score": 1.0, + "content": "the Euclidean norm and", + "type": "text" + }, + { + "bbox": [ + 284, + 387, + 366, + 399 + ], + "score": 0.92, + "content": "[ n ] : = \\{ 1 , 2 , \\dots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 387, + 479, + 400 + ], + "score": 1.0, + "content": ". Given symmetric matrices", + "type": "text" + }, + { + "bbox": [ + 479, + 388, + 501, + 398 + ], + "score": 0.78, + "content": "\\mathbf { A } , \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 387, + 505, + 400 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 410 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 158, + 411 + ], + "score": 0.92, + "content": "\\| \\mathbf { A } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 398, + 318, + 410 + ], + "score": 1.0, + "content": "denote the spectral norm of A. We use", + "type": "text" + }, + { + "bbox": [ + 319, + 400, + 329, + 409 + ], + "score": 0.82, + "content": "\\succeq", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 398, + 462, + 410 + ], + "score": 1.0, + "content": "to denote Loewner ordering, i.e.", + "type": "text" + }, + { + "bbox": [ + 462, + 399, + 495, + 410 + ], + "score": 0.89, + "content": "\\mathbf { A } \\succeq \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 408, + 488, + 422 + ], + "spans": [ + { + "bbox": [ + 107, + 410, + 137, + 420 + ], + "score": 0.81, + "content": "\\mathbf { A } - \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 408, + 295, + 422 + ], + "score": 1.0, + "content": "is positive semi-definite. We denote by", + "type": "text" + }, + { + "bbox": [ + 295, + 410, + 333, + 421 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { m i n } } ( \\mathbf { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 408, + 474, + 422 + ], + "score": 1.0, + "content": "the minimum eigenvalue of matrix", + "type": "text" + }, + { + "bbox": [ + 474, + 410, + 484, + 419 + ], + "score": 0.46, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 408, + 488, + 422 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 387, + 506, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 493, + 435 + ], + "lines": [ + { + "bbox": [ + 104, + 419, + 494, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 195, + 439 + ], + "score": 1.0, + "content": "We consider arbitrary", + "type": "text" + }, + { + "bbox": [ + 195, + 424, + 201, + 434 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 419, + 343, + 439 + ], + "score": 1.0, + "content": "-dimensional non-convex functions", + "type": "text" + }, + { + "bbox": [ + 344, + 423, + 392, + 435 + ], + "score": 0.93, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 419, + 494, + 439 + ], + "score": 1.0, + "content": "satisfying the following:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 419, + 494, + 439 + ] + }, + { + "type": "list", + "bbox": [ + 111, + 437, + 489, + 464 + ], + "lines": [ + { + "bbox": [ + 109, + 437, + 483, + 451 + ], + "spans": [ + { + "bbox": [ + 109, + 437, + 120, + 451 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 438, + 141, + 451 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 437, + 151, + 451 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 152, + 439, + 159, + 449 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 437, + 273, + 451 + ], + "score": 1.0, + "content": "-Lipschitz smooth: meaning", + "type": "text" + }, + { + "bbox": [ + 273, + 438, + 404, + 451 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) - \\nabla f ( y ) \\| \\leq L \\| x - y \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 437, + 437, + 451 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 437, + 437, + 477, + 450 + ], + "score": 0.92, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 437, + 483, + 451 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 109, + 450, + 490, + 465 + ], + "spans": [ + { + "bbox": [ + 109, + 450, + 120, + 465 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 452, + 141, + 464 + ], + "score": 0.89, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 450, + 151, + 465 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 152, + 452, + 164, + 463 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 450, + 256, + 465 + ], + "score": 1.0, + "content": "-second-order smooth:", + "type": "text" + }, + { + "bbox": [ + 257, + 451, + 412, + 464 + ], + "score": 0.91, + "content": "\\| \\nabla ^ { 2 } f ( x ) - \\nabla ^ { 2 } f ( y ) \\| _ { 2 } \\leq L _ { 2 } \\cdot \\| x - y \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 450, + 444, + 465 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 445, + 451, + 485, + 463 + ], + "score": 0.91, + "content": "x , y \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 450, + 490, + 465 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 29.5, + "bbox_fs": [ + 109, + 437, + 490, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 466, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 316, + 478 + ], + "score": 1.0, + "content": "For notational simplicity of the proofs, we assume", + "type": "text" + }, + { + "bbox": [ + 317, + 466, + 398, + 477 + ], + "score": 0.93, + "content": "L = L _ { 2 } = \\mathcal { V } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 465, + 505, + 478 + ], + "score": 1.0, + "content": ".2 Note that we have also", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 474, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 204, + 490 + ], + "score": 1.0, + "content": "assumed the domain of", + "type": "text" + }, + { + "bbox": [ + 204, + 478, + 212, + 489 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 474, + 290, + 490 + ], + "score": 1.0, + "content": "is the entire space", + "type": "text" + }, + { + "bbox": [ + 291, + 476, + 304, + 487 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 474, + 463, + 490 + ], + "score": 1.0, + "content": ". If instead there is a compact domain", + "type": "text" + }, + { + "bbox": [ + 463, + 476, + 501, + 488 + ], + "score": 0.9, + "content": "\\mathcal { X } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 474, + 506, + 490 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "then one can use projected SGD and re-derive similar results of this paper. We choose to present our", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 325, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 325, + 511 + ], + "score": 1.0, + "content": "result in the simplest setting to convey our main ideas.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 465, + 506, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 391, + 525 + ], + "score": 1.0, + "content": "Byzantine non-convex stochastic distributed optimization. We let", + "type": "text" + }, + { + "bbox": [ + 391, + 515, + 401, + 523 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "be the number of worker", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 242, + 538 + ], + "score": 1.0, + "content": "machines and assume at most an", + "type": "text" + }, + { + "bbox": [ + 243, + 527, + 250, + 534 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 523, + 394, + 538 + ], + "score": 1.0, + "content": "fraction of them are Byzantine for", + "type": "text" + }, + { + "bbox": [ + 394, + 523, + 440, + 537 + ], + "score": 0.93, + "content": "\\alpha \\in [ 0 , \\frac { 1 } { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 523, + 505, + 538 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 129, + 549 + ], + "score": 1.0, + "content": "good", + "type": "text" + }, + { + "bbox": [ + 129, + 536, + 156, + 549 + ], + "score": 0.57, + "content": "\\subseteq [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 536, + 504, + 549 + ], + "score": 1.0, + "content": "the set of good (i.e. non-Byzantine) machines, and the algorithm does not know good.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 513, + 505, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 249, + 565 + ], + "score": 1.0, + "content": "Assumption 2.1. In each iteration", + "type": "text" + }, + { + "bbox": [ + 250, + 553, + 255, + 562 + ], + "score": 0.5, + "content": "t _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 550, + 493, + 565 + ], + "score": 1.0, + "content": ", the algorithm (on the master) is allowed to specify a point", + "type": "text" + }, + { + "bbox": [ + 493, + 554, + 504, + 563 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 561, + 459, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 262, + 578 + ], + "score": 1.0, + "content": "and query m machines. 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Our techniques more broadly apply to more", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "complicated methods (e.g. at least to Allen-Zhu (2018a;b)), but we choose to analyze the simplest", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "variant of SGD, since it is the most widely applied method in modern non-convex machine learning.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 294, + 383 + ], + "score": 1.0, + "content": "As illustrated in Algorithm 1, in each iteration", + "type": "text" + }, + { + "bbox": [ + 294, + 370, + 372, + 381 + ], + "score": 0.91, + "content": "t = 0 , 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 369, + 505, + 383 + ], + "score": 1.0, + "content": ", we maintain a set of (allegedly)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 169, + 394 + ], + "score": 1.0, + "content": "good machines", + "type": "text" + }, + { + "bbox": [ + 170, + 381, + 224, + 393 + ], + "score": 0.91, + "content": "\\mathsf { g o o d } _ { t } \\subseteq [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 381, + 291, + 394 + ], + "score": 1.0, + "content": ". 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We choose a learning rate", + "type": "text" + }, + { + "bbox": [ + 336, + 393, + 361, + 404 + ], + "score": 0.9, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 392, + 482, + 405 + ], + "score": 1.0, + "content": ", and perform the SGD update", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 405, + 388, + 422 + ], + "lines": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "spans": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x _ { t + 1 } = x _ { t } + \\xi _ { t } - \\eta \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } } \\nabla _ { t , i } } \\end{array}", + "type": "interline_equation", + "image_path": "fef879fcba28b5759684029a259322eb9f2ace16065ea7970bf4277bf0697f8f.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 425, + 483, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 486, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 133, + 439 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 425, + 196, + 438 + ], + "score": 0.93, + "content": "\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 423, + 486, + 439 + ], + "score": 1.0, + "content": "is a random Gaussian perturbation that is added for theoretical purpose.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 178, + 453 + ], + "score": 1.0, + "content": "For each machine", + "type": "text" + }, + { + "bbox": [ + 179, + 440, + 209, + 452 + ], + "score": 0.92, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 438, + 505, + 453 + ], + "score": 1.0, + "content": ", we keep track of the history of its stochastic gradients up to two windows.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 238, + 468 + ], + "score": 1.0, + "content": "Namely, Ai ← Ptk=last1 k,i|good |", + "type": "text" + }, + { + "bbox": [ + 236, + 453, + 351, + 469 + ], + "score": 1.0, + "content": "and Bi ← P k=last0 k,i|good |", + "type": "text" + }, + { + "bbox": [ + 353, + 452, + 429, + 466 + ], + "score": 1.0, + "content": "for windows sizes", + "type": "text" + }, + { + "bbox": [ + 429, + 453, + 485, + 465 + ], + "score": 0.92, + "content": "T _ { 0 } \\leq T _ { 1 } \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 452, + 505, + 466 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 271, + 478 + ], + "score": 1.0, + "content": "compare among remaining machines in", + "type": "text" + }, + { + "bbox": [ + 272, + 468, + 297, + 479 + ], + "score": 0.91, + "content": "\\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 466, + 433, + 478 + ], + "score": 1.0, + "content": ", and kick out those ones whose", + "type": "text" + }, + { + "bbox": [ + 433, + 467, + 445, + 478 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 459, + 478 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 460, + 467, + 471, + 478 + ], + "score": 0.89, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "deviate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 476, + 467, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 231, + 491 + ], + "score": 1.0, + "content": "“more than usual” to construct", + "type": "text" + }, + { + "bbox": [ + 231, + 478, + 266, + 490 + ], + "score": 0.9, + "content": "\\mathtt { g o o d } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 476, + 467, + 491 + ], + "score": 1.0, + "content": ". Conceptually, we view these two as safe guards.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 506, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 504, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 504, + 504 + ], + "score": 1.0, + "content": "Our theory makes sure that, when the “window sizes” and the thresholds for “more than usual”", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 211, + 516 + ], + "score": 1.0, + "content": "are defined properly, then", + "type": "text" + }, + { + "bbox": [ + 212, + 504, + 237, + 515 + ], + "score": 0.88, + "content": "\\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "shall always include good, and the algorithm shall proceed to find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 338, + 529 + ], + "score": 1.0, + "content": "approximate local minima. Formally, we have (letting the", + "type": "text" + }, + { + "bbox": [ + 338, + 514, + 347, + 526 + ], + "score": 0.86, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "notion to hide polylogarithmic factors)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 502, + 562 + ], + "lines": [ + { + "bbox": [ + 104, + 528, + 501, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 186, + 551 + ], + "score": 1.0, + "content": "Theorem 2.3. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 533, + 248, + 547 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C _ { 3 } = \\alpha ^ { 2 } + \\frac { 1 } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 528, + 335, + 551 + ], + "score": 1.0, + "content": ". Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 336, + 532, + 389, + 546 + ], + "score": 0.85, + "content": "\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 528, + 393, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 531, + 447, + 548 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\eta = \\widetilde { \\Theta } \\big ( \\frac { \\varepsilon ^ { 2 } } { d C _ { 3 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 452, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 452, + 531, + 501, + 549 + ], + "score": 0.92, + "content": "T _ { 0 } = \\widetilde { \\Theta } ( \\textstyle { \\frac { 1 } { \\eta } } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 547, + 354, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 163, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } ( \\frac { 1 } { \\eta \\sqrt { \\varepsilon } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 548, + 168, + 564 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 168, + 547, + 227, + 562 + ], + "score": 0.9, + "content": "{ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 548, + 249, + 564 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 547, + 308, + 562 + ], + "score": 0.93, + "content": "\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 548, + 354, + 564 + ], + "score": 1.0, + "content": ", then after", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 567, + 381, + 588 + ], + "lines": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "spans": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\left( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } ( \\alpha ^ { 2 } + \\frac { 1 } { m } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "2748fee727c58ddaf44a1063d330a1f4039a7906174f66156826a917636e2952.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 590, + 489, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 489, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 409, + 605 + ], + "score": 1.0, + "content": "iterations, with high probability, for at least constant fraction of the indices", + "type": "text" + }, + { + "bbox": [ + 409, + 590, + 439, + 603 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 588, + 489, + 605 + ], + "score": 1.0, + "content": ", they satisfy", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 604, + 394, + 619 + ], + "lines": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "spans": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "score": 0.9, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\le \\varepsilon \\quad a n d \\quad \\nabla ^ { 2 } f ( x _ { t } ) \\succeq - \\sqrt { \\varepsilon } { \\bf I } .", + "type": "interline_equation", + "image_path": "b4d6cf83efa994c492e447ee88cfe8f23a120d1c42deec52db29bb31712342b4.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Remark 2.4. If one only wishes to achieve a significantly simpler goal — finding first-order critical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 631, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 134, + 647 + ], + "score": 1.0, + "content": "points", + "type": "text" + }, + { + "bbox": [ + 134, + 632, + 195, + 645 + ], + "score": 0.88, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 631, + 505, + 647 + ], + "score": 1.0, + "content": "— the analysis becomes much easier (see Section 3.1). In particular, having", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 275, + 657 + ], + "score": 1.0, + "content": "one safe guard without perturbation (i.e.", + "type": "text" + }, + { + "bbox": [ + 275, + 644, + 303, + 654 + ], + "score": 0.87, + "content": "\\nu = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ") suffices, and the iteration complexity reduces to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 652, + 508, + 675 + ], + "spans": [ + { + "bbox": [ + 103, + 652, + 508, + 675 + ], + "score": 1.0, + "content": "T = Oe\u0000 f(x0)−min f(x)ε4 (α2 + 1m )\u0001. Bulusu et al. (2020) achieves this easier goal but requires an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 435, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 435, + 681 + ], + "score": 1.0, + "content": "additional assumption: there is one guaranteed good worker known by the master.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 685, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 698 + ], + "score": 1.0, + "content": "Our contribution. We reiterate our theoretical contributions from three perspectives. 1) When", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 693, + 508, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 155, + 711 + ], + "score": 0.92, + "content": "\\alpha < 1 / \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 693, + 251, + 714 + ], + "score": 1.0, + "content": ", our algorithm requires", + "type": "text" + }, + { + "bbox": [ + 252, + 696, + 367, + 712 + ], + "score": 0.92, + "content": "\\begin{array} { r } { m T = \\widetilde { O } \\big ( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 693, + 508, + 714 + ], + "score": 1.0, + "content": "stochastic gradient computations.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "This matches the best known result (Jin et al., 2019) under our minimal assumptions of the non-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 456, + 733 + ], + "score": 1.0, + "content": "convex objective. 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We begin with", + "type": "text" + }, + { + "bbox": [ + 292, + 382, + 347, + 393 + ], + "score": 0.91, + "content": "\\mathtt { g o o d } _ { 0 } = [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "and start to detect malicious machines", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 392, + 482, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 336, + 405 + ], + "score": 1.0, + "content": "and remove them from the set. We choose a learning rate", + "type": "text" + }, + { + "bbox": [ + 336, + 393, + 361, + 404 + ], + "score": 0.9, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 392, + 482, + 405 + ], + "score": 1.0, + "content": ", and perform the SGD update", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 369, + 505, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 405, + 388, + 422 + ], + "lines": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "spans": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x _ { t + 1 } = x _ { t } + \\xi _ { t } - \\eta \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } } \\nabla _ { t , i } } \\end{array}", + "type": "interline_equation", + "image_path": "fef879fcba28b5759684029a259322eb9f2ace16065ea7970bf4277bf0697f8f.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 405, + 388, + 422 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 425, + 483, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 486, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 133, + 439 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 425, + 196, + 438 + ], + "score": 0.93, + "content": "\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 423, + 486, + 439 + ], + "score": 1.0, + "content": "is a random Gaussian perturbation that is added for theoretical purpose.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 423, + 486, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 178, + 453 + ], + "score": 1.0, + "content": "For each machine", + "type": "text" + }, + { + "bbox": [ + 179, + 440, + 209, + 452 + ], + "score": 0.92, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 438, + 505, + 453 + ], + "score": 1.0, + "content": ", we keep track of the history of its stochastic gradients up to two windows.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 238, + 468 + ], + "score": 1.0, + "content": "Namely, Ai ← Ptk=last1 k,i|good |", + "type": "text" + }, + { + "bbox": [ + 236, + 453, + 351, + 469 + ], + "score": 1.0, + "content": "and Bi ← P k=last0 k,i|good |", + "type": "text" + }, + { + "bbox": [ + 353, + 452, + 429, + 466 + ], + "score": 1.0, + "content": "for windows sizes", + "type": "text" + }, + { + "bbox": [ + 429, + 453, + 485, + 465 + ], + "score": 0.92, + "content": "T _ { 0 } \\leq T _ { 1 } \\leq T", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 452, + 505, + 466 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 271, + 478 + ], + "score": 1.0, + "content": "compare among remaining machines in", + "type": "text" + }, + { + "bbox": [ + 272, + 468, + 297, + 479 + ], + "score": 0.91, + "content": "\\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 466, + 433, + 478 + ], + "score": 1.0, + "content": ", and kick out those ones whose", + "type": "text" + }, + { + "bbox": [ + 433, + 467, + 445, + 478 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 459, + 478 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 460, + 467, + 471, + 478 + ], + "score": 0.89, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "deviate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 476, + 467, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 231, + 491 + ], + "score": 1.0, + "content": "“more than usual” to construct", + "type": "text" + }, + { + "bbox": [ + 231, + 478, + 266, + 490 + ], + "score": 0.9, + "content": "\\mathtt { g o o d } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 476, + 467, + 491 + ], + "score": 1.0, + "content": ". Conceptually, we view these two as safe guards.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 438, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 506, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 504, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 504, + 504 + ], + "score": 1.0, + "content": "Our theory makes sure that, when the “window sizes” and the thresholds for “more than usual”", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 211, + 516 + ], + "score": 1.0, + "content": "are defined properly, then", + "type": "text" + }, + { + "bbox": [ + 212, + 504, + 237, + 515 + ], + "score": 0.88, + "content": "\\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "shall always include good, and the algorithm shall proceed to find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 338, + 529 + ], + "score": 1.0, + "content": "approximate local minima. Formally, we have (letting the", + "type": "text" + }, + { + "bbox": [ + 338, + 514, + 347, + 526 + ], + "score": 0.86, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "notion to hide polylogarithmic factors)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 492, + 505, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 502, + 562 + ], + "lines": [ + { + "bbox": [ + 104, + 528, + 501, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 186, + 551 + ], + "score": 1.0, + "content": "Theorem 2.3. Let", + "type": "text" + }, + { + "bbox": [ + 187, + 533, + 248, + 547 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C _ { 3 } = \\alpha ^ { 2 } + \\frac { 1 } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 528, + 335, + 551 + ], + "score": 1.0, + "content": ". Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 336, + 532, + 389, + 546 + ], + "score": 0.85, + "content": "\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 528, + 393, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 531, + 447, + 548 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\eta = \\widetilde { \\Theta } \\big ( \\frac { \\varepsilon ^ { 2 } } { d C _ { 3 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 452, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 452, + 531, + 501, + 549 + ], + "score": 0.92, + "content": "T _ { 0 } = \\widetilde { \\Theta } ( \\textstyle { \\frac { 1 } { \\eta } } )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 547, + 354, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 163, + 565 + ], + "score": 0.91, + "content": "\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } ( \\frac { 1 } { \\eta \\sqrt { \\varepsilon } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 548, + 168, + 564 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 168, + 547, + 227, + 562 + ], + "score": 0.9, + "content": "{ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 548, + 249, + 564 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 547, + 308, + 562 + ], + "score": 0.93, + "content": "\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 548, + 354, + 564 + ], + "score": 1.0, + "content": ", then after", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 528, + 501, + 565 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 567, + 381, + 588 + ], + "lines": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "spans": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\left( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } ( \\alpha ^ { 2 } + \\frac { 1 } { m } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "2748fee727c58ddaf44a1063d330a1f4039a7906174f66156826a917636e2952.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 229, + 567, + 381, + 588 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 590, + 489, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 489, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 409, + 605 + ], + "score": 1.0, + "content": "iterations, with high probability, for at least constant fraction of the indices", + "type": "text" + }, + { + "bbox": [ + 409, + 590, + 439, + 603 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 588, + 489, + 605 + ], + "score": 1.0, + "content": ", they satisfy", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 588, + 489, + 605 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 604, + 394, + 619 + ], + "lines": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "spans": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "score": 0.9, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\le \\varepsilon \\quad a n d \\quad \\nabla ^ { 2 } f ( x _ { t } ) \\succeq - \\sqrt { \\varepsilon } { \\bf I } .", + "type": "interline_equation", + "image_path": "b4d6cf83efa994c492e447ee88cfe8f23a120d1c42deec52db29bb31712342b4.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 216, + 604, + 394, + 619 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Remark 2.4. If one only wishes to achieve a significantly simpler goal — finding first-order critical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 631, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 134, + 647 + ], + "score": 1.0, + "content": "points", + "type": "text" + }, + { + "bbox": [ + 134, + 632, + 195, + 645 + ], + "score": 0.88, + "content": "\\| \\nabla f ( x _ { t } ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 631, + 505, + 647 + ], + "score": 1.0, + "content": "— the analysis becomes much easier (see Section 3.1). In particular, having", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 275, + 657 + ], + "score": 1.0, + "content": "one safe guard without perturbation (i.e.", + "type": "text" + }, + { + "bbox": [ + 275, + 644, + 303, + 654 + ], + "score": 0.87, + "content": "\\nu = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ") suffices, and the iteration complexity reduces to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 652, + 508, + 675 + ], + "spans": [ + { + "bbox": [ + 103, + 652, + 508, + 675 + ], + "score": 1.0, + "content": "T = Oe\u0000 f(x0)−min f(x)ε4 (α2 + 1m )\u0001. Bulusu et al. (2020) achieves this easier goal but requires an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 435, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 435, + 681 + ], + "score": 1.0, + "content": "additional assumption: there is one guaranteed good worker known by the master.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 103, + 621, + 508, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 685, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 698 + ], + "score": 1.0, + "content": "Our contribution. We reiterate our theoretical contributions from three perspectives. 1) When", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 693, + 508, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 155, + 711 + ], + "score": 0.92, + "content": "\\alpha < 1 / \\sqrt { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 693, + 251, + 714 + ], + "score": 1.0, + "content": ", our algorithm requires", + "type": "text" + }, + { + "bbox": [ + 252, + 696, + 367, + 712 + ], + "score": 0.92, + "content": "\\begin{array} { r } { m T = \\widetilde { O } \\big ( \\frac { ( f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) ) d } { \\varepsilon ^ { 4 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 693, + 508, + 714 + ], + "score": 1.0, + "content": "stochastic gradient computations.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "This matches the best known result (Jin et al., 2019) under our minimal assumptions of the non-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 456, + 733 + ], + "score": 1.0, + "content": "convex objective. (There exist other works in the stochastic setting that break the", + "type": "text" + }, + { + "bbox": [ + 456, + 720, + 473, + 731 + ], + "score": 0.89, + "content": "\\varepsilon ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "barrier", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 91, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 275, + 106 + ], + "score": 1.0, + "content": "and get rid of the dimension dependence", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 275, + 93, + 282, + 102 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 283, + 91, + 451, + 106 + ], + "score": 1.0, + "content": "under stronger assumptions.)5. 2) When", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 451, + 92, + 501, + 105 + ], + "score": 0.92, + "content": "\\alpha < 1 / \\sqrt { m }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 502, + 91, + 505, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "our algorithm enjoys linear parallel speed-up: the parallel time complexity reduces by a factor of", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 107, + 116, + 132, + 128 + ], + "score": 0.91, + "content": "\\Theta ( m )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 132, + 115, + 163, + 129 + ], + "score": 1.0, + "content": ". When", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 164, + 115, + 237, + 128 + ], + "score": 0.93, + "content": "\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 237, + 115, + 366, + 129 + ], + "score": 1.0, + "content": ", our parallel time complexity is", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 366, + 114, + 394, + 128 + ], + "score": 0.93, + "content": "\\widetilde { O } ( \\alpha ^ { 2 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 394, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "times that needed when no", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 327, + 140 + ], + "score": 1.0, + "content": "parallelism is used, still giving noticeable speedup. 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When", + "type": "text" + }, + { + "bbox": [ + 164, + 115, + 237, + 128 + ], + "score": 0.93, + "content": "\\alpha \\in [ 1 / \\sqrt { m } , 1 / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 115, + 366, + 129 + ], + "score": 1.0, + "content": ", our parallel time complexity is", + "type": "text" + }, + { + "bbox": [ + 366, + 114, + 394, + 128 + ], + "score": 0.93, + "content": "\\widetilde { O } ( \\alpha ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "times that needed when no", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 327, + 140 + ], + "score": 1.0, + "content": "parallelism is used, still giving noticeable speedup. The", + "type": "text" + }, + { + "bbox": [ + 328, + 127, + 340, + 137 + ], + "score": 0.87, + "content": "\\textstyle { \\hat { \\alpha } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "factor also appeared in convex Byzantine", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 487, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 487, + 151 + ], + "score": 1.0, + "content": "distributed optimization (and is known to be tight there) (Yin et al., 2018; Alistarh et al., 2018).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 153, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "score": 1.0, + "content": "Comparison to (Yin et al., 2019). Yin et al. (2019) derived three gradient descent-type", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 418, + 176 + ], + "score": 1.0, + "content": "algorithms to find points with a weaker (and less standard) guarantee:", + "type": "text" + }, + { + "bbox": [ + 418, + 164, + 485, + 176 + ], + "score": 0.92, + "content": "\\| \\nabla f ( x ) \\| ~ \\leq ~ \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 208, + 188 + ], + "score": 0.91, + "content": "\\bar { \\nabla ^ { 2 } } f ( x ) ~ \\succeq ~ - ( \\varepsilon ^ { 2 } d ) ^ { \\bar { 1 } / 5 } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 174, + 506, + 190 + ], + "score": 1.0, + "content": ". Despite practical differences (namely, gradient descent may be less", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "favorable comparing to stochastic gradient descent especially in deep learning applications), the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 452, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 452, + 212 + ], + "score": 1.0, + "content": "parallel time complexities derived from their result are also generally larger than ours.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "Their paper focuses on bounding the number of sampled stochastic functions, as opposed to the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "number of stochastic gradient evaluations like we do. When translated to our language, each of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 103, + 233, + 503, + 252 + ], + "spans": [ + { + "bbox": [ + 103, + 233, + 455, + 252 + ], + "score": 1.0, + "content": "workers in their setting needs to evaluate T stochastic gradients, where (1) T = Oe\u0000 α2dε4", + "type": "text" + }, + { + "bbox": [ + 391, + 234, + 503, + 250 + ], + "score": 0.92, + "content": "\\begin{array} { r } { 1 ) T = \\widetilde { O } \\big ( \\frac { \\alpha ^ { 2 } \\overline { { d } } } { \\varepsilon ^ { 4 } } + \\frac { d ^ { 2 } } { \\varepsilon ^ { 4 } m } + \\frac { \\sqrt { d } } { \\varepsilon ^ { 3 } } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 246, + 510, + 284 + ], + "spans": [ + { + "bbox": [ + 102, + 246, + 331, + 267 + ], + "score": 1.0, + "content": "if using coordinate-wise median, (2) T = Oe\u0000 α2d2ε4", + "type": "text" + }, + { + "bbox": [ + 107, + 263, + 183, + 279 + ], + "score": 0.94, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\big ( \\frac { \\alpha } { \\varepsilon ^ { 4 } } + \\frac { d } { \\varepsilon ^ { 4 } m } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 248, + 265, + 284 + ], + "score": 1.0, + "content": "if using iterative filt", + "type": "text" + }, + { + "bbox": [ + 364, + 248, + 510, + 284 + ], + "score": 1.0, + "content": "if using trimmed mean, and (3)(1) and (2) are larger than ours (also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "with a weaker guarantee); the complexity (3) seems incomparable to ours, but when translating to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 102, + 284, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 102, + 284, + 348, + 306 + ], + "score": 1.0, + "content": "the more standard (ε, √ε) guarantee, becomes T = Oe\u0000 αd2ε5", + "type": "text" + }, + { + "bbox": [ + 294, + 288, + 378, + 303 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\widetilde { \\cal O } \\big ( \\frac { \\alpha d ^ { 2 } } { \\varepsilon ^ { 5 } } + \\frac { d ^ { 3 } } { \\varepsilon ^ { 5 } m } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "so is also larger than ours. It is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 498, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 226, + 314 + ], + "score": 1.0, + "content": "worth noting that (3) requires", + "type": "text" + }, + { + "bbox": [ + 226, + 302, + 262, + 314 + ], + "score": 0.91, + "content": "\\alpha < 1 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 302, + 498, + 314 + ], + "score": 1.0, + "content": "so cannot withstand half of the machines being Byzantine.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 466, + 330 + ], + "score": 1.0, + "content": "Resilience against practical attacks. Our algorithm’s filtering is based upon tracking", + "type": "text" + }, + { + "bbox": [ + 466, + 317, + 478, + 328 + ], + "score": 0.87, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 315, + 505, + 330 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 328, + 120, + 339 + ], + "score": 0.8, + "content": "A _ { i , { \\bf \\Pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 328, + 289, + 340 + ], + "score": 1.0, + "content": "), the stochastic gradients of each machine", + "type": "text" + }, + { + "bbox": [ + 290, + 329, + 295, + 338 + ], + "score": 0.5, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 328, + 407, + 340 + ], + "score": 1.0, + "content": "averaged over a window of", + "type": "text" + }, + { + "bbox": [ + 407, + 328, + 418, + 339 + ], + "score": 0.86, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 328, + 445, + 340 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + }, + { + "bbox": [ + 446, + 328, + 457, + 339 + ], + "score": 0.79, + "content": "T _ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 328, + 505, + 340 + ], + "score": 1.0, + "content": ") iterations.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "This is a departure from previous defenses, most of which are history-less, and enables us to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 350, + 499, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 499, + 362 + ], + "score": 1.0, + "content": "provably Byzantine-resilient against state-of-the-art attacks (Baruch et al., 2019; Xie et al., 2020).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 363, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 457, + 376 + ], + "score": 1.0, + "content": "In Baruch et al. (2019), Byzantine workers collude to shift the gradient mean by a factor", + "type": "text" + }, + { + "bbox": [ + 458, + 365, + 466, + 375 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 365, + 504, + 376 + ], + "score": 1.0, + "content": "times the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "standard deviation of the (true stochastic) gradient, while staying within population variance. They", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 138, + 400 + ], + "score": 1.0, + "content": "noticed", + "type": "text" + }, + { + "bbox": [ + 138, + 387, + 146, + 398 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "can be quite large especially in neural network training. Their attack circumvent existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "score": 1.0, + "content": "defenses because those defense algorithms are “historyless”, while their attack is statistically", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "indistinguishable from an honest execution in any single iteration. However, our algorithm can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 433 + ], + "score": 1.0, + "content": "provably defend against this attack since it has memory: Byzantine workers following their strategy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 324, + 443 + ], + "score": 1.0, + "content": "will progressively diverge from the (honest) “median” √", + "type": "text" + }, + { + "bbox": [ + 325, + 430, + 347, + 441 + ], + "score": 0.91, + "content": "B _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 429, + 470, + 443 + ], + "score": 1.0, + "content": "(by an amount proportional to", + "type": "text" + }, + { + "bbox": [ + 470, + 430, + 493, + 442 + ], + "score": 0.91, + "content": "\\Omega ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 115, + 452 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 441, + 212, + 456 + ], + "score": 1.0, + "content": "iterations as opposed to", + "type": "text" + }, + { + "bbox": [ + 212, + 441, + 230, + 453 + ], + "score": 0.89, + "content": "\\sqrt { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "), and be marked as malicious by our algorithm. (See Figure 2(a).) In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "Xie et al. (2020), Byzantine workers deviate in the negative direction of the gradient. However, to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "avoid being caught by our algorithm, the maximum “magnitude” of this attack has to stay within our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 495, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 495, + 488 + ], + "score": 1.0, + "content": "thresholds. We implemented both attacks and showed our algorithm’s robustness experimentally.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Finally, we note that prior “historyless” schemes, such as Krum or median-based schemes, could", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "score": 1.0, + "content": "be thought of as providing stronger guarantees, as they in theory allow Byzantine nodes to change", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "IDs during the computation: such schemes only require an upper bound on the number of Byzantine", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "agents in each round. However, the attack of Baruch et al. (2019) essentially shows that all such", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "schemes are vulnerable to variance attacks, and that such attacks are eminently plausible in practice.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "Thus, this suggests that the use of historical information, which requires that Byzantine nodes cannot", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 434, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 434, + 568 + ], + "score": 1.0, + "content": "change their IDs during the execution, may be necessary for Byzantine resilience.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Tolerating transient failures and node ID relabeling. Our algorithm can also withstand transient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 436, + 595 + ], + "score": 1.0, + "content": "node failures and some degrees of ID relabeling, by resetting the set of good nodes", + "type": "text" + }, + { + "bbox": [ + 437, + 582, + 462, + 593 + ], + "score": 0.89, + "content": "\\mathsf { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "to include", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 592, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 171, + 605 + ], + "score": 1.0, + "content": "all nodes every", + "type": "text" + }, + { + "bbox": [ + 171, + 593, + 182, + 603 + ], + "score": 0.87, + "content": "T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 592, + 504, + 605 + ], + "score": 1.0, + "content": "steps. The algorithm then proceeds as usual. The key observation behind this", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "relaxation is the fact that our analysis only requires that the attack conditions hold inside the current", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 614, + 489, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 489, + 626 + ], + "score": 1.0, + "content": "window. (Please see the Theorem B.1 for details.) We validate this experimentally in Section 5.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 108, + 638, + 291, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 293, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 293, + 653 + ], + "score": 1.0, + "content": "3 WARMUP: SINGLE SAFE GUARD", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "As a warmup, let us first analyze the behavior of perturbed SGD with a single safe guard. Consider", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 270, + 684 + ], + "score": 1.0, + "content": "Algorithm 2, where we start with a point", + "type": "text" + }, + { + "bbox": [ + 270, + 672, + 283, + 681 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 670, + 307, + 684 + ], + "score": 1.0, + "content": ", a set", + "type": "text" + }, + { + "bbox": [ + 307, + 671, + 367, + 682 + ], + "score": 0.8, + "content": "\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 670, + 421, + 684 + ], + "score": 1.0, + "content": ", and perform", + "type": "text" + }, + { + "bbox": [ + 422, + 671, + 430, + 680 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "steps of perturbed", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 180, + 694 + ], + "score": 1.0, + "content": "SGD. (We use the", + "type": "text" + }, + { + "bbox": [ + 180, + 683, + 192, + 692 + ], + "score": 0.85, + "content": "w _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 681, + 287, + 694 + ], + "score": 1.0, + "content": "sequence instead of the", + "type": "text" + }, + { + "bbox": [ + 287, + 683, + 298, + 692 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "sequence to emphasize that we are in Algorithm 2.)", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 117, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "5Works such as (Allen-Zhu, 2018a; Lei et al., 2017; Tripuraneni et al., 2017; Allen-Zhu, 2018b; Fang et al.,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 238, + 723 + ], + "score": 1.0, + "content": "2018; Nguyen et al., 2017) require", + "type": "text" + }, + { + "bbox": [ + 239, + 712, + 320, + 722 + ], + "score": 0.91, + "content": "f ( x ) = \\mathbb { E } _ { s \\sim \\mathcal { D } } [ f _ { s } ( x ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 711, + 367, + 723 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 367, + 712, + 389, + 722 + ], + "score": 0.91, + "content": "f _ { s } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "is second-order smooth and/or", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 424, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 424, + 733 + ], + "score": 1.0, + "content": "Lipschitz smooth. This requirement may be too strong for certain practical applications.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 92, + 505, + 150 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 91, + 505, + 151 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 153, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "score": 1.0, + "content": "Comparison to (Yin et al., 2019). Yin et al. (2019) derived three gradient descent-type", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 418, + 176 + ], + "score": 1.0, + "content": "algorithms to find points with a weaker (and less standard) guarantee:", + "type": "text" + }, + { + "bbox": [ + 418, + 164, + 485, + 176 + ], + "score": 0.92, + "content": "\\| \\nabla f ( x ) \\| ~ \\leq ~ \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 208, + 188 + ], + "score": 0.91, + "content": "\\bar { \\nabla ^ { 2 } } f ( x ) ~ \\succeq ~ - ( \\varepsilon ^ { 2 } d ) ^ { \\bar { 1 } / 5 } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 174, + 506, + 190 + ], + "score": 1.0, + "content": ". Despite practical differences (namely, gradient descent may be less", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "favorable comparing to stochastic gradient descent especially in deep learning applications), the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 452, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 452, + 212 + ], + "score": 1.0, + "content": "parallel time complexities derived from their result are also generally larger than ours.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 152, + 506, + 212 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "Their paper focuses on bounding the number of sampled stochastic functions, as opposed to the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "number of stochastic gradient evaluations like we do. When translated to our language, each of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 103, + 233, + 503, + 252 + ], + "spans": [ + { + "bbox": [ + 103, + 233, + 455, + 252 + ], + "score": 1.0, + "content": "workers in their setting needs to evaluate T stochastic gradients, where (1) T = Oe\u0000 α2dε4", + "type": "text" + }, + { + "bbox": [ + 391, + 234, + 503, + 250 + ], + "score": 0.92, + "content": "\\begin{array} { r } { 1 ) T = \\widetilde { O } \\big ( \\frac { \\alpha ^ { 2 } \\overline { { d } } } { \\varepsilon ^ { 4 } } + \\frac { d ^ { 2 } } { \\varepsilon ^ { 4 } m } + \\frac { \\sqrt { d } } { \\varepsilon ^ { 3 } } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 246, + 510, + 284 + ], + "spans": [ + { + "bbox": [ + 102, + 246, + 331, + 267 + ], + "score": 1.0, + "content": "if using coordinate-wise median, (2) T = Oe\u0000 α2d2ε4", + "type": "text" + }, + { + "bbox": [ + 107, + 263, + 183, + 279 + ], + "score": 0.94, + "content": "\\begin{array} { r } { T = \\widetilde { O } \\big ( \\frac { \\alpha } { \\varepsilon ^ { 4 } } + \\frac { d } { \\varepsilon ^ { 4 } m } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 248, + 265, + 284 + ], + "score": 1.0, + "content": "if using iterative filt", + "type": "text" + }, + { + "bbox": [ + 364, + 248, + 510, + 284 + ], + "score": 1.0, + "content": "if using trimmed mean, and (3)(1) and (2) are larger than ours (also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "with a weaker guarantee); the complexity (3) seems incomparable to ours, but when translating to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 102, + 284, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 102, + 284, + 348, + 306 + ], + "score": 1.0, + "content": "the more standard (ε, √ε) guarantee, becomes T = Oe\u0000 αd2ε5", + "type": "text" + }, + { + "bbox": [ + 294, + 288, + 378, + 303 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\widetilde { \\cal O } \\big ( \\frac { \\alpha d ^ { 2 } } { \\varepsilon ^ { 5 } } + \\frac { d ^ { 3 } } { \\varepsilon ^ { 5 } m } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "so is also larger than ours. It is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 498, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 226, + 314 + ], + "score": 1.0, + "content": "worth noting that (3) requires", + "type": "text" + }, + { + "bbox": [ + 226, + 302, + 262, + 314 + ], + "score": 0.91, + "content": "\\alpha < 1 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 302, + 498, + 314 + ], + "score": 1.0, + "content": "so cannot withstand half of the machines being Byzantine.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 102, + 211, + 510, + 314 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 466, + 330 + ], + "score": 1.0, + "content": "Resilience against practical attacks. Our algorithm’s filtering is based upon tracking", + "type": "text" + }, + { + "bbox": [ + 466, + 317, + 478, + 328 + ], + "score": 0.87, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 315, + 505, + 330 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 328, + 120, + 339 + ], + "score": 0.8, + "content": "A _ { i , { \\bf \\Pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 328, + 289, + 340 + ], + "score": 1.0, + "content": "), the stochastic gradients of each machine", + "type": "text" + }, + { + "bbox": [ + 290, + 329, + 295, + 338 + ], + "score": 0.5, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 328, + 407, + 340 + ], + "score": 1.0, + "content": "averaged over a window of", + "type": "text" + }, + { + "bbox": [ + 407, + 328, + 418, + 339 + ], + "score": 0.86, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 328, + 445, + 340 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + }, + { + "bbox": [ + 446, + 328, + 457, + 339 + ], + "score": 0.79, + "content": "T _ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 328, + 505, + 340 + ], + "score": 1.0, + "content": ") iterations.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "This is a departure from previous defenses, most of which are history-less, and enables us to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 350, + 499, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 499, + 362 + ], + "score": 1.0, + "content": "provably Byzantine-resilient against state-of-the-art attacks (Baruch et al., 2019; Xie et al., 2020).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 315, + 505, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 363, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 457, + 376 + ], + "score": 1.0, + "content": "In Baruch et al. (2019), Byzantine workers collude to shift the gradient mean by a factor", + "type": "text" + }, + { + "bbox": [ + 458, + 365, + 466, + 375 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 365, + 504, + 376 + ], + "score": 1.0, + "content": "times the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "standard deviation of the (true stochastic) gradient, while staying within population variance. They", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 138, + 400 + ], + "score": 1.0, + "content": "noticed", + "type": "text" + }, + { + "bbox": [ + 138, + 387, + 146, + 398 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "can be quite large especially in neural network training. Their attack circumvent existing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 410 + ], + "score": 1.0, + "content": "defenses because those defense algorithms are “historyless”, while their attack is statistically", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "indistinguishable from an honest execution in any single iteration. However, our algorithm can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 433 + ], + "score": 1.0, + "content": "provably defend against this attack since it has memory: Byzantine workers following their strategy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 324, + 443 + ], + "score": 1.0, + "content": "will progressively diverge from the (honest) “median” √", + "type": "text" + }, + { + "bbox": [ + 325, + 430, + 347, + 441 + ], + "score": 0.91, + "content": "B _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 429, + 470, + 443 + ], + "score": 1.0, + "content": "(by an amount proportional to", + "type": "text" + }, + { + "bbox": [ + 470, + 430, + 493, + 442 + ], + "score": 0.91, + "content": "\\Omega ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 115, + 452 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 441, + 212, + 456 + ], + "score": 1.0, + "content": "iterations as opposed to", + "type": "text" + }, + { + "bbox": [ + 212, + 441, + 230, + 453 + ], + "score": 0.89, + "content": "\\sqrt { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "), and be marked as malicious by our algorithm. (See Figure 2(a).) In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "Xie et al. (2020), Byzantine workers deviate in the negative direction of the gradient. However, to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "avoid being caught by our algorithm, the maximum “magnitude” of this attack has to stay within our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 495, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 495, + 488 + ], + "score": 1.0, + "content": "thresholds. We implemented both attacks and showed our algorithm’s robustness experimentally.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 365, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Finally, we note that prior “historyless” schemes, such as Krum or median-based schemes, could", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "score": 1.0, + "content": "be thought of as providing stronger guarantees, as they in theory allow Byzantine nodes to change", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "IDs during the computation: such schemes only require an upper bound on the number of Byzantine", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "agents in each round. However, the attack of Baruch et al. (2019) essentially shows that all such", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "schemes are vulnerable to variance attacks, and that such attacks are eminently plausible in practice.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "Thus, this suggests that the use of historical information, which requires that Byzantine nodes cannot", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 434, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 434, + 568 + ], + "score": 1.0, + "content": "change their IDs during the execution, may be necessary for Byzantine resilience.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 489, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Tolerating transient failures and node ID relabeling. Our algorithm can also withstand transient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 436, + 595 + ], + "score": 1.0, + "content": "node failures and some degrees of ID relabeling, by resetting the set of good nodes", + "type": "text" + }, + { + "bbox": [ + 437, + 582, + 462, + 593 + ], + "score": 0.89, + "content": "\\mathsf { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "to include", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 592, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 171, + 605 + ], + "score": 1.0, + "content": "all nodes every", + "type": "text" + }, + { + "bbox": [ + 171, + 593, + 182, + 603 + ], + "score": 0.87, + "content": "T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 592, + 504, + 605 + ], + "score": 1.0, + "content": "steps. The algorithm then proceeds as usual. The key observation behind this", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "relaxation is the fact that our analysis only requires that the attack conditions hold inside the current", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 614, + 489, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 489, + 626 + ], + "score": 1.0, + "content": "window. (Please see the Theorem B.1 for details.) We validate this experimentally in Section 5.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 570, + 505, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 638, + 291, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 293, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 293, + 653 + ], + "score": 1.0, + "content": "3 WARMUP: SINGLE SAFE GUARD", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "As a warmup, let us first analyze the behavior of perturbed SGD with a single safe guard. Consider", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 270, + 684 + ], + "score": 1.0, + "content": "Algorithm 2, where we start with a point", + "type": "text" + }, + { + "bbox": [ + 270, + 672, + 283, + 681 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 670, + 307, + 684 + ], + "score": 1.0, + "content": ", a set", + "type": "text" + }, + { + "bbox": [ + 307, + 671, + 367, + 682 + ], + "score": 0.8, + "content": "\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 670, + 421, + 684 + ], + "score": 1.0, + "content": ", and perform", + "type": "text" + }, + { + "bbox": [ + 422, + 671, + 430, + 680 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "steps of perturbed", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 180, + 694 + ], + "score": 1.0, + "content": "SGD. 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B _ { i } \\right\\| \\le \\mathfrak { T } \\right\\} \\right| > m / 2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 160, + 501, + 176 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 110, + 174, + 334, + 208 + ], + "spans": [ + { + "bbox": [ + 110, + 175, + 122, + 187 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 137, + 174, + 329, + 208 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } \\colon \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\Big ( \\xi _ { t } + \\frac { 1 } { | \\mathsf { g o o d } _ { t } | } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\Big ) ; } \\end{array}", + "type": "inline_equation", + "image_path": "b7140418ce17fd9368a1f403abc01a88478601ea76209698212f0b43839c272d.jpg" + }, + { + "bbox": [ + 329, + 174, + 334, + 188 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 110, + 190, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 110, + 191, + 122, + 204 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 376, + 190, + 445, + 203 + ], + "score": 1.0, + "content": "\u0005 Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 445, + 191, + 504, + 203 + ], + "score": 0.91, + "content": "\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 105, + 225, + 491, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 493, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 405, + 239 + ], + "score": 1.0, + "content": "Definition 3.1. We make the following definition to simplify notations: let", + "type": "text" + }, + { + "bbox": [ + 405, + 226, + 465, + 236 + ], + "score": 0.92, + "content": "\\Xi _ { t } : = \\sigma _ { t } + \\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 223, + 493, + 239 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 237, + 308, + 273 + ], + "lines": [ + { + "bbox": [ + 120, + 237, + 308, + 273 + ], + "spans": [ + { + "bbox": [ + 120, + 237, + 308, + 273 + ], + "score": 0.34, + "content": "\\begin{array} { r l } & { \\sigma _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\\\ & { \\Delta _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } \\backslash \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "2c08d6eb1052e237c59c211849be5ecaecb5e743c1ca8ee7ebfb5b633f1102c8.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 120, + 237, + 308, + 255.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 120, + 255.0, + 308, + 273.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 445, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 442, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 292, + 290 + ], + "score": 1.0, + "content": "Therefore, we can re-write the SGD update as", + "type": "text" + }, + { + "bbox": [ + 292, + 275, + 442, + 288 + ], + "score": 0.88, + "content": "w _ { t + 1 } = w _ { t } - \\eta ( \\nabla f ( w _ { t } ) + \\xi _ { t } + \\Xi _ { t } )", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 310, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 311, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 311, + 308 + ], + "score": 1.0, + "content": "The following lemma is fairly immediate to prove:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 503, + 334 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 504, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 396, + 325 + ], + "score": 1.0, + "content": "Lemma 3.2 (single safe guard). In Algorithm 2, suppose we choose", + "type": "text" + }, + { + "bbox": [ + 396, + 309, + 501, + 324 + ], + "score": 0.93, + "content": "\\mathfrak { T } = 8 \\sqrt { T \\log ( 1 6 m T / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 308, + 504, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 321, + 378, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 229, + 335 + ], + "score": 1.0, + "content": "Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 229, + 322, + 262, + 334 + ], + "score": 0.91, + "content": "1 - p / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 321, + 304, + 335 + ], + "score": 1.0, + "content": ", for every", + "type": "text" + }, + { + "bbox": [ + 304, + 322, + 373, + 334 + ], + "score": 0.91, + "content": "t = 0 , \\ldots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 321, + 378, + 335 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 336, + 440, + 433 + ], + "lines": [ + { + "bbox": [ + 114, + 349, + 440, + 433 + ], + "spans": [ + { + "bbox": [ + 114, + 349, + 440, + 433 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\bullet \\Vert \\sigma _ { t } \\Vert ^ { 2 } \\leq O ( \\frac { \\log ( T / p ) } { m } ) a n d \\Vert \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } ) } \\\\ & { \\bullet \\Vert \\Delta _ { t } \\Vert ^ { 2 } \\leq \\alpha ^ { 2 } a n d \\Vert \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\alpha ^ { 2 } T \\log ( m T / p ) ) } \\\\ & { \\bullet \\vert \\langle \\nabla f ( w _ { t } ) , \\xi _ { t } \\rangle \\vert \\leq \\Vert \\nabla f ( w _ { t } ) \\Vert \\cdot O ( \\nu \\sqrt { \\log ( T / p ) } ) , } \\\\ & { \\bullet \\Vert \\xi _ { t } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T / p ) ) , \\Vert \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d T \\log ( T / p ) ) } \\\\ & { \\textit { e c a l l t h i s p r o b a b i l i s t i c e n t \\ 5 \\mathrm { v e n t } } \\mathrm { t v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) a n d \\operatorname* { P r } [ \\mathrm { E v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) ] \\geq 1 - p / 4 . } \\end{array}", + "type": "interline_equation", + "image_path": "6b0a758c05f6a45046d4b3f6f38f2b4438c3506da58e940bacc88a861b71fce4.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 336, + 440, + 368.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 368.3333333333333, + 440, + 400.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 400.66666666666663, + 440, + 432.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 320, + 449 + ], + "score": 1.0, + "content": "(The third property above is ensured by our choice of", + "type": "text" + }, + { + "bbox": [ + 320, + 437, + 328, + 447 + ], + "score": 0.78, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "and the use of safe guard, and the rest of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 447, + 498, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 498, + 461 + ], + "score": 1.0, + "content": "properties follow from simple martingale concentration arguments. Details are in Appendix A.1.)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 105, + 467, + 359, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 358, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 358, + 480 + ], + "score": 1.0, + "content": "3.1 CORE TECHNICAL LEMMA 1: OBJECTIVE DECREASE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 299, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 300, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 300, + 498 + ], + "score": 1.0, + "content": "Our first main technical lemma is the following:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 251, + 514 + ], + "score": 1.0, + "content": "Lemma 3.3. Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 252, + 501, + 260, + 510 + ], + "score": 0.68, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 498, + 387, + 514 + ], + "score": 1.0, + "content": "as in Lemma 3.2. Denote by", + "type": "text" + }, + { + "bbox": [ + 387, + 500, + 456, + 513 + ], + "score": 0.92, + "content": "C _ { 1 } = \\log ( T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 498, + 478, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 501, + 505, + 512 + ], + "score": 0.87, + "content": "C _ { 2 } \\ =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 506, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 511, + 194, + 529 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha ^ { 2 } \\log { \\frac { m T } { p } } + { \\frac { \\log ( T / p ) } { m } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 506, + 235, + 531 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 236, + 513, + 322, + 528 + ], + "score": 0.88, + "content": "\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 506, + 326, + 531 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 344, + 512, + 507, + 531 + ], + "score": 1.0, + "content": "1100η(1+√C2) and we start from w0 and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 525, + 346, + 545 + ], + "spans": [ + { + "bbox": [ + 103, + 525, + 259, + 545 + ], + "score": 1.0, + "content": "apply Algorithm 2. Under event Even", + "type": "text" + }, + { + "bbox": [ + 260, + 529, + 298, + 543 + ], + "score": 0.83, + "content": "\\mathbf { \\Pi } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 525, + 346, + 545 + ], + "score": 1.0, + "content": ", it satisfies", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 545, + 494, + 562 + ], + "lines": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "spans": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "cf041560eae4b456c1cd25849fc5d274a7a6d973b5b21586d1f8aa27510fc58b.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 506, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 195, + 581 + ], + "score": 1.0, + "content": "Lemma 3.3 says after", + "type": "text" + }, + { + "bbox": [ + 195, + 568, + 225, + 583 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T \\approx \\frac { 1 } { \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "steps of perturbed SGD, the objective value decreases by, up to some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 102, + 578, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 102, + 578, + 313, + 601 + ], + "score": 1.0, + "content": "small additive error and up to logarithmic factors,", + "type": "text" + }, + { + "bbox": [ + 313, + 582, + 505, + 597 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } ( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 595, + 508, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 128, + 610 + ], + "score": 0.85, + "content": "\\eta C _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 595, + 311, + 616 + ], + "score": 1.0, + "content": ". This immediately implies, if we choose", + "type": "text" + }, + { + "bbox": [ + 312, + 597, + 351, + 612 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\eta \\approx \\frac { \\varepsilon ^ { 2 } } { C _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 595, + 508, + 616 + ], + "score": 1.0, + "content": ", then by repeating this analysis for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 610, + 486, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 196, + 628 + ], + "score": 0.93, + "content": "\\begin{array} { r } { O ( \\frac { C _ { 2 } } { \\varepsilon ^ { 4 } } ) = O ( \\frac { \\alpha ^ { 2 } + 1 / m } { \\varepsilon ^ { 4 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 610, + 486, + 630 + ], + "score": 1.0, + "content": "( α2+1/mε4 ) iterations, we can find approximate critical point x with k∇f (x)k ≤ ε.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 105, + 629, + 504, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "Proof sketch of Lemma 3.3. The full proof is in Appendix A.2 but we illustrate the main idea and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 640, + 402, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 402, + 653 + ], + "score": 1.0, + "content": "difficulties below. 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B _ { i } \\right\\| \\le \\mathfrak { T } \\right\\} \\right| > m / 2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 160, + 501, + 176 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 174, + 334, + 208 + ], + "spans": [ + { + "bbox": [ + 110, + 175, + 122, + 187 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 137, + 174, + 329, + 208 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } \\colon \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\Big ( \\xi _ { t } + \\frac { 1 } { | \\mathsf { g o o d } _ { t } | } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\Big ) ; } \\end{array}", + "type": "inline_equation", + "image_path": "b7140418ce17fd9368a1f403abc01a88478601ea76209698212f0b43839c272d.jpg" + }, + { + "bbox": [ + 329, + 174, + 334, + 188 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 190, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 110, + 191, + 122, + 204 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 376, + 190, + 445, + 203 + ], + "score": 1.0, + "content": "\u0005 Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 445, + 191, + 504, + 203 + ], + "score": 0.91, + "content": "\\xi _ { t } \\sim \\mathcal { N } ( 0 , \\nu ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + } + ], + "index": 7, + "is_list_start_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 106, + 97, + 504, + 208 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 225, + 491, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 493, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 405, + 239 + ], + "score": 1.0, + "content": "Definition 3.1. We make the following definition to simplify notations: let", + "type": "text" + }, + { + "bbox": [ + 405, + 226, + 465, + 236 + ], + "score": 0.92, + "content": "\\Xi _ { t } : = \\sigma _ { t } + \\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 223, + 493, + 239 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 223, + 493, + 239 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 237, + 308, + 273 + ], + "lines": [ + { + "bbox": [ + 120, + 237, + 308, + 273 + ], + "spans": [ + { + "bbox": [ + 120, + 237, + 308, + 273 + ], + "score": 0.34, + "content": "\\begin{array} { r l } & { \\sigma _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\\\ & { \\Delta _ { t } : = \\frac { 1 } { | \\mathrm { g o o d } _ { t } | } \\sum _ { i \\in \\mathrm { g o o d } _ { t } \\backslash \\mathrm { g o o d } } \\left( \\nabla _ { t , i } - \\nabla f ( w _ { t } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "2c08d6eb1052e237c59c211849be5ecaecb5e743c1ca8ee7ebfb5b633f1102c8.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 120, + 237, + 308, + 255.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 120, + 255.0, + 308, + 273.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 445, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 442, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 292, + 290 + ], + "score": 1.0, + "content": "Therefore, we can re-write the SGD update as", + "type": "text" + }, + { + "bbox": [ + 292, + 275, + 442, + 288 + ], + "score": 0.88, + "content": "w _ { t + 1 } = w _ { t } - \\eta ( \\nabla f ( w _ { t } ) + \\xi _ { t } + \\Xi _ { t } )", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 274, + 442, + 290 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 310, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 311, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 311, + 308 + ], + "score": 1.0, + "content": "The following lemma is fairly immediate to prove:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 292, + 311, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 503, + 334 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 504, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 396, + 325 + ], + "score": 1.0, + "content": "Lemma 3.2 (single safe guard). In Algorithm 2, suppose we choose", + "type": "text" + }, + { + "bbox": [ + 396, + 309, + 501, + 324 + ], + "score": 0.93, + "content": "\\mathfrak { T } = 8 \\sqrt { T \\log ( 1 6 m T / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 308, + 504, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 321, + 378, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 229, + 335 + ], + "score": 1.0, + "content": "Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 229, + 322, + 262, + 334 + ], + "score": 0.91, + "content": "1 - p / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 321, + 304, + 335 + ], + "score": 1.0, + "content": ", for every", + "type": "text" + }, + { + "bbox": [ + 304, + 322, + 373, + 334 + ], + "score": 0.91, + "content": "t = 0 , \\ldots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 321, + 378, + 335 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 308, + 504, + 335 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 336, + 440, + 433 + ], + "lines": [ + { + "bbox": [ + 114, + 349, + 440, + 433 + ], + "spans": [ + { + "bbox": [ + 114, + 349, + 440, + 433 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\bullet \\Vert \\sigma _ { t } \\Vert ^ { 2 } \\leq O ( \\frac { \\log ( T / p ) } { m } ) a n d \\Vert \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } ) } \\\\ & { \\bullet \\Vert \\Delta _ { t } \\Vert ^ { 2 } \\leq \\alpha ^ { 2 } a n d \\Vert \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\alpha ^ { 2 } T \\log ( m T / p ) ) } \\\\ & { \\bullet \\vert \\langle \\nabla f ( w _ { t } ) , \\xi _ { t } \\rangle \\vert \\leq \\Vert \\nabla f ( w _ { t } ) \\Vert \\cdot O ( \\nu \\sqrt { \\log ( T / p ) } ) , } \\\\ & { \\bullet \\Vert \\xi _ { t } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T / p ) ) , \\Vert \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\Vert ^ { 2 } \\leq O ( \\nu ^ { 2 } d T \\log ( T / p ) ) } \\\\ & { \\textit { e c a l l t h i s p r o b a b i l i s t i c e n t \\ 5 \\mathrm { v e n t } } \\mathrm { t v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) a n d \\operatorname* { P r } [ \\mathrm { E v e n t } _ { T } ^ { \\mathrm { s i n g l e } } ( w _ { 0 } ) ] \\geq 1 - p / 4 . } \\end{array}", + "type": "interline_equation", + "image_path": "6b0a758c05f6a45046d4b3f6f38f2b4438c3506da58e940bacc88a861b71fce4.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 106, + 336, + 440, + 368.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 106, + 368.3333333333333, + 440, + 400.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 106, + 400.66666666666663, + 440, + 432.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 320, + 449 + ], + "score": 1.0, + "content": "(The third property above is ensured by our choice of", + "type": "text" + }, + { + "bbox": [ + 320, + 437, + 328, + 447 + ], + "score": 0.78, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "and the use of safe guard, and the rest of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 447, + 498, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 498, + 461 + ], + "score": 1.0, + "content": "properties follow from simple martingale concentration arguments. Details are in Appendix A.1.)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 436, + 505, + 461 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 467, + 359, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 358, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 358, + 480 + ], + "score": 1.0, + "content": "3.1 CORE TECHNICAL LEMMA 1: OBJECTIVE DECREASE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 299, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 300, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 300, + 498 + ], + "score": 1.0, + "content": "Our first main technical lemma is the following:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 482, + 300, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 251, + 514 + ], + "score": 1.0, + "content": "Lemma 3.3. Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 252, + 501, + 260, + 510 + ], + "score": 0.68, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 498, + 387, + 514 + ], + "score": 1.0, + "content": "as in Lemma 3.2. Denote by", + "type": "text" + }, + { + "bbox": [ + 387, + 500, + 456, + 513 + ], + "score": 0.92, + "content": "C _ { 1 } = \\log ( T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 498, + 478, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 501, + 505, + 512 + ], + "score": 0.87, + "content": "C _ { 2 } \\ =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 506, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 511, + 194, + 529 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\alpha ^ { 2 } \\log { \\frac { m T } { p } } + { \\frac { \\log ( T / p ) } { m } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 506, + 235, + 531 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 236, + 513, + 322, + 528 + ], + "score": 0.88, + "content": "\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 506, + 326, + 531 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 344, + 512, + 507, + 531 + ], + "score": 1.0, + "content": "1100η(1+√C2) and we start from w0 and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 525, + 346, + 545 + ], + "spans": [ + { + "bbox": [ + 103, + 525, + 259, + 545 + ], + "score": 1.0, + "content": "apply Algorithm 2. Under event Even", + "type": "text" + }, + { + "bbox": [ + 260, + 529, + 298, + 543 + ], + "score": 0.83, + "content": "\\mathbf { \\Pi } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 525, + 346, + 545 + ], + "score": 1.0, + "content": ", it satisfies", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 103, + 498, + 507, + 545 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 545, + 494, + 562 + ], + "lines": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "spans": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "cf041560eae4b456c1cd25849fc5d274a7a6d973b5b21586d1f8aa27510fc58b.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 116, + 545, + 494, + 562 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 506, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 195, + 581 + ], + "score": 1.0, + "content": "Lemma 3.3 says after", + "type": "text" + }, + { + "bbox": [ + 195, + 568, + 225, + 583 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T \\approx \\frac { 1 } { \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "steps of perturbed SGD, the objective value decreases by, up to some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 102, + 578, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 102, + 578, + 313, + 601 + ], + "score": 1.0, + "content": "small additive error and up to logarithmic factors,", + "type": "text" + }, + { + "bbox": [ + 313, + 582, + 505, + 597 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } ( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 595, + 508, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 128, + 610 + ], + "score": 0.85, + "content": "\\eta C _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 595, + 311, + 616 + ], + "score": 1.0, + "content": ". This immediately implies, if we choose", + "type": "text" + }, + { + "bbox": [ + 312, + 597, + 351, + 612 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\eta \\approx \\frac { \\varepsilon ^ { 2 } } { C _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 595, + 508, + 616 + ], + "score": 1.0, + "content": ", then by repeating this analysis for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 610, + 486, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 196, + 628 + ], + "score": 0.93, + "content": "\\begin{array} { r } { O ( \\frac { C _ { 2 } } { \\varepsilon ^ { 4 } } ) = O ( \\frac { \\alpha ^ { 2 } + 1 / m } { \\varepsilon ^ { 4 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 610, + 486, + 630 + ], + "score": 1.0, + "content": "( α2+1/mε4 ) iterations, we can find approximate critical point x with k∇f (x)k ≤ ε.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 102, + 568, + 508, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 629, + 504, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "Proof sketch of Lemma 3.3. The full proof is in Appendix A.2 but we illustrate the main idea and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 640, + 402, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 402, + 653 + ], + "score": 1.0, + "content": "difficulties below. After simple manipulations, it is not hard to derive that", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 629, + 505, + 653 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 653, + 459, + 685 + ], + "lines": [ + { + "bbox": [ + 153, + 653, + 459, + 685 + ], + "spans": [ + { + "bbox": [ + 153, + 653, + 459, + 685 + ], + "score": 0.87, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) - f ( w _ { T } ) \\gtrapprox 0 . 9 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - \\eta \\right) + \\underbrace { \\eta \\sum _ { t = 0 } ^ { T - 1 } \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle } _ { \\mathrm { r e m a i n d e r ~ t e r m s } } } \\end{array}", + "type": "interline_equation", + "image_path": "89c3e604404cfa7fafaf5f0d084d6fa58cc7871c9fe8fdd33578cf56cbb3775b.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 153, + 653, + 459, + 663.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 153, + 663.6666666666666, + 459, + 674.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 153, + 674.3333333333333, + 459, + 684.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 174, + 699 + ], + "score": 1.0, + "content": "where recall that", + "type": "text" + }, + { + "bbox": [ + 174, + 688, + 229, + 699 + ], + "score": 0.92, + "content": "\\Xi _ { t } = \\sigma _ { t } + \\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 687, + 427, + 699 + ], + "score": 1.0, + "content": ". When there are no Byzantine machines, we have", + "type": "text" + }, + { + "bbox": [ + 428, + 687, + 504, + 700 + ], + "score": 0.95, + "content": "\\mathbb { E } [ \\Xi _ { t } ] = \\mathbb { E } [ \\sigma _ { t } ] = 0", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "so the remainder terms must be small by martingale concentration. Therefore, the main technical", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 466, + 722 + ], + "score": 1.0, + "content": "difficulty arises to deal with those Byzantine machines, who can adversarially design their", + "type": "text" + }, + { + "bbox": [ + 467, + 710, + 479, + 721 + ], + "score": 0.88, + "content": "\\nabla _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "(even", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 291, + 734 + ], + "score": 1.0, + "content": "by collusion) so as to negatively correlate with", + "type": "text" + }, + { + "bbox": [ + 292, + 721, + 325, + 732 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "to “maximally destroy” the above inequality.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 687, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 333, + 95 + ], + "score": 1.0, + "content": "Our main idea is to use second-order smoothness to write", + "type": "text" + }, + { + "bbox": [ + 333, + 82, + 501, + 95 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } ) \\approx \\nabla f ( w _ { 0 } ) + \\nabla ^ { 2 } f ( w _ { 0 } ) \\cdot ( w _ { t } - w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "To illustrate our idea, let us ignore the constant vector and assume that the Hessian is the identity:", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 189, + 119 + ], + "score": 1.0, + "content": "that is, imagine as if", + "type": "text" + }, + { + "bbox": [ + 190, + 104, + 270, + 117 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } ) \\mathbf { \\bar { \\approx } } w _ { t } - w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 104, + 300, + 119 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 301, + 104, + 414, + 117 + ], + "score": 0.9, + "content": "\\begin{array} { r } { w _ { t } - w _ { 0 } = - \\sum _ { k < t } \\Xi _ { t } + \\xi _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 104, + 506, + 119 + ], + "score": 1.0, + "content": ", we immediately have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 119, + 450, + 134 + ], + "lines": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "spans": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "score": 0.84, + "content": "\\begin{array} { r } { - \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle \\approx - \\langle w _ { t } - w _ { 0 } , \\Xi _ { t } \\rangle = \\sum _ { k < t } \\langle \\Xi _ { k } , \\Xi _ { t } \\rangle + \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}", + "type": "interline_equation", + "image_path": "3315623a3cd648ed53891363b61039c36b9bf574c6b4ad5106975dd21198c861.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 179 + ], + "lines": [ + { + "bbox": [ + 104, + 136, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 200, + 153 + ], + "score": 1.0, + "content": "For the first partial sum", + "type": "text" + }, + { + "bbox": [ + 200, + 138, + 261, + 151 + ], + "score": 0.92, + "content": "\\textstyle \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 136, + 505, + 153 + ], + "score": 1.0, + "content": "in (3.1), it is easy to bound its magnitude using our safeguard.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 150, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 104, + 150, + 172, + 166 + ], + "score": 1.0, + "content": "Indeed, we have", + "type": "text" + }, + { + "bbox": [ + 172, + 151, + 359, + 165 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big | \\sum _ { t } \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle \\big | \\leq \\| \\sum _ { t } \\Xi _ { t } \\| ^ { 2 } + \\sum _ { t } \\| \\Xi _ { t } \\| ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 150, + 506, + 166 + ], + "score": 1.0, + "content": "so we can apply Lemma 3.2. For the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 184, + 180 + ], + "score": 1.0, + "content": "second partial sum", + "type": "text" + }, + { + "bbox": [ + 184, + 166, + 257, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { t } \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 163, + 506, + 180 + ], + "score": 1.0, + "content": ", we can apply the concentration Proposition 3.4 below. \u0003", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 104, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 184, + 307, + 198 + ], + "score": 1.0, + "content": "Proposition 3.4. Fix the dimension parameter", + "type": "text" + }, + { + "bbox": [ + 307, + 186, + 336, + 197 + ], + "score": 0.91, + "content": "d \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 184, + 383, + 198 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 383, + 185, + 465, + 198 + ], + "score": 0.87, + "content": "\\xi _ { 0 \\underline { { : } } } \\dots , \\xi _ { T - 1 } \\ \\in \\ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "are i.i.d.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 155, + 209 + ], + "score": 1.0, + "content": "drawn from", + "type": "text" + }, + { + "bbox": [ + 155, + 196, + 187, + 209 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , \\bf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 196, + 226, + 209 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + }, + { + "bbox": [ + 226, + 197, + 286, + 208 + ], + "score": 0.9, + "content": "\\Delta _ { 1 } , \\ldots , \\Delta _ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 196, + 382, + 209 + ], + "score": 1.0, + "content": "are arbitrary vectors in", + "type": "text" + }, + { + "bbox": [ + 382, + 197, + 394, + 206 + ], + "score": 0.82, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 196, + 471, + 209 + ], + "score": 1.0, + "content": ". Here, each vector", + "type": "text" + }, + { + "bbox": [ + 472, + 197, + 484, + 208 + ], + "score": 0.88, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 177, + 219 + ], + "score": 0.88, + "content": "t = 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 208, + 242, + 220 + ], + "score": 1.0, + "content": "can depend on", + "type": "text" + }, + { + "bbox": [ + 242, + 209, + 292, + 219 + ], + "score": 0.9, + "content": "\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 208, + 338, + 220 + ], + "score": 1.0, + "content": "but not on", + "type": "text" + }, + { + "bbox": [ + 338, + 208, + 389, + 219 + ], + "score": 0.91, + "content": "\\xi _ { t } , \\dots , \\xi _ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 208, + 505, + 220 + ], + "score": 1.0, + "content": ". Suppose that these vectors", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 217, + 494, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 135, + 232 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + }, + { + "bbox": [ + 135, + 218, + 231, + 231 + ], + "score": 0.91, + "content": "\\| \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\| ^ { 2 } \\leq \\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 217, + 270, + 232 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 270, + 219, + 339, + 230 + ], + "score": 0.91, + "content": "t = 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 217, + 466, + 232 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 466, + 219, + 490, + 230 + ], + "score": 0.89, + "content": "1 - p", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 217, + 494, + 232 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 232, + 419, + 254 + ], + "lines": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "spans": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "4490f7c290ac0e971389f87ab29e6f26d4459fb36fe16ae481c193d32c11963b.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 370, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 370, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 370, + 270 + ], + "score": 1.0, + "content": "3.2 CORE TECHNICAL LEMMA 2: RANDOMNESS COUPLING", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 407, + 287 + ], + "score": 1.0, + "content": "Our next technical lemma studies that, if run Algorithm 2 from a point", + "type": "text" + }, + { + "bbox": [ + 408, + 277, + 420, + 286 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "so that the Hessian", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 107, + 285, + 145, + 298 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 284, + 294, + 298 + ], + "score": 1.0, + "content": "has a eigenvalue which is less than", + "type": "text" + }, + { + "bbox": [ + 294, + 286, + 308, + 296 + ], + "score": 0.87, + "content": "- \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 284, + 348, + 298 + ], + "score": 1.0, + "content": "(think of", + "type": "text" + }, + { + "bbox": [ + 349, + 287, + 361, + 297 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "as a saddle point), then with good", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 296, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 343, + 310 + ], + "score": 1.0, + "content": "probability, after sufficiently many iterations, the sequence", + "type": "text" + }, + { + "bbox": [ + 343, + 298, + 406, + 308 + ], + "score": 0.86, + "content": "w _ { 1 } , w _ { 2 } , \\dots , w _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 296, + 480, + 310 + ], + "score": 1.0, + "content": "shall escape from", + "type": "text" + }, + { + "bbox": [ + 481, + 298, + 493, + 308 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 296, + 506, + 310 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 171, + 321 + ], + "score": 1.0, + "content": "distance at least", + "type": "text" + }, + { + "bbox": [ + 171, + 308, + 180, + 317 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 306, + 260, + 321 + ], + "score": 1.0, + "content": "for some parameter", + "type": "text" + }, + { + "bbox": [ + 260, + 308, + 287, + 318 + ], + "score": 0.89, + "content": "R \\approx \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 306, + 506, + 321 + ], + "score": 1.0, + "content": ". To prove this, motivated by Jin et al. (2017), we study", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "two executions of Algorithm 2 where their randomness are coupled. We then argue that at least one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 460, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 216, + 342 + ], + "score": 1.0, + "content": "of them has to escape from", + "type": "text" + }, + { + "bbox": [ + 217, + 331, + 229, + 340 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 329, + 294, + 342 + ], + "score": 1.0, + "content": ". For any vector", + "type": "text" + }, + { + "bbox": [ + 294, + 331, + 300, + 339 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 329, + 317, + 342 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 317, + 329, + 331, + 342 + ], + "score": 0.9, + "content": "[ v ] _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 329, + 376, + 342 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + }, + { + "bbox": [ + 377, + 330, + 381, + 339 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 329, + 449, + 342 + ], + "score": 1.0, + "content": "-th coordinate of", + "type": "text" + }, + { + "bbox": [ + 450, + 332, + 455, + 339 + ], + "score": 0.78, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 329, + 460, + 342 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 253, + 358 + ], + "score": 1.0, + "content": "Lemma 3.5. Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 253, + 346, + 262, + 355 + ], + "score": 0.72, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 344, + 357, + 358 + ], + "score": 1.0, + "content": "as in Lemma 3.2 and", + "type": "text" + }, + { + "bbox": [ + 357, + 345, + 386, + 357 + ], + "score": 0.92, + "content": "C _ { 1 } , C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "as in Lemma 3.3. Suppose", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 147, + 367 + ], + "score": 0.91, + "content": "\\boldsymbol { w } _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 354, + 185, + 369 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 185, + 356, + 281, + 368 + ], + "score": 0.92, + "content": "\\dot { \\lambda } _ { \\mathrm { m i n } } ( \\nabla ^ { 2 } f ( w _ { 0 } ) ) = - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 354, + 322, + 369 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 322, + 356, + 350, + 367 + ], + "score": 0.9, + "content": "\\delta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 354, + 479, + 369 + ], + "score": 1.0, + "content": ". Without loss of generality let", + "type": "text" + }, + { + "bbox": [ + 479, + 357, + 491, + 367 + ], + "score": 0.85, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 180, + 381 + ], + "score": 1.0, + "content": "the eigenvector of", + "type": "text" + }, + { + "bbox": [ + 180, + 367, + 218, + 379 + ], + "score": 0.9, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 366, + 505, + 381 + ], + "score": 1.0, + "content": "with smallest eigenvalue. Consider now two executions of Algorithm 2,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 375, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 182, + 393 + ], + "score": 1.0, + "content": "both starting from", + "type": "text" + }, + { + "bbox": [ + 182, + 379, + 246, + 390 + ], + "score": 0.9, + "content": "w _ { 0 } ^ { \\mathsf { a } } = w _ { 0 } ^ { \\mathsf { b } } = w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 375, + 375, + 393 + ], + "score": 1.0, + "content": ", and suppose their randomness", + "type": "text" + }, + { + "bbox": [ + 376, + 378, + 399, + 390 + ], + "score": 0.92, + "content": "\\{ \\xi _ { t } ^ { \\mathsf { a } } \\} _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 375, + 419, + 393 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 419, + 378, + 442, + 390 + ], + "score": 0.92, + "content": "\\{ \\xi _ { t } ^ { \\mathrm { b } } \\} _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 375, + 507, + 393 + ], + "score": 1.0, + "content": "are coupled so", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 387, + 507, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 125, + 403 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 389, + 185, + 401 + ], + "score": 0.94, + "content": "[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } = - [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 202, + 403 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 202, + 389, + 252, + 402 + ], + "score": 0.89, + "content": "[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { i } = [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 387, + 268, + 403 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 268, + 390, + 291, + 400 + ], + "score": 0.85, + "content": "i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 387, + 507, + 403 + ], + "score": 1.0, + "content": ". In words, the randomness is the same orthogonal to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 508, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 117, + 414 + ], + "score": 0.8, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 397, + 162, + 420 + ], + "score": 1.0, + "content": ", but along", + "type": "text" + }, + { + "bbox": [ + 162, + 405, + 173, + 414 + ], + "score": 0.85, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 397, + 400, + 420 + ], + "score": 1.0, + "content": ", the two have opposite signs. Now, suppose we perform", + "type": "text" + }, + { + "bbox": [ + 400, + 401, + 480, + 417 + ], + "score": 0.95, + "content": "\\begin{array} { r } { T = \\Theta \\big ( \\frac { 1 } { \\eta \\delta } \\log \\frac { R ^ { 2 } \\delta } { \\eta \\nu ^ { 2 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 397, + 508, + 420 + ], + "score": 1.0, + "content": "steps", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 395, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 201, + 430 + ], + "score": 1.0, + "content": "of perturbed SGD from", + "type": "text" + }, + { + "bbox": [ + 201, + 416, + 230, + 429 + ], + "score": 0.9, + "content": "w _ { 0 } ^ { \\mathsf { a } } , w _ { 0 } ^ { \\mathsf { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 416, + 395, + 430 + ], + "score": 1.0, + "content": "respectively using Algorithm 2. Suppose", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 431, + 421, + 450 + ], + "lines": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "spans": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "score": 0.9, + "content": "\\begin{array} { r } { R \\leq { \\cal O } \\big ( { \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( R ^ { 2 } \\delta / \\eta \\nu ^ { 2 } ) } } \\big ) \\quad a n d \\quad \\nu ^ { 2 } \\geq \\Omega \\big ( C _ { 2 } \\log { \\frac { R ^ { 2 } \\delta } { \\eta \\nu } } \\big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6dc82af2029d61380e159b33aa634e018e7b1f60fefef4261df05b4c31f7d919.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 103, + 447, + 509, + 471 + ], + "spans": [ + { + "bbox": [ + 103, + 447, + 213, + 471 + ], + "score": 1.0, + "content": "Then, under events Even", + "type": "text" + }, + { + "bbox": [ + 213, + 452, + 254, + 466 + ], + "score": 0.8, + "content": "\\mathbf { t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 447, + 282, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 452, + 339, + 466 + ], + "score": 0.85, + "content": "\\mathsf {bar { c } v e n t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { b } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 447, + 509, + 471 + ], + "score": 1.0, + "content": ", with probability at least 0.98, either", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 464, + 326, + 481 + ], + "spans": [ + { + "bbox": [ + 107, + 466, + 174, + 479 + ], + "score": 0.88, + "content": "\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } \\| > R", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 464, + 187, + 481 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 187, + 466, + 254, + 478 + ], + "score": 0.89, + "content": "\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } \\| > R", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 464, + 292, + 481 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 293, + 466, + 321, + 478 + ], + "score": 0.92, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 464, + 326, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "Proof details in Appendix A.4. The main proof difficulty is to analyze a noisy version of the power", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "method, where the noise comes from (1) Gaussian perturbation (which is the good noise), (2)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 502, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 502, + 519 + ], + "score": 1.0, + "content": "stochastic gradients (which has zero mean), and (3) Byzantine workers (which can be adversarial).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 105, + 530, + 469, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 470, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 470, + 545 + ], + "score": 1.0, + "content": "4 FROM WARMUP TO FINAL THEOREM WITH DOUBLE SAFE GUARDS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "At a high level, Lemma 3.3 ensures that if we keep encountering points with large gradient", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 150, + 574 + ], + "score": 0.93, + "content": "\\| \\nabla f ( w _ { t } ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 561, + 506, + 576 + ], + "score": 1.0, + "content": ", then the objective should sufficiently decrease; in contrast, Lemma 3.5 says that if", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 383, + 586 + ], + "score": 1.0, + "content": "we keep encountering points with negative Hessian directions (i.e.,", + "type": "text" + }, + { + "bbox": [ + 383, + 573, + 481, + 585 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( w _ { t } ) ) < \\dot { - } \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 572, + 506, + 586 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 585, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 296, + 596 + ], + "score": 1.0, + "content": "the points must move a lot (i.e., by more than", + "type": "text" + }, + { + "bbox": [ + 297, + 585, + 306, + 594 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 585, + 318, + 596 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 318, + 585, + 327, + 594 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 585, + 506, + 596 + ], + "score": 1.0, + "content": "iterations, which can also lead to sufficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "objective decrease, see Lemma B.4). Therefore, at a high level, when the two lemmas are combined,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 605, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 298, + 620 + ], + "score": 1.0, + "content": "they tell that we must not encounter points with", + "type": "text" + }, + { + "bbox": [ + 299, + 606, + 337, + 618 + ], + "score": 0.93, + "content": "\\| \\nabla f ( x ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 605, + 398, + 620 + ], + "score": 1.0, + "content": "being large, or", + "type": "text" + }, + { + "bbox": [ + 399, + 606, + 459, + 618 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 605, + 505, + 620 + ], + "score": 1.0, + "content": "being very", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 617, + 485, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 485, + 630 + ], + "score": 1.0, + "content": "negative, for too many iterations. Therefore, the algorithm can find approximate local minima.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 411, + 644 + ], + "score": 1.0, + "content": "The reason we need two safe guards, is because the number of rounds", + "type": "text" + }, + { + "bbox": [ + 411, + 632, + 420, + 641 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "for Lemma 3.3 and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "Lemma 3.5 differ by a factor. We need two safe guards with different window sizes to ensure the two", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 653, + 503, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 503, + 666 + ], + "score": 1.0, + "content": "lemmas simultaneously hold. We encourage the reader to examine the full analysis in Appendix B.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 273, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 276, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 276, + 693 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL VALIDATION", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We evaluate the convergence of SafeguardSGD to examine its practical performance against prior", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "works. We perform the non-convex task of training a residual network ResNet-20 (He et al., 2016)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 489, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 489, + 734 + ], + "score": 1.0, + "content": "on the CIFAR-10/100 datasets (Krizhevsky et al., 2014). More details are given in Appendix C.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 333, + 95 + ], + "score": 1.0, + "content": "Our main idea is to use second-order smoothness to write", + "type": "text" + }, + { + "bbox": [ + 333, + 82, + 501, + 95 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } ) \\approx \\nabla f ( w _ { 0 } ) + \\nabla ^ { 2 } f ( w _ { 0 } ) \\cdot ( w _ { t } - w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "To illustrate our idea, let us ignore the constant vector and assume that the Hessian is the identity:", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 189, + 119 + ], + "score": 1.0, + "content": "that is, imagine as if", + "type": "text" + }, + { + "bbox": [ + 190, + 104, + 270, + 117 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } ) \\mathbf { \\bar { \\approx } } w _ { t } - w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 104, + 300, + 119 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + }, + { + "bbox": [ + 301, + 104, + 414, + 117 + ], + "score": 0.9, + "content": "\\begin{array} { r } { w _ { t } - w _ { 0 } = - \\sum _ { k < t } \\Xi _ { t } + \\xi _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 104, + 506, + 119 + ], + "score": 1.0, + "content": ", we immediately have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 119 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 119, + 450, + 134 + ], + "lines": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "spans": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "score": 0.84, + "content": "\\begin{array} { r } { - \\langle \\nabla f ( w _ { t } ) , \\Xi _ { t } \\rangle \\approx - \\langle w _ { t } - w _ { 0 } , \\Xi _ { t } \\rangle = \\sum _ { k < t } \\langle \\Xi _ { k } , \\Xi _ { t } \\rangle + \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}", + "type": "interline_equation", + "image_path": "3315623a3cd648ed53891363b61039c36b9bf574c6b4ad5106975dd21198c861.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 119, + 450, + 134 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 179 + ], + "lines": [ + { + "bbox": [ + 104, + 136, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 200, + 153 + ], + "score": 1.0, + "content": "For the first partial sum", + "type": "text" + }, + { + "bbox": [ + 200, + 138, + 261, + 151 + ], + "score": 0.92, + "content": "\\textstyle \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 136, + 505, + 153 + ], + "score": 1.0, + "content": "in (3.1), it is easy to bound its magnitude using our safeguard.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 150, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 104, + 150, + 172, + 166 + ], + "score": 1.0, + "content": "Indeed, we have", + "type": "text" + }, + { + "bbox": [ + 172, + 151, + 359, + 165 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big | \\sum _ { t } \\langle \\sum _ { k < t } \\Xi _ { k } , \\Xi _ { t } \\rangle \\big | \\leq \\| \\sum _ { t } \\Xi _ { t } \\| ^ { 2 } + \\sum _ { t } \\| \\Xi _ { t } \\| ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 150, + 506, + 166 + ], + "score": 1.0, + "content": "so we can apply Lemma 3.2. For the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 184, + 180 + ], + "score": 1.0, + "content": "second partial sum", + "type": "text" + }, + { + "bbox": [ + 184, + 166, + 257, + 180 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { t } \\sum _ { k < t } \\langle \\xi _ { k } , \\Xi _ { t } \\rangle } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 163, + 506, + 180 + ], + "score": 1.0, + "content": ", we can apply the concentration Proposition 3.4 below. \u0003", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 136, + 506, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 104, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 184, + 307, + 198 + ], + "score": 1.0, + "content": "Proposition 3.4. Fix the dimension parameter", + "type": "text" + }, + { + "bbox": [ + 307, + 186, + 336, + 197 + ], + "score": 0.91, + "content": "d \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 184, + 383, + 198 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 383, + 185, + 465, + 198 + ], + "score": 0.87, + "content": "\\xi _ { 0 \\underline { { : } } } \\dots , \\xi _ { T - 1 } \\ \\in \\ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "are i.i.d.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 155, + 209 + ], + "score": 1.0, + "content": "drawn from", + "type": "text" + }, + { + "bbox": [ + 155, + 196, + 187, + 209 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , \\bf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 196, + 226, + 209 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + }, + { + "bbox": [ + 226, + 197, + 286, + 208 + ], + "score": 0.9, + "content": "\\Delta _ { 1 } , \\ldots , \\Delta _ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 196, + 382, + 209 + ], + "score": 1.0, + "content": "are arbitrary vectors in", + "type": "text" + }, + { + "bbox": [ + 382, + 197, + 394, + 206 + ], + "score": 0.82, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 196, + 471, + 209 + ], + "score": 1.0, + "content": ". Here, each vector", + "type": "text" + }, + { + "bbox": [ + 472, + 197, + 484, + 208 + ], + "score": 0.88, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 177, + 219 + ], + "score": 0.88, + "content": "t = 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 208, + 242, + 220 + ], + "score": 1.0, + "content": "can depend on", + "type": "text" + }, + { + "bbox": [ + 242, + 209, + 292, + 219 + ], + "score": 0.9, + "content": "\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 208, + 338, + 220 + ], + "score": 1.0, + "content": "but not on", + "type": "text" + }, + { + "bbox": [ + 338, + 208, + 389, + 219 + ], + "score": 0.91, + "content": "\\xi _ { t } , \\dots , \\xi _ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 208, + 505, + 220 + ], + "score": 1.0, + "content": ". Suppose that these vectors", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 217, + 494, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 135, + 232 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + }, + { + "bbox": [ + 135, + 218, + 231, + 231 + ], + "score": 0.91, + "content": "\\| \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\| ^ { 2 } \\leq \\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 217, + 270, + 232 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 270, + 219, + 339, + 230 + ], + "score": 0.91, + "content": "t = 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 217, + 466, + 232 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 466, + 219, + 490, + 230 + ], + "score": 0.89, + "content": "1 - p", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 217, + 494, + 232 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 184, + 505, + 232 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 232, + 419, + 254 + ], + "lines": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "spans": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "4490f7c290ac0e971389f87ab29e6f26d4459fb36fe16ae481c193d32c11963b.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 191, + 232, + 419, + 254 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 370, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 370, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 370, + 270 + ], + "score": 1.0, + "content": "3.2 CORE TECHNICAL LEMMA 2: RANDOMNESS COUPLING", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 257, + 370, + 270 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 407, + 287 + ], + "score": 1.0, + "content": "Our next technical lemma studies that, if run Algorithm 2 from a point", + "type": "text" + }, + { + "bbox": [ + 408, + 277, + 420, + 286 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "so that the Hessian", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 107, + 285, + 145, + 298 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 284, + 294, + 298 + ], + "score": 1.0, + "content": "has a eigenvalue which is less than", + "type": "text" + }, + { + "bbox": [ + 294, + 286, + 308, + 296 + ], + "score": 0.87, + "content": "- \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 284, + 348, + 298 + ], + "score": 1.0, + "content": "(think of", + "type": "text" + }, + { + "bbox": [ + 349, + 287, + 361, + 297 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "as a saddle point), then with good", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 296, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 343, + 310 + ], + "score": 1.0, + "content": "probability, after sufficiently many iterations, the sequence", + "type": "text" + }, + { + "bbox": [ + 343, + 298, + 406, + 308 + ], + "score": 0.86, + "content": "w _ { 1 } , w _ { 2 } , \\dots , w _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 296, + 480, + 310 + ], + "score": 1.0, + "content": "shall escape from", + "type": "text" + }, + { + "bbox": [ + 481, + 298, + 493, + 308 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 296, + 506, + 310 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 171, + 321 + ], + "score": 1.0, + "content": "distance at least", + "type": "text" + }, + { + "bbox": [ + 171, + 308, + 180, + 317 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 306, + 260, + 321 + ], + "score": 1.0, + "content": "for some parameter", + "type": "text" + }, + { + "bbox": [ + 260, + 308, + 287, + 318 + ], + "score": 0.89, + "content": "R \\approx \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 306, + 506, + 321 + ], + "score": 1.0, + "content": ". To prove this, motivated by Jin et al. (2017), we study", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "two executions of Algorithm 2 where their randomness are coupled. We then argue that at least one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 460, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 216, + 342 + ], + "score": 1.0, + "content": "of them has to escape from", + "type": "text" + }, + { + "bbox": [ + 217, + 331, + 229, + 340 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 329, + 294, + 342 + ], + "score": 1.0, + "content": ". For any vector", + "type": "text" + }, + { + "bbox": [ + 294, + 331, + 300, + 339 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 329, + 317, + 342 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 317, + 329, + 331, + 342 + ], + "score": 0.9, + "content": "[ v ] _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 329, + 376, + 342 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + }, + { + "bbox": [ + 377, + 330, + 381, + 339 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 329, + 449, + 342 + ], + "score": 1.0, + "content": "-th coordinate of", + "type": "text" + }, + { + "bbox": [ + 450, + 332, + 455, + 339 + ], + "score": 0.78, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 329, + 460, + 342 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 273, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 253, + 358 + ], + "score": 1.0, + "content": "Lemma 3.5. Suppose we choose", + "type": "text" + }, + { + "bbox": [ + 253, + 346, + 262, + 355 + ], + "score": 0.72, + "content": "\\mathfrak { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 344, + 357, + 358 + ], + "score": 1.0, + "content": "as in Lemma 3.2 and", + "type": "text" + }, + { + "bbox": [ + 357, + 345, + 386, + 357 + ], + "score": 0.92, + "content": "C _ { 1 } , C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "as in Lemma 3.3. Suppose", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 147, + 367 + ], + "score": 0.91, + "content": "\\boldsymbol { w } _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 354, + 185, + 369 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 185, + 356, + 281, + 368 + ], + "score": 0.92, + "content": "\\dot { \\lambda } _ { \\mathrm { m i n } } ( \\nabla ^ { 2 } f ( w _ { 0 } ) ) = - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 354, + 322, + 369 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 322, + 356, + 350, + 367 + ], + "score": 0.9, + "content": "\\delta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 354, + 479, + 369 + ], + "score": 1.0, + "content": ". Without loss of generality let", + "type": "text" + }, + { + "bbox": [ + 479, + 357, + 491, + 367 + ], + "score": 0.85, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 180, + 381 + ], + "score": 1.0, + "content": "the eigenvector of", + "type": "text" + }, + { + "bbox": [ + 180, + 367, + 218, + 379 + ], + "score": 0.9, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 366, + 505, + 381 + ], + "score": 1.0, + "content": "with smallest eigenvalue. Consider now two executions of Algorithm 2,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 375, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 182, + 393 + ], + "score": 1.0, + "content": "both starting from", + "type": "text" + }, + { + "bbox": [ + 182, + 379, + 246, + 390 + ], + "score": 0.9, + "content": "w _ { 0 } ^ { \\mathsf { a } } = w _ { 0 } ^ { \\mathsf { b } } = w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 375, + 375, + 393 + ], + "score": 1.0, + "content": ", and suppose their randomness", + "type": "text" + }, + { + "bbox": [ + 376, + 378, + 399, + 390 + ], + "score": 0.92, + "content": "\\{ \\xi _ { t } ^ { \\mathsf { a } } \\} _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 375, + 419, + 393 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 419, + 378, + 442, + 390 + ], + "score": 0.92, + "content": "\\{ \\xi _ { t } ^ { \\mathrm { b } } \\} _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 375, + 507, + 393 + ], + "score": 1.0, + "content": "are coupled so", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 387, + 507, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 125, + 403 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 389, + 185, + 401 + ], + "score": 0.94, + "content": "[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } = - [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 387, + 202, + 403 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 202, + 389, + 252, + 402 + ], + "score": 0.89, + "content": "[ \\xi _ { t } ^ { \\mathsf { a } } ] _ { i } = [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 387, + 268, + 403 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 268, + 390, + 291, + 400 + ], + "score": 0.85, + "content": "i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 387, + 507, + 403 + ], + "score": 1.0, + "content": ". In words, the randomness is the same orthogonal to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 508, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 117, + 414 + ], + "score": 0.8, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 397, + 162, + 420 + ], + "score": 1.0, + "content": ", but along", + "type": "text" + }, + { + "bbox": [ + 162, + 405, + 173, + 414 + ], + "score": 0.85, + "content": "\\mathbf { e } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 397, + 400, + 420 + ], + "score": 1.0, + "content": ", the two have opposite signs. Now, suppose we perform", + "type": "text" + }, + { + "bbox": [ + 400, + 401, + 480, + 417 + ], + "score": 0.95, + "content": "\\begin{array} { r } { T = \\Theta \\big ( \\frac { 1 } { \\eta \\delta } \\log \\frac { R ^ { 2 } \\delta } { \\eta \\nu ^ { 2 } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 397, + 508, + 420 + ], + "score": 1.0, + "content": "steps", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 395, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 201, + 430 + ], + "score": 1.0, + "content": "of perturbed SGD from", + "type": "text" + }, + { + "bbox": [ + 201, + 416, + 230, + 429 + ], + "score": 0.9, + "content": "w _ { 0 } ^ { \\mathsf { a } } , w _ { 0 } ^ { \\mathsf { b } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 416, + 395, + 430 + ], + "score": 1.0, + "content": "respectively using Algorithm 2. Suppose", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 344, + 508, + 430 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 431, + 421, + 450 + ], + "lines": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "spans": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "score": 0.9, + "content": "\\begin{array} { r } { R \\leq { \\cal O } \\big ( { \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( R ^ { 2 } \\delta / \\eta \\nu ^ { 2 } ) } } \\big ) \\quad a n d \\quad \\nu ^ { 2 } \\geq \\Omega \\big ( C _ { 2 } \\log { \\frac { R ^ { 2 } \\delta } { \\eta \\nu } } \\big ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6dc82af2029d61380e159b33aa634e018e7b1f60fefef4261df05b4c31f7d919.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 189, + 431, + 421, + 450 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 103, + 447, + 509, + 471 + ], + "spans": [ + { + "bbox": [ + 103, + 447, + 213, + 471 + ], + "score": 1.0, + "content": "Then, under events Even", + "type": "text" + }, + { + "bbox": [ + 213, + 452, + 254, + 466 + ], + "score": 0.8, + "content": "\\mathbf { t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 447, + 282, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 452, + 339, + 466 + ], + "score": 0.85, + "content": "\\mathsf {bar { c } v e n t } _ { T } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } ^ { \\mathsf { b } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 447, + 509, + 471 + ], + "score": 1.0, + "content": ", with probability at least 0.98, either", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 464, + 326, + 481 + ], + "spans": [ + { + "bbox": [ + 107, + 466, + 174, + 479 + ], + "score": 0.88, + "content": "\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } \\| > R", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 464, + 187, + 481 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 187, + 466, + 254, + 478 + ], + "score": 0.89, + "content": "\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } \\| > R", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 464, + 292, + 481 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 293, + 466, + 321, + 478 + ], + "score": 0.92, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 464, + 326, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 103, + 447, + 509, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "Proof details in Appendix A.4. The main proof difficulty is to analyze a noisy version of the power", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "method, where the noise comes from (1) Gaussian perturbation (which is the good noise), (2)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 502, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 502, + 519 + ], + "score": 1.0, + "content": "stochastic gradients (which has zero mean), and (3) Byzantine workers (which can be adversarial).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 483, + 506, + 519 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 530, + 469, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 470, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 470, + 545 + ], + "score": 1.0, + "content": "4 FROM WARMUP TO FINAL THEOREM WITH DOUBLE SAFE GUARDS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "At a high level, Lemma 3.3 ensures that if we keep encountering points with large gradient", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 150, + 574 + ], + "score": 0.93, + "content": "\\| \\nabla f ( w _ { t } ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 561, + 506, + 576 + ], + "score": 1.0, + "content": ", then the objective should sufficiently decrease; in contrast, Lemma 3.5 says that if", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 383, + 586 + ], + "score": 1.0, + "content": "we keep encountering points with negative Hessian directions (i.e.,", + "type": "text" + }, + { + "bbox": [ + 383, + 573, + 481, + 585 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( w _ { t } ) ) < \\dot { - } \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 572, + 506, + 586 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 585, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 296, + 596 + ], + "score": 1.0, + "content": "the points must move a lot (i.e., by more than", + "type": "text" + }, + { + "bbox": [ + 297, + 585, + 306, + 594 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 585, + 318, + 596 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 318, + 585, + 327, + 594 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 585, + 506, + 596 + ], + "score": 1.0, + "content": "iterations, which can also lead to sufficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "objective decrease, see Lemma B.4). Therefore, at a high level, when the two lemmas are combined,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 605, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 298, + 620 + ], + "score": 1.0, + "content": "they tell that we must not encounter points with", + "type": "text" + }, + { + "bbox": [ + 299, + 606, + 337, + 618 + ], + "score": 0.93, + "content": "\\| \\nabla f ( x ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 605, + 398, + 620 + ], + "score": 1.0, + "content": "being large, or", + "type": "text" + }, + { + "bbox": [ + 399, + 606, + 459, + 618 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m i n } } ( \\nabla ^ { 2 } f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 605, + 505, + 620 + ], + "score": 1.0, + "content": "being very", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 617, + 485, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 485, + 630 + ], + "score": 1.0, + "content": "negative, for too many iterations. Therefore, the algorithm can find approximate local minima.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 550, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 411, + 644 + ], + "score": 1.0, + "content": "The reason we need two safe guards, is because the number of rounds", + "type": "text" + }, + { + "bbox": [ + 411, + 632, + 420, + 641 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "for Lemma 3.3 and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "Lemma 3.5 differ by a factor. We need two safe guards with different window sizes to ensure the two", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 653, + 503, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 503, + 666 + ], + "score": 1.0, + "content": "lemmas simultaneously hold. We encourage the reader to examine the full analysis in Appendix B.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 631, + 506, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 273, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 276, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 276, + 693 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL VALIDATION", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We evaluate the convergence of SafeguardSGD to examine its practical performance against prior", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "works. We perform the non-convex task of training a residual network ResNet-20 (He et al., 2016)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 489, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 489, + 734 + ], + "score": 1.0, + "content": "on the CIFAR-10/100 datasets (Krizhevsky et al., 2014). More details are given in Appendix C.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 74, + 506, + 267 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 74, + 506, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 74, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 506, + 267 + ], + "score": 0.973, + "type": "image", + "image_path": "fd2edb2655662c1193ffe6a690559d8147cb61fcf659521ef7f5a1fb281c36fa.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 74, + 506, + 138.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 138.33333333333331, + 506, + 202.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 202.66666666666663, + 506, + 266.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 275, + 503, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "Figure 1: Convergence comparison (CIFAR-10 test accuracy) under different attacks. (In Appendix C.2, one", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 284, + 435, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 435, + 296 + ], + "score": 1.0, + "content": "can find additional CIFAR-100 experiments, more discussions, and bigger plots.)", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 164, + 331 + ], + "score": 1.0, + "content": "We instantiate", + "type": "text" + }, + { + "bbox": [ + 164, + 320, + 198, + 330 + ], + "score": 0.9, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 320, + 504, + 331 + ], + "score": 1.0, + "content": "workers and one master node executing data-parallel SGD for 140 passes (i.e.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "epochs) over the training dataset. The results for higher number of workers and epochs are similar,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "and therefore omitted. We compare against Geometric Median (Chen et al., 2017), Coordinate-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "wise Median (Yin et al., 2018; 2019), Krum (Blanchard et al., 2017), and Zeno (Xie et al., 2018b).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "Overall, our experimental setup is very similar to Zeno (Xie et al., 2018b) but with additional attacks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "We implemented the approach of Yang et al. (2019), but found it very sensitive to hyper-parameter", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "values and were unable to make it converge across all attacks even after significant tuning of its", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 114, + 411 + ], + "score": 0.73, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "parameter. We also implemented the convex algorithm of Alistarh et al. (2018), and executed it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 409, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 506, + 424 + ], + "score": 1.0, + "content": "in our non-convex setting. We found their algorithm can be easily attacked on our ResNet training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "tasks. There exists a simple attack, described in Appendix C.4 which causes their algorithm to either", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "mislabel most good workers as Byzantine, or diverge, or converge to very poor solutions. This is not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "surprising, since their algorithm is designed for, and only guaranteed to work in, the convex setting.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "To make the comparison stronger, when implementing SafeguardSGD, we have chosen fixed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "score": 1.0, + "content": "window sizes", + "type": "text" + }, + { + "bbox": [ + 163, + 469, + 195, + 480 + ], + "score": 0.91, + "content": "T _ { 0 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 469, + 239, + 481 + ], + "score": 1.0, + "content": "epoch and", + "type": "text" + }, + { + "bbox": [ + 240, + 469, + 271, + 480 + ], + "score": 0.93, + "content": "T _ { 1 } = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "epochs across all experiments, and adopted an automated", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 176, + 493 + ], + "score": 1.0, + "content": "process to select", + "type": "text" + }, + { + "bbox": [ + 177, + 480, + 204, + 491 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 0 } , \\mathfrak { T } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 479, + 506, + 493 + ], + "score": 1.0, + "content": ". Determining these thresholds requires being able to pre-run the task on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "an honest worker. We have also implemented a single safeguard variant of SafeguardSGD, with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 501, + 219, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 158, + 515 + ], + "score": 1.0, + "content": "window size", + "type": "text" + }, + { + "bbox": [ + 159, + 502, + 186, + 513 + ], + "score": 0.9, + "content": "T = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 501, + 219, + 515 + ], + "score": 1.0, + "content": "epochs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 183, + 529 + ], + "score": 1.0, + "content": "Attacks. We set", + "type": "text" + }, + { + "bbox": [ + 183, + 516, + 220, + 527 + ], + "score": 0.88, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 516, + 505, + 529 + ], + "score": 1.0, + "content": ", which means that there are 4 Byzantine workers. (This exceeds the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 526, + 449, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 449, + 540 + ], + "score": 1.0, + "content": "fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 111, + 543, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 111, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 111, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "• LABEL-FLIPPING ATTACK: each Byzantine worker computes its gradient based on the cross-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 119, + 554, + 465, + 566 + ], + "spans": [ + { + "bbox": [ + 119, + 554, + 334, + 566 + ], + "score": 1.0, + "content": "entropy loss with flipped labels: for CIFAR-10, label", + "type": "text" + }, + { + "bbox": [ + 334, + 554, + 388, + 566 + ], + "score": 0.94, + "content": "\\ell \\in \\{ 0 , . . . , 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 554, + 439, + 566 + ], + "score": 1.0, + "content": "is flipped to", + "type": "text" + }, + { + "bbox": [ + 439, + 554, + 462, + 565 + ], + "score": 0.85, + "content": "9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 554, + 465, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 112, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 112, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "• DELAYED-GRADIENT ATTACK: each Byzantine worker sends an old gradient to master. In our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 119, + 578, + 319, + 590 + ], + "spans": [ + { + "bbox": [ + 119, + 578, + 233, + 590 + ], + "score": 1.0, + "content": "experiments, the delay is of", + "type": "text" + }, + { + "bbox": [ + 233, + 578, + 276, + 588 + ], + "score": 0.9, + "content": "D = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 578, + 319, + 590 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 112, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 112, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "• VARIANCE ATTACK (Baruch et al., 2019): Byzantine workers measure the mean and the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 119, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 119, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "standard-deviation of gradients at each round, and collude to move the mean by the largest value", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 119, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 119, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "which still operates within population variance. (For our parameter settings, this is 0.3 times the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 119, + 624, + 474, + 636 + ], + "spans": [ + { + "bbox": [ + 119, + 624, + 474, + 636 + ], + "score": 1.0, + "content": "standard deviation. We discuss results for additional parameter values in the Appendix.)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 111, + 636, + 486, + 649 + ], + "spans": [ + { + "bbox": [ + 111, + 636, + 486, + 649 + ], + "score": 1.0, + "content": "• SIGN-FLIPPING ATTACK: each Byzantine worker sends the negative gradient to the master.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 110, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 110, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "• SAFEGUARD ATTACK: each Byzantine workers sends a negative but re-scaled gradient to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 120, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 120, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "master. We use re-scale factors 0.6 and 0.7 in our experiments. The re-scale factor 0.6 avoids", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 119, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 119, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "triggering the safe-guard conditions at the master, and the re-scale factor 0.7 occasionally triggers", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 120, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 120, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "the safe-guard conditions. This attack is an instantiation of the inner-product attack (Xie et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 694, + 449, + 707 + ], + "spans": [ + { + "bbox": [ + 119, + 694, + 449, + 707 + ], + "score": 1.0, + "content": "2020), customized specifically to maximally affect our SafeguardSGD algorithm.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 328, + 722 + ], + "score": 1.0, + "content": "Main experimental results. The ideal test accuracy is", + "type": "text" + }, + { + "bbox": [ + 328, + 709, + 355, + 720 + ], + "score": 0.87, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", which corresponds to applying SGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "using only the stochastic gradients from the honest workers. Figure 1 compares the performances", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 74, + 506, + 267 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 74, + 506, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 74, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 74, + 506, + 267 + ], + "score": 0.973, + "type": "image", + "image_path": "fd2edb2655662c1193ffe6a690559d8147cb61fcf659521ef7f5a1fb281c36fa.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 74, + 506, + 138.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 138.33333333333331, + 506, + 202.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 202.66666666666663, + 506, + 266.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 275, + 503, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "Figure 1: Convergence comparison (CIFAR-10 test accuracy) under different attacks. (In Appendix C.2, one", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 284, + 435, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 435, + 296 + ], + "score": 1.0, + "content": "can find additional CIFAR-100 experiments, more discussions, and bigger plots.)", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 164, + 331 + ], + "score": 1.0, + "content": "We instantiate", + "type": "text" + }, + { + "bbox": [ + 164, + 320, + 198, + 330 + ], + "score": 0.9, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 320, + 504, + 331 + ], + "score": 1.0, + "content": "workers and one master node executing data-parallel SGD for 140 passes (i.e.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "epochs) over the training dataset. The results for higher number of workers and epochs are similar,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "and therefore omitted. We compare against Geometric Median (Chen et al., 2017), Coordinate-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "wise Median (Yin et al., 2018; 2019), Krum (Blanchard et al., 2017), and Zeno (Xie et al., 2018b).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "Overall, our experimental setup is very similar to Zeno (Xie et al., 2018b) but with additional attacks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 320, + 505, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "We implemented the approach of Yang et al. (2019), but found it very sensitive to hyper-parameter", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "values and were unable to make it converge across all attacks even after significant tuning of its", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 114, + 411 + ], + "score": 0.73, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "parameter. We also implemented the convex algorithm of Alistarh et al. (2018), and executed it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 409, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 506, + 424 + ], + "score": 1.0, + "content": "in our non-convex setting. We found their algorithm can be easily attacked on our ResNet training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "tasks. There exists a simple attack, described in Appendix C.4 which causes their algorithm to either", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "mislabel most good workers as Byzantine, or diverge, or converge to very poor solutions. This is not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "surprising, since their algorithm is designed for, and only guaranteed to work in, the convex setting.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 377, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "To make the comparison stronger, when implementing SafeguardSGD, we have chosen fixed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 163, + 481 + ], + "score": 1.0, + "content": "window sizes", + "type": "text" + }, + { + "bbox": [ + 163, + 469, + 195, + 480 + ], + "score": 0.91, + "content": "T _ { 0 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 469, + 239, + 481 + ], + "score": 1.0, + "content": "epoch and", + "type": "text" + }, + { + "bbox": [ + 240, + 469, + 271, + 480 + ], + "score": 0.93, + "content": "T _ { 1 } = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "epochs across all experiments, and adopted an automated", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 176, + 493 + ], + "score": 1.0, + "content": "process to select", + "type": "text" + }, + { + "bbox": [ + 177, + 480, + 204, + 491 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 0 } , \\mathfrak { T } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 479, + 506, + 493 + ], + "score": 1.0, + "content": ". Determining these thresholds requires being able to pre-run the task on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "an honest worker. We have also implemented a single safeguard variant of SafeguardSGD, with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 501, + 219, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 158, + 515 + ], + "score": 1.0, + "content": "window size", + "type": "text" + }, + { + "bbox": [ + 159, + 502, + 186, + 513 + ], + "score": 0.9, + "content": "T = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 501, + 219, + 515 + ], + "score": 1.0, + "content": "epochs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 457, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 183, + 529 + ], + "score": 1.0, + "content": "Attacks. We set", + "type": "text" + }, + { + "bbox": [ + 183, + 516, + 220, + 527 + ], + "score": 0.88, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 516, + 505, + 529 + ], + "score": 1.0, + "content": ", which means that there are 4 Byzantine workers. (This exceeds the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 526, + 449, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 449, + 540 + ], + "score": 1.0, + "content": "fault-tolerance of Krum, and so we also tested Krum with only 3 Byzantine workers.)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 516, + 505, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 543, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 111, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 111, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "• LABEL-FLIPPING ATTACK: each Byzantine worker computes its gradient based on the cross-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 119, + 554, + 465, + 566 + ], + "spans": [ + { + "bbox": [ + 119, + 554, + 334, + 566 + ], + "score": 1.0, + "content": "entropy loss with flipped labels: for CIFAR-10, label", + "type": "text" + }, + { + "bbox": [ + 334, + 554, + 388, + 566 + ], + "score": 0.94, + "content": "\\ell \\in \\{ 0 , . . . , 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 554, + 439, + 566 + ], + "score": 1.0, + "content": "is flipped to", + "type": "text" + }, + { + "bbox": [ + 439, + 554, + 462, + 565 + ], + "score": 0.85, + "content": "9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 554, + 465, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 112, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 112, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "• DELAYED-GRADIENT ATTACK: each Byzantine worker sends an old gradient to master. In our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 119, + 578, + 319, + 590 + ], + "spans": [ + { + "bbox": [ + 119, + 578, + 233, + 590 + ], + "score": 1.0, + "content": "experiments, the delay is of", + "type": "text" + }, + { + "bbox": [ + 233, + 578, + 276, + 588 + ], + "score": 0.9, + "content": "D = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 578, + 319, + 590 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 112, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 112, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "• VARIANCE ATTACK (Baruch et al., 2019): Byzantine workers measure the mean and the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 119, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 119, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "standard-deviation of gradients at each round, and collude to move the mean by the largest value", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 119, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 119, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "which still operates within population variance. (For our parameter settings, this is 0.3 times the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 119, + 624, + 474, + 636 + ], + "spans": [ + { + "bbox": [ + 119, + 624, + 474, + 636 + ], + "score": 1.0, + "content": "standard deviation. We discuss results for additional parameter values in the Appendix.)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 111, + 636, + 486, + 649 + ], + "spans": [ + { + "bbox": [ + 111, + 636, + 486, + 649 + ], + "score": 1.0, + "content": "• SIGN-FLIPPING ATTACK: each Byzantine worker sends the negative gradient to the master.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 110, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 110, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "• SAFEGUARD ATTACK: each Byzantine workers sends a negative but re-scaled gradient to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 120, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 120, + 662, + 505, + 673 + ], + "score": 1.0, + "content": "master. We use re-scale factors 0.6 and 0.7 in our experiments. The re-scale factor 0.6 avoids", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 119, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 119, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "triggering the safe-guard conditions at the master, and the re-scale factor 0.7 occasionally triggers", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 120, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 120, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "the safe-guard conditions. This attack is an instantiation of the inner-product attack (Xie et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 694, + 449, + 707 + ], + "spans": [ + { + "bbox": [ + 119, + 694, + 449, + 707 + ], + "score": 1.0, + "content": "2020), customized specifically to maximally affect our SafeguardSGD algorithm.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 30.5, + "bbox_fs": [ + 110, + 542, + 506, + 707 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 328, + 722 + ], + "score": 1.0, + "content": "Main experimental results. The ideal test accuracy is", + "type": "text" + }, + { + "bbox": [ + 328, + 709, + 355, + 720 + ], + "score": 0.87, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", which corresponds to applying SGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "using only the stochastic gradients from the honest workers. Figure 1 compares the performances", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "in test accuracy. 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Below we summarize our main findings for the experiments, and we defer detailed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 243, + 398, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 398, + 254 + ], + "score": 1.0, + "content": "discussions (and additional experiments for CIFAR-100) to Appendix C.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 111, + 257, + 506, + 534 + ], + "lines": [ + { + "bbox": [ + 111, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 111, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "• SafeguardSGD generally outperforms all the previous methods in test accuracy. The test", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 119, + 267, + 494, + 280 + ], + "spans": [ + { + "bbox": [ + 119, + 268, + 229, + 280 + ], + "score": 1.0, + "content": "accuracy difference can be", + "type": "text" + }, + { + "bbox": [ + 230, + 268, + 253, + 279 + ], + "score": 0.87, + "content": "{ } ^ { 6 6 } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 268, + 268, + 280 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 269, + 267, + 300, + 279 + ], + "score": 0.86, + "content": "< 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 268, + 494, + 280 + ], + "score": 1.0, + "content": "” between our algorithm and the best prior work.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 111, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 111, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "• The variance attack is indeed very strong, in that it severely affects the accuracy of all prior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 119, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 119, + 291, + 206, + 304 + ], + "score": 1.0, + "content": "works (test accuracy", + "type": "text" + }, + { + "bbox": [ + 206, + 291, + 238, + 303 + ], + "score": 0.9, + "content": "< 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "). This is because thesese defenses are “historyless.” By contrast,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 119, + 303, + 407, + 315 + ], + "spans": [ + { + "bbox": [ + 119, + 303, + 407, + 315 + ], + "score": 1.0, + "content": "our algorithm not only provably but also empirically defends against it.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 112, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 112, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "• Our safeguard attack (especially with re-scale factor 0.7) is as strong as the variance attack, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 119, + 326, + 454, + 339 + ], + "spans": [ + { + "bbox": [ + 119, + 326, + 454, + 339 + ], + "score": 1.0, + "content": "even stronger on the CIFAR-100 dataset; please see the results in Appendix C.2.5.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 110, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 110, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "• The label-flipping attack is rather weak: although some defenses, such as Zeno, did not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 119, + 351, + 473, + 363 + ], + "spans": [ + { + "bbox": [ + 119, + 351, + 473, + 363 + ], + "score": 1.0, + "content": "determine which of the workers are malicious, they still converge well under this attack.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 113, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 113, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "• The sign-flipping and delayed-gradient attacks are moderate: the best prior works can achieve", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 119, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 119, + 373, + 159, + 388 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + }, + { + "bbox": [ + 159, + 374, + 214, + 385 + ], + "score": 0.92, + "content": "6 0 \\% \\sim 7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 373, + 506, + 388 + ], + "score": 1.0, + "content": ". It is worth noting that the sign-flipping attack can already nullify the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 119, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 119, + 385, + 234, + 398 + ], + "score": 1.0, + "content": "Zeno defence (test accuracy", + "type": "text" + }, + { + "bbox": [ + 234, + 385, + 254, + 396 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "). The issue seems to be that it can be very hard for Zeno to use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 119, + 397, + 448, + 409 + ], + "spans": [ + { + "bbox": [ + 119, + 397, + 448, + 409 + ], + "score": 1.0, + "content": "relatively few samples to determine if the gradient direction is flipped to negative.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 112, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 112, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "• SafeguardSGD can easily catch all the bad workers under sign-flipping and variance attacks,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 119, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 119, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "and thus leads to gives ideal performance. It cannot catch any bad worker for label-flipping and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 119, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "delayed-gradient attacks, but there is no performance loss anyways if we use such bad gradients.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 112, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 112, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "• The safeguard attacks, designed to maximally impact the performance of our SafeguardSGD,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 119, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 119, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "can indeed affect our performance. Specifically, under re-scale factor 0.6, the test accuracy drops", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 120, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 120, + 465, + 142, + 478 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 142, + 465, + 169, + 476 + ], + "score": 0.86, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 465, + 180, + 478 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 180, + 466, + 207, + 476 + ], + "score": 0.9, + "content": "8 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "because SafeguardSGD cannot catch any bad worker; however, under re-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 119, + 476, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 119, + 476, + 504, + 489 + ], + "score": 1.0, + "content": "scale factor 0.7, the test accuracy no longer drops because SafeguardSGD can begin to catch", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 119, + 488, + 493, + 500 + ], + "spans": [ + { + "bbox": [ + 119, + 488, + 493, + 500 + ], + "score": 1.0, + "content": "some bad workers (it can catch between 0 and 4 bad workers depending on the randomness.)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 112, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 112, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "• In most cases, the single-safeguard algorithm is close to double-safeguard, except for the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 120, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 120, + 512, + 160, + 524 + ], + "score": 1.0, + "content": "safeguard", + "type": "text" + }, + { + "bbox": [ + 160, + 511, + 185, + 522 + ], + "score": 0.55, + "content": "( \\mathrm { x 0 . 7 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "attack, in which using double-safeguard one can more easily catch bad workers.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 120, + 522, + 425, + 535 + ], + "spans": [ + { + "bbox": [ + 120, + 522, + 425, + 535 + ], + "score": 1.0, + "content": "(This is more apparent in the CIFAR-100 experiment, see Appendix C.2.5.)", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 464, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 465, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 465, + 551 + ], + "score": 1.0, + "content": "We conclude that SafeguardSGD can be practical, and outperforms previous approaches.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 504, + 563 + ], + "score": 1.0, + "content": "A deeper dive: how the algorithm works. Let us explain the inner workings of our algorithm", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 562, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 504, + 574 + ], + "score": 1.0, + "content": "in the context of a “delayed” attack, where the Byzantine nodes collude to execute an attack only", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "after a specific, given point in the execution (in this case, the first half-epoch). Figure 2(a) presents", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 294, + 597 + ], + "score": 1.0, + "content": "the results from the perspective of the value of", + "type": "text" + }, + { + "bbox": [ + 295, + 584, + 348, + 596 + ], + "score": 0.92, + "content": "\\| B _ { i } - B _ { \\mathrm { m e d } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "registered at the master server, for two", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 332, + 608 + ], + "score": 1.0, + "content": "nodes, an honest one, and a Byzantine one. The value of √", + "type": "text" + }, + { + "bbox": [ + 333, + 595, + 386, + 607 + ], + "score": 0.93, + "content": "\\| B _ { i } - B _ { \\sf m e d } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "increases for all the nodes (at", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 175, + 620 + ], + "score": 1.0, + "content": "a rate of roughly", + "type": "text" + }, + { + "bbox": [ + 176, + 606, + 189, + 619 + ], + "score": 0.9, + "content": "\\sqrt { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 608, + 218, + 620 + ], + "score": 1.0, + "content": "at step", + "type": "text" + }, + { + "bbox": [ + 218, + 609, + 223, + 618 + ], + "score": 0.45, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "); but, once the attack starts, the statistic for the Byzantine node grows", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 619, + 259, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 150, + 630 + ], + "score": 1.0, + "content": "linearly in", + "type": "text" + }, + { + "bbox": [ + 150, + 619, + 155, + 628 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 619, + 259, + 630 + ], + "score": 1.0, + "content": ", leading to fast detection.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "Transient attacks and node ID relabeling. Finally, in Figure 2(b) we analyze the behaviour of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "our algorithm when it periodically (every 3 epochs for single safeguard and 6 epochs for double", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "score": 1.0, + "content": "safeguard) resets the set of good nodes to include all nodes, restarting the detection process from", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "scratch. Our theoretical result still applies after this relaxation. This relaxation has two benefits.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "First, it benefits from bad workers that under transient failures (e.g., the node fails for 10 epochs", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "score": 1.0, + "content": "but resumes to work correctly after a while), and thus benefits from the data stored on this worker.", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Second, it can defend against certain degree of node ID relabeling: it supports the case when good", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "and bad workers exchange their IDs every 6 epochs. 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The test", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 119, + 267, + 494, + 280 + ], + "spans": [ + { + "bbox": [ + 119, + 268, + 229, + 280 + ], + "score": 1.0, + "content": "accuracy difference can be", + "type": "text" + }, + { + "bbox": [ + 230, + 268, + 253, + 279 + ], + "score": 0.87, + "content": "{ } ^ { 6 6 } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 268, + 268, + 280 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 269, + 267, + 300, + 279 + ], + "score": 0.86, + "content": "< 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 268, + 494, + 280 + ], + "score": 1.0, + "content": "” between our algorithm and the best prior work.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 111, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 111, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "• The variance attack is indeed very strong, in that it severely affects the accuracy of all prior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 119, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 119, + 291, + 206, + 304 + ], + "score": 1.0, + "content": "works (test accuracy", + "type": "text" + }, + { + "bbox": [ + 206, + 291, + 238, + 303 + ], + "score": 0.9, + "content": "< 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "). This is because thesese defenses are “historyless.” By contrast,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 119, + 303, + 407, + 315 + ], + "spans": [ + { + "bbox": [ + 119, + 303, + 407, + 315 + ], + "score": 1.0, + "content": "our algorithm not only provably but also empirically defends against it.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 112, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 112, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "• Our safeguard attack (especially with re-scale factor 0.7) is as strong as the variance attack, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 119, + 326, + 454, + 339 + ], + "spans": [ + { + "bbox": [ + 119, + 326, + 454, + 339 + ], + "score": 1.0, + "content": "even stronger on the CIFAR-100 dataset; please see the results in Appendix C.2.5.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 110, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 110, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "• The label-flipping attack is rather weak: although some defenses, such as Zeno, did not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 119, + 351, + 473, + 363 + ], + "spans": [ + { + "bbox": [ + 119, + 351, + 473, + 363 + ], + "score": 1.0, + "content": "determine which of the workers are malicious, they still converge well under this attack.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 113, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 113, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "• The sign-flipping and delayed-gradient attacks are moderate: the best prior works can achieve", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 119, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 119, + 373, + 159, + 388 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + }, + { + "bbox": [ + 159, + 374, + 214, + 385 + ], + "score": 0.92, + "content": "6 0 \\% \\sim 7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 373, + 506, + 388 + ], + "score": 1.0, + "content": ". It is worth noting that the sign-flipping attack can already nullify the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 119, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 119, + 385, + 234, + 398 + ], + "score": 1.0, + "content": "Zeno defence (test accuracy", + "type": "text" + }, + { + "bbox": [ + 234, + 385, + 254, + 396 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "). The issue seems to be that it can be very hard for Zeno to use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 119, + 397, + 448, + 409 + ], + "spans": [ + { + "bbox": [ + 119, + 397, + 448, + 409 + ], + "score": 1.0, + "content": "relatively few samples to determine if the gradient direction is flipped to negative.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 112, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 112, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "• SafeguardSGD can easily catch all the bad workers under sign-flipping and variance attacks,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 119, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 119, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "and thus leads to gives ideal performance. It cannot catch any bad worker for label-flipping and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 119, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "delayed-gradient attacks, but there is no performance loss anyways if we use such bad gradients.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 112, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 112, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "• The safeguard attacks, designed to maximally impact the performance of our SafeguardSGD,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 119, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 119, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "can indeed affect our performance. Specifically, under re-scale factor 0.6, the test accuracy drops", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 120, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 120, + 465, + 142, + 478 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 142, + 465, + 169, + 476 + ], + "score": 0.86, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 465, + 180, + 478 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 180, + 466, + 207, + 476 + ], + "score": 0.9, + "content": "8 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "because SafeguardSGD cannot catch any bad worker; however, under re-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 119, + 476, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 119, + 476, + 504, + 489 + ], + "score": 1.0, + "content": "scale factor 0.7, the test accuracy no longer drops because SafeguardSGD can begin to catch", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 119, + 488, + 493, + 500 + ], + "spans": [ + { + "bbox": [ + 119, + 488, + 493, + 500 + ], + "score": 1.0, + "content": "some bad workers (it can catch between 0 and 4 bad workers depending on the randomness.)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 112, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 112, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "• In most cases, the single-safeguard algorithm is close to double-safeguard, except for the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 120, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 120, + 512, + 160, + 524 + ], + "score": 1.0, + "content": "safeguard", + "type": "text" + }, + { + "bbox": [ + 160, + 511, + 185, + 522 + ], + "score": 0.55, + "content": "( \\mathrm { x 0 . 7 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "attack, in which using double-safeguard one can more easily catch bad workers.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 120, + 522, + 425, + 535 + ], + "spans": [ + { + "bbox": [ + 120, + 522, + 425, + 535 + ], + "score": 1.0, + "content": "(This is more apparent in the CIFAR-100 experiment, see Appendix C.2.5.)", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 33.5, + "bbox_fs": [ + 110, + 256, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 464, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 465, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 465, + 551 + ], + "score": 1.0, + "content": "We conclude that SafeguardSGD can be practical, and outperforms previous approaches.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 534, + 465, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 504, + 563 + ], + "score": 1.0, + "content": "A deeper dive: how the algorithm works. Let us explain the inner workings of our algorithm", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 562, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 504, + 574 + ], + "score": 1.0, + "content": "in the context of a “delayed” attack, where the Byzantine nodes collude to execute an attack only", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "after a specific, given point in the execution (in this case, the first half-epoch). Figure 2(a) presents", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 294, + 597 + ], + "score": 1.0, + "content": "the results from the perspective of the value of", + "type": "text" + }, + { + "bbox": [ + 295, + 584, + 348, + 596 + ], + "score": 0.92, + "content": "\\| B _ { i } - B _ { \\mathrm { m e d } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "registered at the master server, for two", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 332, + 608 + ], + "score": 1.0, + "content": "nodes, an honest one, and a Byzantine one. The value of √", + "type": "text" + }, + { + "bbox": [ + 333, + 595, + 386, + 607 + ], + "score": 0.93, + "content": "\\| B _ { i } - B _ { \\sf m e d } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "increases for all the nodes (at", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 175, + 620 + ], + "score": 1.0, + "content": "a rate of roughly", + "type": "text" + }, + { + "bbox": [ + 176, + 606, + 189, + 619 + ], + "score": 0.9, + "content": "\\sqrt { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 608, + 218, + 620 + ], + "score": 1.0, + "content": "at step", + "type": "text" + }, + { + "bbox": [ + 218, + 609, + 223, + 618 + ], + "score": 0.45, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "); but, once the attack starts, the statistic for the Byzantine node grows", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 619, + 259, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 150, + 630 + ], + "score": 1.0, + "content": "linearly in", + "type": "text" + }, + { + "bbox": [ + 150, + 619, + 155, + 628 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 619, + 259, + 630 + ], + "score": 1.0, + "content": ", leading to fast detection.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 552, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "Transient attacks and node ID relabeling. Finally, in Figure 2(b) we analyze the behaviour of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "our algorithm when it periodically (every 3 epochs for single safeguard and 6 epochs for double", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "score": 1.0, + "content": "safeguard) resets the set of good nodes to include all nodes, restarting the detection process from", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "scratch. Our theoretical result still applies after this relaxation. This relaxation has two benefits.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "First, it benefits from bad workers that under transient failures (e.g., the node fails for 10 epochs", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "score": 1.0, + "content": "but resumes to work correctly after a while), and thus benefits from the data stored on this worker.", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Second, it can defend against certain degree of node ID relabeling: it supports the case when good", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 724 + ], + "score": 1.0, + "content": "and bad workers exchange their IDs every 6 epochs. In Figure 2(b), we see even under the (very", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 720, + 394, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 394, + 734 + ], + "score": 1.0, + "content": "strong) variance attack, relaxed safeguard maintains good performance.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 58, + "bbox_fs": [ + 105, + 632, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 200, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 200, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 99, + 504, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 505, + 111 + ], + "score": 1.0, + "content": "F. E. and D. 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In International Conference on Machine Learning, pp. 7074–7084,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 344, + 140, + 357 + ], + "spans": [ + { + "bbox": [ + 115, + 344, + 140, + 357 + ], + "score": 1.0, + "content": "2019.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 11.5 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 105, + 82, + 506, + 356 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 83, + 506, + 357 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 250, + 80, + 361, + 101 + ], + "lines": [ + { + "bbox": [ + 248, + 77, + 365, + 106 + ], + "spans": [ + { + "bbox": [ + 248, + 77, + 365, + 106 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 113, + 301, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 112, + 301, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 301, + 128 + ], + "score": 1.0, + "content": "A MISSING PROOFS FOR SECTION 3", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 230, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 230, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 230, + 147 + ], + "score": 1.0, + "content": "A.1 PROOF OF LEMMA 3.2", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 248, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 248, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 248, + 165 + ], + "score": 1.0, + "content": "Recall the following, useful inequality.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 504, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 321, + 179 + ], + "score": 1.0, + "content": "Lemma A.1 (Pinelis’ 1994 inequality (Pinelis, 1994)). Let", + "type": "text" + }, + { + "bbox": [ + 322, + 165, + 390, + 176 + ], + "score": 0.86, + "content": "X _ { 1 } , \\dots , X _ { T } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "be a random process satisfying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 174, + 474, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 176, + 201, + 187 + ], + "score": 0.9, + "content": "\\mathbb { E } [ X _ { t } | X _ { 1 } , \\dots , X _ { t - 1 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 174, + 219, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 177, + 262, + 187 + ], + "score": 0.91, + "content": "\\| X _ { t } \\| \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 174, + 288, + 189 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 288, + 176, + 469, + 188 + ], + "score": 0.86, + "content": "\\operatorname* { P r } \\left[ \\| X _ { 1 } + \\cdot \\cdot \\cdot + X _ { T } \\| ^ { 2 } > 2 \\log ( 2 / \\delta ) M ^ { \\hat { 2 } } T \\right] \\leq \\delta .", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 174, + 474, + 189 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "Lemma 3.2 is in fact a direct corollary of the following claim, whose proof is quite classical. Denote by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 202, + 223, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 204, + 179, + 215 + ], + "score": 0.92, + "content": "C = \\log ( 1 6 m T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 202, + 223, + 216 + ], + "score": 1.0, + "content": ". Denote by", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 217, + 460, + 241 + ], + "lines": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "spans": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "score": 0.9, + "content": "B _ { i } ^ { ( t ) } : = \\frac { \\nabla _ { 0 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla _ { t - 1 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } \\quad \\mathrm { a n d } \\quad B _ { \\star } ^ { ( t ) } : = \\frac { \\nabla f ( w _ { 0 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla f ( w _ { t - 1 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } .", + "type": "interline_equation", + "image_path": "b9561823442ca9dc63974e4c38f59f46c4ad4c4846fd7b870b52044459c1b8bf.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 244, + 503, + 271 + ], + "lines": [ + { + "bbox": [ + 102, + 238, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 102, + 238, + 172, + 263 + ], + "score": 1.0, + "content": "Recall at iteration", + "type": "text" + }, + { + "bbox": [ + 173, + 247, + 192, + 256 + ], + "score": 0.8, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 238, + 279, + 263 + ], + "score": 1.0, + "content": ", Algorithm 2 computes", + "type": "text" + }, + { + "bbox": [ + 280, + 244, + 343, + 257 + ], + "score": 0.93, + "content": "\\{ B _ { 1 } ^ { ( t ) } , \\ldots , B _ { m } ^ { ( t ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 238, + 402, + 263 + ], + "score": 1.0, + "content": "as well as some", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 451, + 258 + ], + "score": 0.93, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 238, + 476, + 263 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 476, + 247, + 480, + 255 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 238, + 505, + 263 + ], + "score": 1.0, + "content": "is any", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 255, + 441, + 273 + ], + "spans": [ + { + "bbox": [ + 103, + 255, + 148, + 273 + ], + "score": 1.0, + "content": "machine in", + "type": "text" + }, + { + "bbox": [ + 148, + 259, + 181, + 271 + ], + "score": 0.82, + "content": "\\tt g o o d _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 255, + 270, + 273 + ], + "score": 1.0, + "content": "such that at least half of", + "type": "text" + }, + { + "bbox": [ + 270, + 259, + 300, + 270 + ], + "score": 0.92, + "content": "j \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 255, + 332, + 273 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 332, + 257, + 434, + 271 + ], + "score": 0.92, + "content": "\\lVert B _ { j } ^ { ( t ) } - B _ { i } ^ { ( t ) } \\rVert \\leq 8 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 255, + 441, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 105, + 274, + 420, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 419, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 167, + 288 + ], + "score": 1.0, + "content": "Claim A.2. Let", + "type": "text" + }, + { + "bbox": [ + 167, + 275, + 238, + 286 + ], + "score": 0.92, + "content": "C = \\log ( 1 6 m T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 274, + 353, + 288 + ], + "score": 1.0, + "content": ". 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B _ { \\mathrm { m e d } } ^ { ( t ) } \\rVert \\leq 1 6 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 320, + 354, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 354, + 321, + 463, + 335 + ], + "score": 0.8, + "content": "\\| B _ { \\star } ^ { ( t ) } - B _ { \\sf m e d } ^ { ( t ) } \\| \\le 1 2 \\sqrt { t C } / m", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 333, + 327, + 351 + ], + "spans": [ + { + "bbox": [ + 103, + 333, + 145, + 351 + ], + "score": 1.0, + "content": "(d) for all", + "type": "text" + }, + { + "bbox": [ + 145, + 337, + 159, + 347 + ], + "score": 0.83, + "content": "i \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 333, + 196, + 351 + ], + "score": 1.0, + "content": "good and", + "type": "text" + }, + { + "bbox": [ + 197, + 336, + 224, + 348 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 333, + 276, + 351 + ], + "score": 1.0, + "content": ", we also have", + "type": "text" + }, + { + "bbox": [ + 276, + 337, + 322, + 349 + ], + "score": 0.91, + "content": "i \\in \\mathbf { g } \\circ \\circ \\mathsf { d } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 333, + 327, + 351 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 351, + 316, + 366 + ], + "spans": [ + { + "bbox": [ + 103, + 351, + 120, + 366 + ], + "score": 1.0, + "content": "(e)", + "type": "text" + }, + { + "bbox": [ + 120, + 351, + 316, + 366 + ], + "score": 0.65, + "content": "\\begin{array} { r } { \\big \\| \\sum _ { i \\in \\mathrm { g o o d } } \\big ( B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) . } \\end{array}", + "type": "inline_equation", + "image_path": "f2225a4ffacf300efa72c62a56e5b383d8274f9942e8531c51fb6901d45779bf.jpg" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 384, + 491, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 491, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 385, + 396 + ], + "score": 1.0, + "content": "Proof of Claim A.2. We prove by induction. Suppose the statements hold for", + "type": "text" + }, + { + "bbox": [ + 385, + 385, + 405, + 394 + ], + "score": 0.84, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 384, + 483, + 396 + ], + "score": 1.0, + "content": "and we now move to", + "type": "text" + }, + { + "bbox": [ + 483, + 386, + 487, + 393 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 384, + 491, + 396 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 506, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 395, + 509, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 120, + 432 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 158, + 401, + 176, + 411 + ], + "score": 0.82, + "content": "\\textit { i } \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 400, + 285, + 411 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\nabla _ { t , i } ] ~ = ~ \\nabla _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 395, + 305, + 432 + ], + "score": 1.0, + "content": "and n thu", + "type": "text" + }, + { + "bbox": [ + 306, + 400, + 381, + 412 + ], + "score": 0.92, + "content": "\\| \\nabla _ { t , i } - \\nabla _ { t } \\| \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 395, + 406, + 432 + ], + "score": 1.0, + "content": ". Let to the", + "type": "text" + }, + { + "bbox": [ + 407, + 397, + 471, + 414 + ], + "score": 0.94, + "content": "\\begin{array} { r } { { \\boldsymbol { X } } _ { t } ~ = ~ \\frac { \\nabla _ { t , i } - \\nabla _ { t } } { | { \\bf g } \\circ { \\bf o d } _ { t } | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 395, + 509, + 432 + ], + "score": 1.0, + "content": ", so that a union", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 412, + 419, + 427 + ], + "spans": [ + { + "bbox": [ + 120, + 412, + 254, + 427 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Vert X _ { t } \\Vert ~ \\leq ~ \\frac { 1 } { \\vert \\mathbf { g } \\circ \\mathbf { o } \\mathbf { d } _ { t } \\vert } ~ \\leq ~ \\frac { 1 } { ( 1 - \\alpha ) m } ~ \\leq ~ \\frac { 2 } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 414, + 419, + 423 + ], + "score": 0.88, + "content": "X _ { t }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 117, + 424, + 508, + 442 + ], + "spans": [ + { + "bbox": [ + 117, + 424, + 174, + 442 + ], + "score": 1.0, + "content": "bound over all", + "type": "text" + }, + { + "bbox": [ + 175, + 428, + 210, + 438 + ], + "score": 0.45, + "content": "i \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 424, + 325, + 442 + ], + "score": 1.0, + "content": ". Thus, with probability at least", + "type": "text" + }, + { + "bbox": [ + 325, + 428, + 354, + 440 + ], + "score": 0.9, + "content": "1 - { \\frac { p } { 8 T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 424, + 387, + 442 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 387, + 426, + 491, + 439 + ], + "score": 0.93, + "content": "\\lVert B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\rVert \\leq 4 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 424, + 508, + 442 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 438, + 387, + 450 + ], + "spans": [ + { + "bbox": [ + 120, + 438, + 132, + 450 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 132, + 439, + 167, + 449 + ], + "score": 0.29, + "content": "i \\in { \\bf g } { \\sf o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 438, + 355, + 450 + ], + "score": 1.0, + "content": ". The result follows from a further union bound over", + "type": "text" + }, + { + "bbox": [ + 356, + 438, + 383, + 449 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 438, + 387, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 104, + 452, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 103, + 450, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 103, + 450, + 228, + 467 + ], + "score": 1.0, + "content": "(b) Claim A.2a implies for every", + "type": "text" + }, + { + "bbox": [ + 228, + 455, + 272, + 465 + ], + "score": 0.65, + "content": "i , j \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 450, + 305, + 467 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 305, + 451, + 408, + 466 + ], + "score": 0.94, + "content": "\\| B _ { i } ^ { ( t ) } - B _ { j } ^ { ( t ) } \\| \\leq 8 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 450, + 468, + 467 + ], + "score": 1.0, + "content": ". Therefore each", + "type": "text" + }, + { + "bbox": [ + 469, + 455, + 483, + 464 + ], + "score": 0.81, + "content": "i \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 450, + 506, + 467 + ], + "score": 1.0, + "content": "good", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 117, + 460, + 272, + 484 + ], + "spans": [ + { + "bbox": [ + 117, + 460, + 219, + 484 + ], + "score": 1.0, + "content": "is a valid choice for setting", + "type": "text" + }, + { + "bbox": [ + 220, + 465, + 268, + 479 + ], + "score": 0.92, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 460, + 272, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 483, + 376, + 496 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 376, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 354, + 496 + ], + "score": 1.0, + "content": "(c) This is a consequence of the previous items and the definition of", + "type": "text" + }, + { + "bbox": [ + 354, + 482, + 373, + 495 + ], + "score": 0.9, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 482, + 376, + 496 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 277, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 277, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 277, + 510 + ], + "score": 1.0, + "content": "(d) This is a consequence of the previous item.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 101, + 504, + 509, + 531 + ], + "spans": [ + { + "bbox": [ + 101, + 504, + 241, + 531 + ], + "score": 1.0, + "content": "(e) We can apply Lemma A.1 with", + "type": "text" + }, + { + "bbox": [ + 241, + 511, + 445, + 527 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\{ X _ { 1 } , X _ { 2 } , \\dots , X _ { t \\mid \\operatorname { g o o d } \\mid } \\} ~ = ~ \\{ { \\frac { \\nabla _ { k , i } - \\nabla f ( w _ { k } ) } { | \\operatorname { g o o d } _ { k } | } } \\} _ { k \\in [ t ] , i \\in \\operatorname { g o o d } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 504, + 509, + 531 + ], + "score": 1.0, + "content": ". It holds with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 119, + 526, + 433, + 541 + ], + "spans": [ + { + "bbox": [ + 119, + 526, + 190, + 541 + ], + "score": 1.0, + "content": "probability at least", + "type": "text" + }, + { + "bbox": [ + 190, + 528, + 218, + 540 + ], + "score": 0.89, + "content": "\\textstyle 1 - { \\frac { p } { 8 T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 526, + 235, + 541 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 235, + 527, + 429, + 541 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big \\| \\sum _ { i \\in \\mathrm { g o o d } } \\big ( B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 526, + 433, + 541 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 363, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 363, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 234, + 589 + ], + "score": 1.0, + "content": "Proof of Lemma 3.2. The property", + "type": "text" + }, + { + "bbox": [ + 234, + 577, + 289, + 588 + ], + "score": 0.81, + "content": "\\mathbf { g } _ { 0 0 } \\mathsf { d } _ { t } \\supseteq \\mathsf { g } _ { 0 0 } \\mathsf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 576, + 363, + 589 + ], + "score": 1.0, + "content": "is from Claim A.2d.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 109, + 591, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 157, + 605 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 158, + 590, + 241, + 604 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\sigma _ { t } \\| ^ { 2 } \\le { \\cal O } ( \\frac { \\log ( T / p ) } { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 588, + 506, + 605 + ], + "score": 1.0, + "content": "is by standard concentration inequalities for sums of bounded random", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 602, + 138, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 138, + 614 + ], + "score": 1.0, + "content": "vectors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 370, + 628 + ], + "lines": [ + { + "bbox": [ + 104, + 609, + 371, + 630 + ], + "spans": [ + { + "bbox": [ + 104, + 609, + 156, + 630 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 156, + 614, + 294, + 628 + ], + "score": 0.93, + "content": "\\| \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 609, + 371, + 630 + ], + "score": 1.0, + "content": "is from Claim A.2e.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 421, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 420, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 155, + 641 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 156, + 629, + 195, + 641 + ], + "score": 0.92, + "content": "\\| \\Delta _ { t } \\| \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 628, + 304, + 641 + ], + "score": 1.0, + "content": "is obvious as we have at most", + "type": "text" + }, + { + "bbox": [ + 304, + 631, + 312, + 639 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 628, + 420, + 641 + ], + "score": 1.0, + "content": "fraction of the bad machines.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 109, + 642, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 107, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 107, + 642, + 161, + 654 + ], + "score": 1.0, + "content": "The bound on", + "type": "text" + }, + { + "bbox": [ + 161, + 642, + 243, + 654 + ], + "score": 0.89, + "content": "\\| \\Delta _ { 0 } + \\cdots + \\Delta _ { t - 1 } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 642, + 382, + 654 + ], + "score": 1.0, + "content": "can be derived as follows. For every", + "type": "text" + }, + { + "bbox": [ + 383, + 642, + 442, + 654 + ], + "score": 0.84, + "content": "i \\in [ m ] \\ \\backslash \\ \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 642, + 458, + 654 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 459, + 644, + 464, + 652 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 642, + 506, + 654 + ], + "score": 1.0, + "content": "be the last", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 651, + 333, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 138, + 665 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 139, + 654, + 143, + 662 + ], + "score": 0.7, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 651, + 174, + 665 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 175, + 654, + 212, + 664 + ], + "score": 0.88, + "content": "i \\in \\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 651, + 333, + 665 + ], + "score": 1.0, + "content": ". 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combining this with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 700, + 454, + 718 + ], + "spans": [ + { + "bbox": [ + 107, + 702, + 216, + 716 + ], + "score": 0.91, + "content": "\\| B _ { \\star } ^ { ( t ) } - B _ { \\sf m e d } ^ { ( t ) } \\| \\le 1 2 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 700, + 380, + 718 + ], + "score": 1.0, + "content": ", and summing up over all such bad machines", + "type": "text" + }, + { + "bbox": [ + 380, + 706, + 384, + 713 + ], + "score": 0.78, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 700, + 454, + 718 + ], + "score": 1.0, + "content": "finishes the proof.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 721, + 409, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 721, + 409, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 409, + 733 + ], + "score": 1.0, + "content": "The final two properties follow from standard facts about Gaussian random vectors.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 491, + 543, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 492, + 543, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 492, + 543, + 504, + 558 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 491, + 718, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 492, + 718, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 492, + 718, + 504, + 732 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 250, + 80, + 361, + 101 + ], + "lines": [ + { + "bbox": [ + 248, + 77, + 365, + 106 + ], + "spans": [ + { + "bbox": [ + 248, + 77, + 365, + 106 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 113, + 301, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 112, + 301, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 301, + 128 + ], + "score": 1.0, + "content": "A MISSING PROOFS FOR SECTION 3", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 230, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 230, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 230, + 147 + ], + "score": 1.0, + "content": "A.1 PROOF OF LEMMA 3.2", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 135, + 230, + 147 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 248, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 248, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 248, + 165 + ], + "score": 1.0, + "content": "Recall the following, useful inequality.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 149, + 248, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 504, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 321, + 179 + ], + "score": 1.0, + "content": "Lemma A.1 (Pinelis’ 1994 inequality (Pinelis, 1994)). Let", + "type": "text" + }, + { + "bbox": [ + 322, + 165, + 390, + 176 + ], + "score": 0.86, + "content": "X _ { 1 } , \\dots , X _ { T } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "be a random process satisfying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 174, + 474, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 176, + 201, + 187 + ], + "score": 0.9, + "content": "\\mathbb { E } [ X _ { t } | X _ { 1 } , \\dots , X _ { t - 1 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 174, + 219, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 177, + 262, + 187 + ], + "score": 0.91, + "content": "\\| X _ { t } \\| \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 174, + 288, + 189 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 288, + 176, + 469, + 188 + ], + "score": 0.86, + "content": "\\operatorname* { P r } \\left[ \\| X _ { 1 } + \\cdot \\cdot \\cdot + X _ { T } \\| ^ { 2 } > 2 \\log ( 2 / \\delta ) M ^ { \\hat { 2 } } T \\right] \\leq \\delta .", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 174, + 474, + 189 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 164, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "Lemma 3.2 is in fact a direct corollary of the following claim, whose proof is quite classical. Denote by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 202, + 223, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 204, + 179, + 215 + ], + "score": 0.92, + "content": "C = \\log ( 1 6 m T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 202, + 223, + 216 + ], + "score": 1.0, + "content": ". Denote by", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 191, + 505, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 217, + 460, + 241 + ], + "lines": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "spans": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "score": 0.9, + "content": "B _ { i } ^ { ( t ) } : = \\frac { \\nabla _ { 0 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla _ { t - 1 , i } } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } \\quad \\mathrm { a n d } \\quad B _ { \\star } ^ { ( t ) } : = \\frac { \\nabla f ( w _ { 0 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { 0 } | } + \\cdot \\cdot \\cdot + \\frac { \\nabla f ( w _ { t - 1 } ) } { | \\mathbf { g } \\mathsf { o o d } _ { t - 1 } | } .", + "type": "interline_equation", + "image_path": "b9561823442ca9dc63974e4c38f59f46c4ad4c4846fd7b870b52044459c1b8bf.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 150, + 217, + 460, + 241 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 244, + 503, + 271 + ], + "lines": [ + { + "bbox": [ + 102, + 238, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 102, + 238, + 172, + 263 + ], + "score": 1.0, + "content": "Recall at iteration", + "type": "text" + }, + { + "bbox": [ + 173, + 247, + 192, + 256 + ], + "score": 0.8, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 238, + 279, + 263 + ], + "score": 1.0, + "content": ", Algorithm 2 computes", + "type": "text" + }, + { + "bbox": [ + 280, + 244, + 343, + 257 + ], + "score": 0.93, + "content": "\\{ B _ { 1 } ^ { ( t ) } , \\ldots , B _ { m } ^ { ( t ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 238, + 402, + 263 + ], + "score": 1.0, + "content": "as well as some", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 451, + 258 + ], + "score": 0.93, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 238, + 476, + 263 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 476, + 247, + 480, + 255 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 238, + 505, + 263 + ], + "score": 1.0, + "content": "is any", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 255, + 441, + 273 + ], + "spans": [ + { + "bbox": [ + 103, + 255, + 148, + 273 + ], + "score": 1.0, + "content": "machine in", + "type": "text" + }, + { + "bbox": [ + 148, + 259, + 181, + 271 + ], + "score": 0.82, + "content": "\\tt g o o d _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 255, + 270, + 273 + ], + "score": 1.0, + "content": "such that at least half of", + "type": "text" + }, + { + "bbox": [ + 270, + 259, + 300, + 270 + ], + "score": 0.92, + "content": "j \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 255, + 332, + 273 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 332, + 257, + 434, + 271 + ], + "score": 0.92, + "content": "\\lVert B _ { j } ^ { ( t ) } - B _ { i } ^ { ( t ) } \\rVert \\leq 8 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 255, + 441, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 102, + 238, + 505, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 274, + 420, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 419, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 167, + 288 + ], + "score": 1.0, + "content": "Claim A.2. Let", + "type": "text" + }, + { + "bbox": [ + 167, + 275, + 238, + 286 + ], + "score": 0.92, + "content": "C = \\log ( 1 6 m T / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 274, + 353, + 288 + ], + "score": 1.0, + "content": ". 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B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) . } \\end{array}", + "type": "inline_equation", + "image_path": "f2225a4ffacf300efa72c62a56e5b383d8274f9942e8531c51fb6901d45779bf.jpg" + } + ], + "index": 16, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 14, + "bbox_fs": [ + 100, + 288, + 463, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 384, + 491, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 491, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 385, + 396 + ], + "score": 1.0, + "content": "Proof of Claim A.2. We prove by induction. Suppose the statements hold for", + "type": "text" + }, + { + "bbox": [ + 385, + 385, + 405, + 394 + ], + "score": 0.84, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 384, + 483, + 396 + ], + "score": 1.0, + "content": "and we now move to", + "type": "text" + }, + { + "bbox": [ + 483, + 386, + 487, + 393 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 384, + 491, + 396 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 384, + 491, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 399, + 506, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 395, + 509, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 120, + 432 + ], + "score": 1.0, + "content": "(a)", + "type": "text" + }, + { + "bbox": [ + 158, + 401, + 176, + 411 + ], + "score": 0.82, + "content": "\\textit { i } \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 400, + 285, + 411 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\nabla _ { t , i } ] ~ = ~ \\nabla _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 395, + 305, + 432 + ], + "score": 1.0, + "content": "and n thu", + "type": "text" + }, + { + "bbox": [ + 306, + 400, + 381, + 412 + ], + "score": 0.92, + "content": "\\| \\nabla _ { t , i } - \\nabla _ { t } \\| \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 395, + 406, + 432 + ], + "score": 1.0, + "content": ". Let to the", + "type": "text" + }, + { + "bbox": [ + 407, + 397, + 471, + 414 + ], + "score": 0.94, + "content": "\\begin{array} { r } { { \\boldsymbol { X } } _ { t } ~ = ~ \\frac { \\nabla _ { t , i } - \\nabla _ { t } } { | { \\bf g } \\circ { \\bf o d } _ { t } | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 395, + 509, + 432 + ], + "score": 1.0, + "content": ", so that a union", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 412, + 419, + 427 + ], + "spans": [ + { + "bbox": [ + 120, + 412, + 254, + 427 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Vert X _ { t } \\Vert ~ \\leq ~ \\frac { 1 } { \\vert \\mathbf { g } \\circ \\mathbf { o } \\mathbf { d } _ { t } \\vert } ~ \\leq ~ \\frac { 1 } { ( 1 - \\alpha ) m } ~ \\leq ~ \\frac { 2 } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 414, + 419, + 423 + ], + "score": 0.88, + "content": "X _ { t }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 117, + 424, + 508, + 442 + ], + "spans": [ + { + "bbox": [ + 117, + 424, + 174, + 442 + ], + "score": 1.0, + "content": "bound over all", + "type": "text" + }, + { + "bbox": [ + 175, + 428, + 210, + 438 + ], + "score": 0.45, + "content": "i \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 424, + 325, + 442 + ], + "score": 1.0, + "content": ". Thus, with probability at least", + "type": "text" + }, + { + "bbox": [ + 325, + 428, + 354, + 440 + ], + "score": 0.9, + "content": "1 - { \\frac { p } { 8 T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 424, + 387, + 442 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 387, + 426, + 491, + 439 + ], + "score": 0.93, + "content": "\\lVert B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\rVert \\leq 4 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 424, + 508, + 442 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 120, + 438, + 387, + 450 + ], + "spans": [ + { + "bbox": [ + 120, + 438, + 132, + 450 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 132, + 439, + 167, + 449 + ], + "score": 0.29, + "content": "i \\in { \\bf g } { \\sf o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 438, + 355, + 450 + ], + "score": 1.0, + "content": ". The result follows from a further union bound over", + "type": "text" + }, + { + "bbox": [ + 356, + 438, + 383, + 449 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 438, + 387, + 450 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 395, + 509, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 452, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 103, + 450, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 103, + 450, + 228, + 467 + ], + "score": 1.0, + "content": "(b) Claim A.2a implies for every", + "type": "text" + }, + { + "bbox": [ + 228, + 455, + 272, + 465 + ], + "score": 0.65, + "content": "i , j \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 450, + 305, + 467 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 305, + 451, + 408, + 466 + ], + "score": 0.94, + "content": "\\| B _ { i } ^ { ( t ) } - B _ { j } ^ { ( t ) } \\| \\leq 8 \\sqrt { t C } / m", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 450, + 468, + 467 + ], + "score": 1.0, + "content": ". Therefore each", + "type": "text" + }, + { + "bbox": [ + 469, + 455, + 483, + 464 + ], + "score": 0.81, + "content": "i \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 450, + 506, + 467 + ], + "score": 1.0, + "content": "good", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 117, + 460, + 272, + 484 + ], + "spans": [ + { + "bbox": [ + 117, + 460, + 219, + 484 + ], + "score": 1.0, + "content": "is a valid choice for setting", + "type": "text" + }, + { + "bbox": [ + 220, + 465, + 268, + 479 + ], + "score": 0.92, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) } = B _ { i } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 460, + 272, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 103, + 450, + 506, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 483, + 376, + 496 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 376, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 354, + 496 + ], + "score": 1.0, + "content": "(c) This is a consequence of the previous items and the definition of", + "type": "text" + }, + { + "bbox": [ + 354, + 482, + 373, + 495 + ], + "score": 0.9, + "content": "B _ { \\mathrm { m e d } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 482, + 376, + 496 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 482, + 376, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 277, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 277, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 277, + 510 + ], + "score": 1.0, + "content": "(d) This is a consequence of the previous item.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 497, + 277, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 101, + 504, + 509, + 531 + ], + "spans": [ + { + "bbox": [ + 101, + 504, + 241, + 531 + ], + "score": 1.0, + "content": "(e) We can apply Lemma A.1 with", + "type": "text" + }, + { + "bbox": [ + 241, + 511, + 445, + 527 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\{ X _ { 1 } , X _ { 2 } , \\dots , X _ { t \\mid \\operatorname { g o o d } \\mid } \\} ~ = ~ \\{ { \\frac { \\nabla _ { k , i } - \\nabla f ( w _ { k } ) } { | \\operatorname { g o o d } _ { k } | } } \\} _ { k \\in [ t ] , i \\in \\operatorname { g o o d } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 504, + 509, + 531 + ], + "score": 1.0, + "content": ". It holds with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 119, + 526, + 433, + 541 + ], + "spans": [ + { + "bbox": [ + 119, + 526, + 190, + 541 + ], + "score": 1.0, + "content": "probability at least", + "type": "text" + }, + { + "bbox": [ + 190, + 528, + 218, + 540 + ], + "score": 0.89, + "content": "\\textstyle 1 - { \\frac { p } { 8 T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 526, + 235, + 541 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 235, + 527, + 429, + 541 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\big \\| \\sum _ { i \\in \\mathrm { g o o d } } \\big ( B _ { i } ^ { ( t ) } - B _ { \\star } ^ { ( t ) } \\big ) \\big \\| \\leq O ( \\sqrt { t \\log ( T / p ) } / \\sqrt { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 526, + 433, + 541 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 101, + 504, + 509, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 363, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 363, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 234, + 589 + ], + "score": 1.0, + "content": "Proof of Lemma 3.2. The property", + "type": "text" + }, + { + "bbox": [ + 234, + 577, + 289, + 588 + ], + "score": 0.81, + "content": "\\mathbf { g } _ { 0 0 } \\mathsf { d } _ { t } \\supseteq \\mathsf { g } _ { 0 0 } \\mathsf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 576, + 363, + 589 + ], + "score": 1.0, + "content": "is from Claim A.2d.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 576, + 363, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 591, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 157, + 605 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 158, + 590, + 241, + 604 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\sigma _ { t } \\| ^ { 2 } \\le { \\cal O } ( \\frac { \\log ( T / p ) } { m } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 588, + 506, + 605 + ], + "score": 1.0, + "content": "is by standard concentration inequalities for sums of bounded random", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 602, + 138, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 138, + 614 + ], + "score": 1.0, + "content": "vectors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 588, + 506, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 370, + 628 + ], + "lines": [ + { + "bbox": [ + 104, + 609, + 371, + 630 + ], + "spans": [ + { + "bbox": [ + 104, + 609, + 156, + 630 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 156, + 614, + 294, + 628 + ], + "score": 0.93, + "content": "\\| \\sigma _ { 0 } + \\cdot \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T \\log ( T / p ) } { m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 609, + 371, + 630 + ], + "score": 1.0, + "content": "is from Claim A.2e.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 609, + 371, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 421, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 420, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 155, + 641 + ], + "score": 1.0, + "content": "The property", + "type": "text" + }, + { + "bbox": [ + 156, + 629, + 195, + 641 + ], + "score": 0.92, + "content": "\\| \\Delta _ { t } \\| \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 628, + 304, + 641 + ], + "score": 1.0, + "content": "is obvious as we have at most", + "type": "text" + }, + { + "bbox": [ + 304, + 631, + 312, + 639 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 628, + 420, + 641 + ], + "score": 1.0, + "content": "fraction of the bad machines.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 628, + 420, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 642, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 107, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 107, + 642, + 161, + 654 + ], + "score": 1.0, + "content": "The bound on", + "type": "text" + }, + { + "bbox": [ + 161, + 642, + 243, + 654 + ], + "score": 0.89, + "content": "\\| \\Delta _ { 0 } + \\cdots + \\Delta _ { t - 1 } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 642, + 382, + 654 + ], + "score": 1.0, + "content": "can be derived as follows. 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f ( w _ { T } ) \\geq 0 . 8 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - O ( \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad - C _ { 2 } \\cdot O ( \\eta + \\eta ^ { 2 } T + \\eta ^ { 3 } T ^ { 2 } + \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \\cdot O ( T \\eta ^ { 2 } \\nu ^ { 2 } d + T ^ { 2 } \\eta ^ { 3 } \\nu ^ { 2 } \\sqrt { C _ { 2 } } + \\eta T \\nu ^ { 2 } ( \\eta d + \\frac { 1 } { T } ) ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "73d20642e74334221ecebb186b6e94d4f02ee4cdcefe6880ff755824495c98e2.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 124, + 523, + 527, + 540.6666666666666 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 124, + 540.6666666666666, + 527, + 558.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 124, + 558.3333333333333, + 527, + 575.9999999999999 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 357, + 593 + ], + "lines": [ + { + "bbox": [ + 156, + 571, + 321, + 599 + ], + "spans": [ + { + "bbox": [ + 156, + 578, + 224, + 593 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 571, + 240, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 578, + 321, + 592 + ], + "score": 0.92, + "content": "\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 156, + 571, + 321, + 599 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 595, + 514, + 691 + ], + "lines": [ + { + "bbox": [ + 111, + 595, + 514, + 691 + ], + "spans": [ + { + "bbox": [ + 111, + 595, + 514, + 691 + ], + "score": 0.91, + "content": "\\begin{array} { l } { { \\displaystyle { \\dot { \\tau } } ( w _ { 0 } ) - f ( w _ { T } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - C _ { 2 } \\cdot O ( \\eta + \\eta ^ { 2 } T + \\eta ^ { 3 } T ^ { 2 } + \\sqrt { C _ { 2 } } \\eta ^ { 3 } T ^ { 2 } ) - C _ { 1 } \\cdot O ( T \\eta \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ^ { 2 } ) } } \\\\ { { \\displaystyle \\qquad = 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - C _ { 2 } \\cdot O ( \\frac { 1 } { T } + \\eta + \\eta ^ { 2 } T + \\sqrt { C _ { 2 } } \\eta ^ { 2 } T ) - C _ { 1 } \\cdot O ( \\eta T \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) } } \\\\ { { \\displaystyle \\qquad \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) \\ . } } \\end{array}", + "type": "interline_equation", + "image_path": "b02c5875e9361055b04e9aec5db6929204e34db32506261376def8049a9823c1.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 111, + 595, + 514, + 627.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 111, + 627.0, + 514, + 659.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 111, + 659.0, + 514, + 691.0 + ], + "spans": [], + "index": 22 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 254, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 255, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 255, + 95 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 3.4", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 506, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 283, + 113 + ], + "score": 1.0, + "content": "Proposition 3.4. 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Suppose that these vectors satisfy", + "type": "text" + }, + { + "bbox": [ + 405, + 121, + 494, + 132 + ], + "score": 0.87, + "content": "\\| \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\| ^ { 2 } \\leq \\mathfrak { T } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 117, + 507, + 135 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 130, + 332, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 128, + 144 + ], + "score": 1.0, + "content": "every", + "type": "text" + }, + { + "bbox": [ + 128, + 132, + 192, + 142 + ], + "score": 0.9, + "content": "t = 1 , \\dots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 130, + 307, + 144 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 307, + 132, + 329, + 142 + ], + "score": 0.87, + "content": "1 - p ,", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 130, + 332, + 144 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 144, + 410, + 162 + ], + "lines": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "spans": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "906cf685822d4e99239c9192719e45f6f34f058ec802f69f6d93ad2cd650ce5e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 171, + 300, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 301, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 301, + 184 + ], + "score": 1.0, + "content": "Proof of Proposition 3.4. Using the identity formula6", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 184, + 469, + 203 + ], + "lines": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "spans": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle = \\left( \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right) \\left( \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right) - \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle = \\Delta _ { 0 } ^ { 2 } \\Delta _ { t } . } \\end{array}", + "type": "interline_equation", + "image_path": "033543a9e2d0e7c570956a509a551f90a74b038f4d51ed3baf87b4812a32529a.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 138, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 139, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 139, + 215 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 213, + 482, + 254 + ], + "lines": [ + { + "bbox": [ + 129, + 213, + 482, + 254 + ], + "spans": [ + { + "bbox": [ + 129, + 213, + 482, + 254 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta t \\rangle \\right| \\leq \\left\\| \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right\\| \\cdot \\left\\| \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right\\| + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| } \\\\ & { \\qquad \\leq O ( \\sqrt { d T ^ { \\sum } \\log ( T / p ) } ) + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\ . } \\end{array}", + "type": "interline_equation", + "image_path": "9307e5d36c5cb0f6c1fa33fe3f6999cf01db532521918c99303003456b056e31.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 129, + 213, + 482, + 226.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 129, + 226.66666666666666, + 482, + 240.33333333333331 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 129, + 240.33333333333331, + 482, + 253.99999999999997 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 218, + 270 + ], + "score": 1.0, + "content": "where the last inequality uses", + "type": "text" + }, + { + "bbox": [ + 219, + 256, + 376, + 268 + ], + "score": 0.87, + "content": "\\| \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { T - 2 } \\| \\le O ( \\sqrt { d T \\log ( 1 / p ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 256, + 469, + 270 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 470, + 257, + 501, + 268 + ], + "score": 0.91, + "content": "1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 256, + 505, + 270 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 205, + 281 + ], + "score": 1.0, + "content": "Furthermore, we note that", + "type": "text" + }, + { + "bbox": [ + 205, + 268, + 214, + 277 + ], + "score": 0.87, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 264, + 282, + 281 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 282, + 268, + 376, + 277 + ], + "score": 0.76, + "content": "\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 } , \\Delta _ { 1 } , \\ldots , \\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 264, + 393, + 281 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 267, + 433, + 278 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\xi _ { t } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 264, + 506, + 281 + ], + "score": 1.0, + "content": ". Therefore, letting", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 107, + 277, + 204, + 288 + ], + "score": 0.9, + "content": "S _ { t } = \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 275, + 239, + 290 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 239, + 277, + 325, + 288 + ], + "score": 0.91, + "content": "\\mathbb { E } [ S _ { t } | \\xi _ { 0 } , \\dots , \\xi _ { t - 1 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 275, + 463, + 290 + ], + "score": 1.0, + "content": "; furthermore, with probability at least", + "type": "text" + }, + { + "bbox": [ + 464, + 277, + 494, + 288 + ], + "score": 0.91, + "content": "1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 275, + 506, + 290 + ], + "score": 1.0, + "content": ", it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 288, + 425, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 137, + 300 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 138, + 288, + 235, + 300 + ], + "score": 0.9, + "content": "| S _ { t } | \\leq O ( \\sqrt { d \\mathfrak { T } \\log ( T / p ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 288, + 271, + 300 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 271, + 290, + 276, + 298 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 288, + 425, + 300 + ], + "score": 1.0, + "content": ". Finally, by Azuma’s inequality, we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 302, + 408, + 322 + ], + "lines": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "spans": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "db940c92d791db75fe64be42e15a5df5fd004067d7b79dd9f36aa34ed1facaa6.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 330, + 229, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 230, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 230, + 343 + ], + "score": 1.0, + "content": "A.4 PROOF OF LEMMA 3.5", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 348, + 482, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 482, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 247, + 365 + ], + "score": 1.0, + "content": "Proof of Lemma 3.5. Let us denote by", + "type": "text" + }, + { + "bbox": [ + 247, + 348, + 321, + 363 + ], + "score": 0.94, + "content": "\\begin{array} { r } { r _ { t } = \\frac { [ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } } { 2 } = - \\frac { [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 } } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 347, + 372, + 365 + ], + "score": 1.0, + "content": "and we know", + "type": "text" + }, + { + "bbox": [ + 373, + 349, + 428, + 363 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { t } \\sim \\mathcal { N } ( 0 , \\frac { \\nu ^ { 2 } } { 4 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 347, + 482, + 365 + ], + "score": 1.0, + "content": ". We can write", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 365, + 445, + 379 + ], + "lines": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "spans": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "score": 0.82, + "content": "w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } - \\eta ( \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } )", + "type": "interline_equation", + "image_path": "a1f34a11af70377a67f341f5253d4a55947b8704dad796c1881cf38f5c3e6f83.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 271, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 271, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 271, + 392 + ], + "score": 1.0, + "content": "Using the second-order smoothness, we have", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 392, + 430, + 432 + ], + "lines": [ + { + "bbox": [ + 181, + 392, + 430, + 432 + ], + "spans": [ + { + "bbox": [ + 181, + 392, + 430, + 432 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) = \\displaystyle \\int _ { \\tau = 0 } ^ { 1 } \\nabla ^ { 2 } f \\big ( { w } _ { t } ^ { \\mathsf { a } } + \\tau ( w _ { t } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { a } } ) \\big ) \\big ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\big ) d \\tau } \\\\ & { \\qquad = \\nabla ^ { 2 } f \\big ( { w } _ { 0 } \\big ) \\cdot \\big ( { w } _ { t } ^ { \\mathsf { a } } - { w } _ { t } ^ { \\mathsf { b } } \\big ) + \\theta _ { t } } \\end{array}", + "type": "interline_equation", + "image_path": "e6b952ac6a13dc351897ac31a436c7f5dffe8bfb39ca83bc1cb258ae0560d2b9.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 181, + 392, + 430, + 405.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 181, + 405.3333333333333, + 430, + 418.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 181, + 418.66666666666663, + 430, + 431.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 434, + 434, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 434, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 164, + 447 + ], + "score": 1.0, + "content": "for some vector", + "type": "text" + }, + { + "bbox": [ + 164, + 434, + 358, + 446 + ], + "score": 0.87, + "content": "\\| \\theta _ { t } \\| \\leq \\operatorname* { m a x } \\{ \\| w _ { 0 } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { a } } \\| , \\| w _ { 0 } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { b } } \\| \\} \\cdot \\| w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 433, + 434, + 447 + ], + "score": 1.0, + "content": ". Therefore, we have", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 448, + 441, + 462 + ], + "lines": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "spans": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "score": 0.78, + "content": "w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } )", + "type": "interline_equation", + "image_path": "30ff4dbae63bb63312ce28242825b7f508e71385711c567d855da5d5d81e5a41.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 291, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 293, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 152, + 477 + ], + "score": 1.0, + "content": "Now, giving", + "type": "text" + }, + { + "bbox": [ + 153, + 464, + 204, + 476 + ], + "score": 0.92, + "content": "\\psi _ { 0 } = \\overline { { \\psi } } _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 464, + 293, + 477 + ], + "score": 1.0, + "content": ", imagine two sequences", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 112, + 478, + 396, + 507 + ], + "lines": [ + { + "bbox": [ + 111, + 477, + 276, + 491 + ], + "spans": [ + { + "bbox": [ + 111, + 477, + 119, + 491 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 119, + 477, + 259, + 491 + ], + "score": 0.85, + "content": "\\psi _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) \\psi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 477, + 276, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 110, + 491, + 396, + 509 + ], + "spans": [ + { + "bbox": [ + 110, + 491, + 119, + 509 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 120, + 492, + 396, + 506 + ], + "score": 0.83, + "content": "\\overline { { \\psi } } _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) \\overline { { \\psi } } _ { t } - \\eta \\big ( \\Xi _ { t } ^ { 3 } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } \\big ) = w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } }", + "type": "inline_equation", + "image_path": "b8c6e48c046f72e0f446e72874995ffe01d9b92ea9140f374103b2e557e246ae.jpg" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 204, + 523 + ], + "score": 1.0, + "content": "We will inductively prove", + "type": "text" + }, + { + "bbox": [ + 204, + 509, + 284, + 522 + ], + "score": 0.93, + "content": "\\lVert \\psi _ { t } - \\overline { { \\psi } } _ { t } \\rVert \\leq \\frac 1 2 \\lVert \\psi _ { t } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 509, + 416, + 523 + ], + "score": 1.0, + "content": ". On one hand, it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 417, + 511, + 427, + 520 + ], + "score": 0.88, + "content": "\\psi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "is zero except in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 517, + 497, + 541 + ], + "spans": [ + { + "bbox": [ + 102, + 517, + 497, + 541 + ], + "score": 1.0, + "content": "first coordinate, in which it behaves as a Gaussian with zero mean and variance Pt−1k=0(1 + ηδ)2k · η2ν24", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 111, + 532, + 333, + 554 + ], + "spans": [ + { + "bbox": [ + 111, + 532, + 333, + 554 + ], + "score": 1.0, + "content": "Θ \u0010 (1+ηδ)2tηδ · η2ν2\u0011. By Gaussian tail bounds, we know that", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 111, + 555, + 405, + 591 + ], + "lines": [ + { + "bbox": [ + 110, + 552, + 405, + 573 + ], + "spans": [ + { + "bbox": [ + 110, + 552, + 266, + 573 + ], + "score": 1.0, + "content": "• with probability at least 0.99, it satisfies", + "type": "text" + }, + { + "bbox": [ + 267, + 555, + 363, + 572 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { t } \\| \\leq O ( \\frac { \\sqrt { \\eta C _ { 1 } } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 552, + 400, + 573 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 401, + 560, + 405, + 568 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 108, + 573, + 370, + 595 + ], + "spans": [ + { + "bbox": [ + 108, + 573, + 267, + 595 + ], + "score": 1.0, + "content": "• with probability at least 0.99, it satisfies", + "type": "text" + }, + { + "bbox": [ + 267, + 575, + 370, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { T } \\| \\ge \\frac { 1 } { 1 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { T } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 105, + 594, + 504, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "In the rest of the proof, we condition on this event happens. We prove towards contradiction by assuming", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 603, + 303, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 169, + 615 + ], + "score": 0.92, + "content": "\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } ^ { \\mathsf { a } } \\| \\leq R", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 603, + 185, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 185, + 603, + 247, + 615 + ], + "score": 0.93, + "content": "\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } ^ { \\mathsf { b } } \\| \\leq R", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 603, + 272, + 615 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 272, + 604, + 299, + 615 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 603, + 303, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 441, + 634 + ], + "lines": [ + { + "bbox": [ + 102, + 614, + 444, + 636 + ], + "spans": [ + { + "bbox": [ + 102, + 614, + 217, + 636 + ], + "score": 1.0, + "content": "We will inductively prove that", + "type": "text" + }, + { + "bbox": [ + 217, + 617, + 336, + 634 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 614, + 444, + 636 + ], + "score": 1.0, + "content": ". We calculate the difference", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 636, + 411, + 666 + ], + "lines": [ + { + "bbox": [ + 203, + 636, + 411, + 666 + ], + "spans": [ + { + "bbox": [ + 203, + 636, + 411, + 666 + ], + "score": 0.9, + "content": "\\psi _ { t } - { \\overline { { \\psi } } } _ { t } = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\left( \\mathbf { I } - \\eta { \\boldsymbol \\nabla } ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } ( { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { a } } - { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { b } } + { \\boldsymbol \\theta } _ { i } )", + "type": "interline_equation", + "image_path": "83d83c17d711d93c9eb740d9b432eb48f9f512cd2553ebe5e21bcc0a79f92e19.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 636, + 411, + 651.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 203, + 651.0, + 411, + 666.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 669, + 430, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 430, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 138, + 683 + ], + "score": 1.0, + "content": "Let g =", + "type": "text" + }, + { + "bbox": [ + 121, + 668, + 171, + 685 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g = \\frac { \\psi _ { t } - \\overline { { \\psi } } _ { t } } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 667, + 376, + 687 + ], + "score": 1.0, + "content": ", then we can inner-product the above equation by vector", + "type": "text" + }, + { + "bbox": [ + 376, + 675, + 381, + 682 + ], + "score": 0.76, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 667, + 430, + 687 + ], + "score": 1.0, + "content": ", which gives", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 686, + 341, + 717 + ], + "lines": [ + { + "bbox": [ + 109, + 686, + 341, + 717 + ], + "spans": [ + { + "bbox": [ + 109, + 686, + 341, + 717 + ], + "score": 0.73, + "content": "\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\Big \\langle \\Xi _ { i } ^ { \\mathrm { a } } - \\Xi _ { i } ^ { \\mathrm { b } } + \\theta _ { i } , \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) ^ { t - 1 - i } g \\Big \\rangle", + "type": "interline_equation", + "image_path": "a650dcb471bce83dc72faedab74503e39191a292b0a2ab01a7b7dfb6ea9fa774.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 109, + 686, + 341, + 701.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 109, + 701.5, + 341, + 717.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 117, + 722, + 449, + 734 + ], + "lines": [ + { + "bbox": [ + 118, + 721, + 450, + 736 + ], + "spans": [ + { + "bbox": [ + 118, + 721, + 450, + 736 + ], + "score": 1.0, + "content": "6We thank an anonymous reviewer on openreview who pointed out this simpler proof to us.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 491, + 303, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 492, + 303, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 492, + 303, + 504, + 318 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 254, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 255, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 255, + 95 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 3.4", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 506, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 283, + 113 + ], + "score": 1.0, + "content": "Proposition 3.4. Fix the dimension parameter", + "type": "text" + }, + { + "bbox": [ + 284, + 101, + 309, + 111 + ], + "score": 0.9, + "content": "d \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 100, + 348, + 113 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 348, + 100, + 422, + 112 + ], + "score": 0.91, + "content": "\\xi _ { 0 } , \\ldots , \\xi _ { T - 1 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "are i.i.d. drawn from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 109, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 107, + 111, + 136, + 122 + ], + "score": 0.9, + "content": "\\mathcal { N } ( 0 , \\bf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 109, + 171, + 123 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + }, + { + "bbox": [ + 172, + 111, + 227, + 121 + ], + "score": 0.91, + "content": "\\Delta _ { 1 } , \\ldots , \\Delta _ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 109, + 314, + 123 + ], + "score": 1.0, + "content": "are arbitrary vectors in", + "type": "text" + }, + { + "bbox": [ + 314, + 111, + 326, + 120 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 109, + 395, + 123 + ], + "score": 1.0, + "content": ". 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Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 307, + 132, + 329, + 142 + ], + "score": 0.87, + "content": "1 - p ,", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 130, + 332, + 144 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 100, + 507, + 144 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 144, + 410, + 162 + ], + "lines": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "spans": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "906cf685822d4e99239c9192719e45f6f34f058ec802f69f6d93ad2cd650ce5e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 200, + 144, + 410, + 162 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 171, + 300, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 301, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 301, + 184 + ], + "score": 1.0, + "content": "Proof of Proposition 3.4. Using the identity formula6", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 171, + 301, + 184 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 184, + 469, + 203 + ], + "lines": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "spans": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta _ { t } \\rangle = \\left( \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right) \\left( \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right) - \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle = \\Delta _ { 0 } ^ { 2 } \\Delta _ { t } . } \\end{array}", + "type": "interline_equation", + "image_path": "033543a9e2d0e7c570956a509a551f90a74b038f4d51ed3baf87b4812a32529a.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 141, + 184, + 469, + 203 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 138, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 139, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 139, + 215 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 204, + 139, + 215 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 213, + 482, + 254 + ], + "lines": [ + { + "bbox": [ + 129, + 213, + 482, + 254 + ], + "spans": [ + { + "bbox": [ + 129, + 213, + 482, + 254 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\left| \\sum _ { t = 1 } ^ { T - 1 } \\langle \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } , \\Delta t \\rangle \\right| \\leq \\left\\| \\sum _ { t = 0 } ^ { T - 2 } \\xi _ { t } \\right\\| \\cdot \\left\\| \\sum _ { t = 1 } ^ { T - 1 } \\Delta _ { t } \\right\\| + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| } \\\\ & { \\qquad \\leq O ( \\sqrt { d T ^ { \\sum } \\log ( T / p ) } ) + \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\ . } \\end{array}", + "type": "interline_equation", + "image_path": "9307e5d36c5cb0f6c1fa33fe3f6999cf01db532521918c99303003456b056e31.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 129, + 213, + 482, + 226.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 129, + 226.66666666666666, + 482, + 240.33333333333331 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 129, + 240.33333333333331, + 482, + 253.99999999999997 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 218, + 270 + ], + "score": 1.0, + "content": "where the last inequality uses", + "type": "text" + }, + { + "bbox": [ + 219, + 256, + 376, + 268 + ], + "score": 0.87, + "content": "\\| \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { T - 2 } \\| \\le O ( \\sqrt { d T \\log ( 1 / p ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 256, + 469, + 270 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 470, + 257, + 501, + 268 + ], + "score": 0.91, + "content": "1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 256, + 505, + 270 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 205, + 281 + ], + "score": 1.0, + "content": "Furthermore, we note that", + "type": "text" + }, + { + "bbox": [ + 205, + 268, + 214, + 277 + ], + "score": 0.87, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 264, + 282, + 281 + ], + "score": 1.0, + "content": "is independent of", + "type": "text" + }, + { + "bbox": [ + 282, + 268, + 376, + 277 + ], + "score": 0.76, + "content": "\\xi _ { 0 } , \\ldots , \\xi _ { t - 1 } , \\Delta _ { 1 } , \\ldots , \\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 264, + 393, + 281 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 267, + 433, + 278 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\xi _ { t } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 264, + 506, + 281 + ], + "score": 1.0, + "content": ". Therefore, letting", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 107, + 277, + 204, + 288 + ], + "score": 0.9, + "content": "S _ { t } = \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 275, + 239, + 290 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 239, + 277, + 325, + 288 + ], + "score": 0.91, + "content": "\\mathbb { E } [ S _ { t } | \\xi _ { 0 } , \\dots , \\xi _ { t - 1 } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 275, + 463, + 290 + ], + "score": 1.0, + "content": "; furthermore, with probability at least", + "type": "text" + }, + { + "bbox": [ + 464, + 277, + 494, + 288 + ], + "score": 0.91, + "content": "1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 275, + 506, + 290 + ], + "score": 1.0, + "content": ", it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 288, + 425, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 137, + 300 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 138, + 288, + 235, + 300 + ], + "score": 0.9, + "content": "| S _ { t } | \\leq O ( \\sqrt { d \\mathfrak { T } \\log ( T / p ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 288, + 271, + 300 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 271, + 290, + 276, + 298 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 288, + 425, + 300 + ], + "score": 1.0, + "content": ". Finally, by Azuma’s inequality, we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 256, + 506, + 300 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 302, + 408, + 322 + ], + "lines": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "spans": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\left| \\sum _ { t = 1 } ^ { T - 2 } \\langle \\xi _ { t } , \\Delta _ { 1 } + \\cdot \\cdot \\cdot + \\Delta _ { t } \\rangle \\right| \\leq O ( \\sqrt { d T \\mathfrak { T } \\log ( T / p ) } ) } \\end{array} .", + "type": "interline_equation", + "image_path": "db940c92d791db75fe64be42e15a5df5fd004067d7b79dd9f36aa34ed1facaa6.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 203, + 302, + 408, + 322 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 330, + 229, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 230, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 230, + 343 + ], + "score": 1.0, + "content": "A.4 PROOF OF LEMMA 3.5", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 348, + 482, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 482, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 247, + 365 + ], + "score": 1.0, + "content": "Proof of Lemma 3.5. Let us denote by", + "type": "text" + }, + { + "bbox": [ + 247, + 348, + 321, + 363 + ], + "score": 0.94, + "content": "\\begin{array} { r } { r _ { t } = \\frac { [ \\xi _ { t } ^ { \\mathsf { a } } ] _ { 1 } } { 2 } = - \\frac { [ \\xi _ { t } ^ { \\mathsf { b } } ] _ { 1 } } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 347, + 372, + 365 + ], + "score": 1.0, + "content": "and we know", + "type": "text" + }, + { + "bbox": [ + 373, + 349, + 428, + 363 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { t } \\sim \\mathcal { N } ( 0 , \\frac { \\nu ^ { 2 } } { 4 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 347, + 482, + 365 + ], + "score": 1.0, + "content": ". We can write", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 347, + 482, + 365 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 365, + 445, + 379 + ], + "lines": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "spans": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "score": 0.82, + "content": "w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } - \\eta ( \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } )", + "type": "interline_equation", + "image_path": "a1f34a11af70377a67f341f5253d4a55947b8704dad796c1881cf38f5c3e6f83.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 165, + 365, + 445, + 379 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 271, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 271, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 271, + 392 + ], + "score": 1.0, + "content": "Using the second-order smoothness, we have", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 379, + 271, + 392 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 392, + 430, + 432 + ], + "lines": [ + { + "bbox": [ + 181, + 392, + 430, + 432 + ], + "spans": [ + { + "bbox": [ + 181, + 392, + 430, + 432 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\nabla f ( w _ { t } ^ { \\mathsf { a } } ) - \\nabla f ( w _ { t } ^ { \\mathsf { b } } ) = \\displaystyle \\int _ { \\tau = 0 } ^ { 1 } \\nabla ^ { 2 } f \\big ( { w } _ { t } ^ { \\mathsf { a } } + \\tau ( w _ { t } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { a } } ) \\big ) \\big ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\big ) d \\tau } \\\\ & { \\qquad = \\nabla ^ { 2 } f \\big ( { w } _ { 0 } \\big ) \\cdot \\big ( { w } _ { t } ^ { \\mathsf { a } } - { w } _ { t } ^ { \\mathsf { b } } \\big ) + \\theta _ { t } } \\end{array}", + "type": "interline_equation", + "image_path": "e6b952ac6a13dc351897ac31a436c7f5dffe8bfb39ca83bc1cb258ae0560d2b9.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 181, + 392, + 430, + 405.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 181, + 405.3333333333333, + 430, + 418.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 181, + 418.66666666666663, + 430, + 431.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 434, + 434, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 434, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 164, + 447 + ], + "score": 1.0, + "content": "for some vector", + "type": "text" + }, + { + "bbox": [ + 164, + 434, + 358, + 446 + ], + "score": 0.87, + "content": "\\| \\theta _ { t } \\| \\leq \\operatorname* { m a x } \\{ \\| w _ { 0 } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { a } } \\| , \\| w _ { 0 } ^ { \\mathsf { b } } - w _ { t } ^ { \\mathsf { b } } \\| \\} \\cdot \\| w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 433, + 434, + 447 + ], + "score": 1.0, + "content": ". Therefore, we have", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 433, + 434, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 448, + 441, + 462 + ], + "lines": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "spans": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "score": 0.78, + "content": "w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ( w _ { t } ^ { \\mathsf { a } } - w _ { t } ^ { \\mathsf { b } } ) - \\eta ( \\Xi _ { t } ^ { \\mathsf { a } } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } )", + "type": "interline_equation", + "image_path": "30ff4dbae63bb63312ce28242825b7f508e71385711c567d855da5d5d81e5a41.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 168, + 448, + 441, + 462 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 291, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 293, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 152, + 477 + ], + "score": 1.0, + "content": "Now, giving", + "type": "text" + }, + { + "bbox": [ + 153, + 464, + 204, + 476 + ], + "score": 0.92, + "content": "\\psi _ { 0 } = \\overline { { \\psi } } _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 464, + 293, + 477 + ], + "score": 1.0, + "content": ", imagine two sequences", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 464, + 293, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 112, + 478, + 396, + 507 + ], + "lines": [ + { + "bbox": [ + 111, + 477, + 276, + 491 + ], + "spans": [ + { + "bbox": [ + 111, + 477, + 119, + 491 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 119, + 477, + 259, + 491 + ], + "score": 0.85, + "content": "\\psi _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) \\psi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 477, + 276, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 110, + 491, + 396, + 509 + ], + "spans": [ + { + "bbox": [ + 110, + 491, + 119, + 509 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 120, + 492, + 396, + 506 + ], + "score": 0.83, + "content": "\\overline { { \\psi } } _ { t + 1 } = \\eta r _ { t } \\mathbf { e } _ { 1 } + \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) \\overline { { \\psi } } _ { t } - \\eta \\big ( \\Xi _ { t } ^ { 3 } - \\Xi _ { t } ^ { \\mathsf { b } } + \\theta _ { t } \\big ) = w _ { t + 1 } ^ { \\mathsf { a } } - w _ { t + 1 } ^ { \\mathsf { b } }", + "type": "inline_equation", + "image_path": "b8c6e48c046f72e0f446e72874995ffe01d9b92ea9140f374103b2e557e246ae.jpg" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 110, + 477, + 396, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 204, + 523 + ], + "score": 1.0, + "content": "We will inductively prove", + "type": "text" + }, + { + "bbox": [ + 204, + 509, + 284, + 522 + ], + "score": 0.93, + "content": "\\lVert \\psi _ { t } - \\overline { { \\psi } } _ { t } \\rVert \\leq \\frac 1 2 \\lVert \\psi _ { t } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 509, + 416, + 523 + ], + "score": 1.0, + "content": ". On one hand, it is easy to see that", + "type": "text" + }, + { + "bbox": [ + 417, + 511, + 427, + 520 + ], + "score": 0.88, + "content": "\\psi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "is zero except in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 517, + 497, + 541 + ], + "spans": [ + { + "bbox": [ + 102, + 517, + 497, + 541 + ], + "score": 1.0, + "content": "first coordinate, in which it behaves as a Gaussian with zero mean and variance Pt−1k=0(1 + ηδ)2k · η2ν24", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 111, + 532, + 333, + 554 + ], + "spans": [ + { + "bbox": [ + 111, + 532, + 333, + 554 + ], + "score": 1.0, + "content": "Θ \u0010 (1+ηδ)2tηδ · η2ν2\u0011. By Gaussian tail bounds, we know that", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 102, + 509, + 505, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 555, + 405, + 591 + ], + "lines": [ + { + "bbox": [ + 110, + 552, + 405, + 573 + ], + "spans": [ + { + "bbox": [ + 110, + 552, + 266, + 573 + ], + "score": 1.0, + "content": "• with probability at least 0.99, it satisfies", + "type": "text" + }, + { + "bbox": [ + 267, + 555, + 363, + 572 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { t } \\| \\leq O ( \\frac { \\sqrt { \\eta C _ { 1 } } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 552, + 400, + 573 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 401, + 560, + 405, + 568 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 108, + 573, + 370, + 595 + ], + "spans": [ + { + "bbox": [ + 108, + 573, + 267, + 595 + ], + "score": 1.0, + "content": "• with probability at least 0.99, it satisfies", + "type": "text" + }, + { + "bbox": [ + 267, + 575, + 370, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { T } \\| \\ge \\frac { 1 } { 1 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { T } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 108, + 552, + 405, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 594, + 504, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "In the rest of the proof, we condition on this event happens. We prove towards contradiction by assuming", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 603, + 303, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 169, + 615 + ], + "score": 0.92, + "content": "\\| w _ { t } ^ { \\mathsf { a } } - w _ { 0 } ^ { \\mathsf { a } } \\| \\leq R", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 603, + 185, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 185, + 603, + 247, + 615 + ], + "score": 0.93, + "content": "\\| w _ { t } ^ { \\mathsf { b } } - w _ { 0 } ^ { \\mathsf { b } } \\| \\leq R", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 603, + 272, + 615 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 272, + 604, + 299, + 615 + ], + "score": 0.91, + "content": "t \\in [ T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 603, + 303, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 592, + 505, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 441, + 634 + ], + "lines": [ + { + "bbox": [ + 102, + 614, + 444, + 636 + ], + "spans": [ + { + "bbox": [ + 102, + 614, + 217, + 636 + ], + "score": 1.0, + "content": "We will inductively prove that", + "type": "text" + }, + { + "bbox": [ + 217, + 617, + 336, + 634 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 614, + 444, + 636 + ], + "score": 1.0, + "content": ". We calculate the difference", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 102, + 614, + 444, + 636 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 636, + 411, + 666 + ], + "lines": [ + { + "bbox": [ + 203, + 636, + 411, + 666 + ], + "spans": [ + { + "bbox": [ + 203, + 636, + 411, + 666 + ], + "score": 0.9, + "content": "\\psi _ { t } - { \\overline { { \\psi } } } _ { t } = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\left( \\mathbf { I } - \\eta { \\boldsymbol \\nabla } ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } ( { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { a } } - { \\boldsymbol \\Xi } _ { i } ^ { \\mathrm { b } } + { \\boldsymbol \\theta } _ { i } )", + "type": "interline_equation", + "image_path": "83d83c17d711d93c9eb740d9b432eb48f9f512cd2553ebe5e21bcc0a79f92e19.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 636, + 411, + 651.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 203, + 651.0, + 411, + 666.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 669, + 430, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 430, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 138, + 683 + ], + "score": 1.0, + "content": "Let g =", + "type": "text" + }, + { + "bbox": [ + 121, + 668, + 171, + 685 + ], + "score": 0.94, + "content": "\\begin{array} { r } { g = \\frac { \\psi _ { t } - \\overline { { \\psi } } _ { t } } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 667, + 376, + 687 + ], + "score": 1.0, + "content": ", then we can inner-product the above equation by vector", + "type": "text" + }, + { + "bbox": [ + 376, + 675, + 381, + 682 + ], + "score": 0.76, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 667, + 430, + 687 + ], + "score": 1.0, + "content": ", which gives", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 667, + 430, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 686, + 341, + 717 + ], + "lines": [ + { + "bbox": [ + 109, + 686, + 341, + 717 + ], + "spans": [ + { + "bbox": [ + 109, + 686, + 341, + 717 + ], + "score": 0.73, + "content": "\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| = \\eta \\sum _ { i = 0 } ^ { t - 1 } \\Big \\langle \\Xi _ { i } ^ { \\mathrm { a } } - \\Xi _ { i } ^ { \\mathrm { b } } + \\theta _ { i } , \\big ( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\big ) ^ { t - 1 - i } g \\Big \\rangle", + "type": "interline_equation", + "image_path": "a650dcb471bce83dc72faedab74503e39191a292b0a2ab01a7b7dfb6ea9fa774.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 109, + 686, + 341, + 701.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 109, + 701.5, + 341, + 717.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 117, + 722, + 449, + 734 + ], + "lines": [ + { + "bbox": [ + 118, + 721, + 450, + 736 + ], + "spans": [ + { + "bbox": [ + 118, + 721, + 450, + 736 + ], + "score": 1.0, + "content": "6We thank an anonymous reviewer on openreview who pointed out this simpler proof to us.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 118, + 721, + 450, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 149, + 90, + 502, + 150 + ], + "lines": [ + { + "bbox": [ + 149, + 90, + 502, + 150 + ], + "spans": [ + { + "bbox": [ + 149, + 90, + 502, + 150 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\overset { \\circledast } { \\leq } \\eta \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\left( \\left. \\Xi _ { i } ^ { \\mathsf { a } } - \\Xi _ { i } ^ { \\mathsf { b } } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. + R \\cdot O ( \\frac { \\sqrt { \\eta C _ { 1 } } \\nu \\left( 1 + \\eta \\delta \\right) ^ { i } } { \\sqrt { \\delta } } ) \\cdot ( 1 + \\eta \\delta ) ^ { t - 1 - i } \\right) } \\\\ & { \\leq \\eta \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\left. \\Xi _ { i } ^ { \\mathsf { a } } - \\Xi _ { i } ^ { \\mathsf { b } } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. + O ( R \\eta T \\frac { \\sqrt { \\eta C _ { 1 } } } { \\sqrt { \\delta } } \\nu ( 1 + \\eta \\delta ) ^ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "cdf21f4c57748ce7ff780d02f57dd2438b5f85ee7d62154dde9928a0e85504ef.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 149, + 90, + 502, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 149, + 110.0, + 502, + 130.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 130.0, + 502, + 150.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 104, + 151, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 104, + 151, + 184, + 170 + ], + "score": 1.0, + "content": "where the inequality", + "type": "text" + }, + { + "bbox": [ + 184, + 156, + 192, + 165 + ], + "score": 0.84, + "content": "\\textcircled{1}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 151, + 212, + 170 + ], + "score": 1.0, + "content": "uses", + "type": "text" + }, + { + "bbox": [ + 213, + 155, + 385, + 168 + ], + "score": 0.9, + "content": "\\| \\theta _ { i } \\| \\leq R \\cdot \\| \\overline { { \\psi } } _ { i } \\| \\leq R \\cdot ( \\| \\psi _ { i } \\| + \\| \\psi _ { i } - \\overline { { \\psi } } _ { i } \\| )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 151, + 391, + 170 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 391, + 154, + 506, + 169 + ], + "score": 0.85, + "content": "\\begin{array} { r l } { \\Vert \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\Vert } & { { } \\leq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 165, + 454, + 181 + ], + "spans": [ + { + "bbox": [ + 107, + 167, + 161, + 179 + ], + "score": 0.93, + "content": "( 1 + \\eta \\delta ) ^ { t - 1 - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 165, + 314, + 181 + ], + "score": 1.0, + "content": ", and the inductive assumption. Let us call", + "type": "text" + }, + { + "bbox": [ + 314, + 168, + 400, + 180 + ], + "score": 0.91, + "content": "M = \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 165, + 454, + 181 + ], + "score": 1.0, + "content": ", and focus on", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 183, + 452, + 324 + ], + "lines": [ + { + "bbox": [ + 160, + 183, + 452, + 324 + ], + "spans": [ + { + "bbox": [ + 160, + 183, + 452, + 324 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left| \\displaystyle \\sum _ { i = 0 } ^ { \\ell - 1 } \\left. \\Xi _ { i } ^ { 3 } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. \\right| } \\\\ & { = \\left| \\displaystyle \\langle \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } , g \\rangle + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\left. \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } , M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\right. \\right| } \\\\ & { \\leq \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } \\| \\cdot \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } \\| \\cdot \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\left( \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| \\right) } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\cdot ( 1 + \\eta \\delta ) ^ { t - 1 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "09b1d9101bd28e7dee275dbbc230f22bc627008d052872758af55e104207bd2b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 160, + 183, + 452, + 230.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 160, + 230.0, + 452, + 277.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 160, + 277.0, + 452, + 324.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 174, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 175, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 175, + 336 + ], + "score": 1.0, + "content": "Together, we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 338, + 431, + 362 + ], + "lines": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "spans": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "score": 0.9, + "content": "\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq O ( \\eta \\sqrt { T C _ { 2 } } ) \\cdot ( 1 + \\eta \\delta ) ^ { t } + O ( R \\eta T \\frac { \\sqrt { \\eta C _ { 1 } } } { \\sqrt { \\delta } } \\nu ( 1 + \\eta \\delta ) ^ { t } )", + "type": "interline_equation", + "image_path": "3b7d722923f62180d19941de86c94aa7a00331c3eec18295687b3457afce6f05.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 101, + 363, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 101, + 363, + 221, + 384 + ], + "score": 1.0, + "content": "Under our assumption, we have", + "type": "text" + }, + { + "bbox": [ + 221, + 365, + 337, + 382 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| < \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 363, + 388, + 384 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 388, + 367, + 505, + 380 + ], + "score": 0.91, + "content": "\\lVert \\overline { { \\psi } } _ { T } \\rVert \\ge \\lVert \\psi _ { T } \\rVert - \\lVert \\psi _ { T } - \\overline { { \\psi } } _ { T } \\rVert \\ge", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 379, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 120, + 379, + 456, + 398 + ], + "score": 1.0, + "content": "0 ( √ην(1+ηδ)t√ ). Thus, within T iterations, we have kψtk > R and this gives a contradiction.", + "type": "text" + }, + { + "bbox": [ + 491, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 284, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 285, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 285, + 424 + ], + "score": 1.0, + "content": "B FINAL: DOUBLE SAFE GUARD", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 506, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "We now come to our final Algorithm 3 which is our perturbed SGD algorithm with two safeguards. The two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 377, + 452 + ], + "score": 1.0, + "content": "safeguard algorithm naturally divides itself into epochs, each consisting of", + "type": "text" + }, + { + "bbox": [ + 378, + 441, + 388, + 450 + ], + "score": 0.88, + "content": "T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "iterations. We will demonstrate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 396, + 462 + ], + "score": 1.0, + "content": "that within most epochs, we make good progress. Thus, consider some iterate", + "type": "text" + }, + { + "bbox": [ + 396, + 452, + 417, + 461 + ], + "score": 0.87, + "content": "x _ { m T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 450, + 457, + 462 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 457, + 450, + 501, + 461 + ], + "score": 0.92, + "content": "m < T / T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 450, + 505, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 395, + 474 + ], + "score": 1.0, + "content": "Our goal will be to argue that we make good function value progress by iterate", + "type": "text" + }, + { + "bbox": [ + 395, + 462, + 431, + 471 + ], + "score": 0.9, + "content": "x _ { ( m + 1 ) T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 459, + 506, + 474 + ], + "score": 1.0, + "content": ", and that we do not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 331, + 482 + ], + "score": 1.0, + "content": "settle into any saddle points. To slightly simplify notation, let", + "type": "text" + }, + { + "bbox": [ + 331, + 471, + 376, + 481 + ], + "score": 0.89, + "content": "w _ { 0 } = x _ { m T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 469, + 506, + 482 + ], + "score": 1.0, + "content": ", and let the sequence of iterates be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 480, + 480, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 482, + 163, + 491 + ], + "score": 0.89, + "content": "w _ { 0 } , \\dots , w _ { T _ { 1 } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 480, + 193, + 492 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + }, + { + "bbox": [ + 193, + 482, + 276, + 492 + ], + "score": 0.9, + "content": "w _ { T _ { 1 } - 1 } = x _ { ( m + 1 ) T _ { 1 } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 480, + 480, + 492 + ], + "score": 1.0, + "content": ". For completeness’ sake we rewrite this as Algorithm 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 407, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 408, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 408, + 528 + ], + "score": 1.0, + "content": "Algorithm 3 Perturbed SGD with double safe guard (for analysis purpose)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 529, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 491, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 138, + 543 + ], + "score": 1.0, + "content": "Input:", + "type": "text" + }, + { + "bbox": [ + 138, + 529, + 175, + 541 + ], + "score": 0.9, + "content": "w _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 528, + 192, + 543 + ], + "score": 1.0, + "content": ", set", + "type": "text" + }, + { + "bbox": [ + 193, + 531, + 253, + 542 + ], + "score": 0.85, + "content": "\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 528, + 274, + 543 + ], + "score": 1.0, + "content": ", rate", + "type": "text" + }, + { + "bbox": [ + 274, + 531, + 299, + 542 + ], + "score": 0.87, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 528, + 334, + 543 + ], + "score": 1.0, + "content": ", lengths", + "type": "text" + }, + { + "bbox": [ + 334, + 531, + 387, + 542 + ], + "score": 0.9, + "content": "T _ { 1 } \\ge T _ { 0 } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 528, + 430, + 543 + ], + "score": 1.0, + "content": ", threshold", + "type": "text" + }, + { + "bbox": [ + 431, + 531, + 486, + 542 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 1 } > \\mathfrak { T } _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 528, + 491, + 543 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 540, + 222, + 553 + ], + "spans": [ + { + "bbox": [ + 110, + 540, + 140, + 553 + ], + "score": 1.0, + "content": "1: for", + "type": "text" + }, + { + "bbox": [ + 140, + 542, + 167, + 551 + ], + "score": 0.86, + "content": "t \\gets 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 540, + 178, + 553 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 178, + 542, + 207, + 552 + ], + "score": 0.89, + "content": "T _ { 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 540, + 222, + 553 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 110, + 550, + 320, + 565 + ], + "spans": [ + { + "bbox": [ + 110, + 551, + 121, + 563 + ], + "score": 1.0, + "content": "2:", + "type": "text" + }, + { + "bbox": [ + 138, + 552, + 240, + 564 + ], + "score": 0.87, + "content": "l a s t \\gets \\operatorname* { m a x } \\{ t _ { 0 } \\in [ t ] \\colon t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 550, + 303, + 565 + ], + "score": 1.0, + "content": "is a multiple of", + "type": "text" + }, + { + "bbox": [ + 304, + 552, + 320, + 564 + ], + "score": 0.54, + "content": "T _ { 0 } \\}", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 110, + 563, + 231, + 575 + ], + "spans": [ + { + "bbox": [ + 110, + 563, + 121, + 574 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 137, + 563, + 175, + 575 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 175, + 564, + 216, + 574 + ], + "score": 0.87, + "content": "i \\in \\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 563, + 231, + 575 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 110, + 572, + 296, + 586 + ], + "spans": [ + { + "bbox": [ + 110, + 574, + 121, + 586 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 151, + 572, + 184, + 586 + ], + "score": 1.0, + "content": "receive", + "type": "text" + }, + { + "bbox": [ + 184, + 574, + 226, + 586 + ], + "score": 0.87, + "content": "\\nabla _ { t , i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 572, + 285, + 586 + ], + "score": 1.0, + "content": "from machine", + "type": "text" + }, + { + "bbox": [ + 286, + 575, + 290, + 584 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 572, + 296, + 586 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 110, + 586, + 349, + 604 + ], + "spans": [ + { + "bbox": [ + 110, + 587, + 122, + 601 + ], + "score": 1.0, + "content": "5:", + "type": "text" + }, + { + "bbox": [ + 152, + 587, + 237, + 603 + ], + "score": 1.0, + "content": "Ai ← Ptk=0 ∇k,i|good", + "type": "text" + }, + { + "bbox": [ + 232, + 586, + 349, + 604 + ], + "score": 1.0, + "content": "| and Bi ← Ptk=last ∇k,i|good | ;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 110, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 110, + 604, + 121, + 616 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 138, + 604, + 187, + 615 + ], + "score": 0.9, + "content": "A _ { \\sf m e d } A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 603, + 214, + 618 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 215, + 605, + 256, + 616 + ], + "score": 0.92, + "content": "i \\in { \\mathfrak { g o o d } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 603, + 334, + 618 + ], + "score": 1.0, + "content": "is any machine s.t.", + "type": "text" + }, + { + "bbox": [ + 334, + 603, + 501, + 617 + ], + "score": 0.87, + "content": "\\left| \\left\\{ j \\in \\operatorname { g o o d } _ { t } \\colon \\left\\| A _ { j } - A _ { i } \\right\\| \\leq \\mathfrak { T } _ { 1 } \\right\\} \\right| > m / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 603, + 505, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 110, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 110, + 617, + 121, + 629 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 138, + 617, + 187, + 628 + ], + "score": 0.87, + "content": "B _ { \\sf m e d } B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 615, + 215, + 630 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 215, + 618, + 256, + 629 + ], + "score": 0.9, + "content": "i \\in \\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 615, + 335, + 630 + ], + "score": 1.0, + "content": "is any machine s.t.", + "type": "text" + }, + { + "bbox": [ + 335, + 617, + 501, + 630 + ], + "score": 0.84, + "content": "\\left| \\left\\{ j \\in \\mathbf { g 0 0 d } _ { t } \\colon \\left\\| B _ { j } - B _ { i } \\right\\| \\le \\mathfrak { T } _ { 0 } \\right\\} \\right| > m / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 615, + 505, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 110, + 628, + 433, + 663 + ], + "spans": [ + { + "bbox": [ + 110, + 630, + 121, + 642 + ], + "score": 1.0, + "content": "8:", + "type": "text" + }, + { + "bbox": [ + 137, + 629, + 426, + 663 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } : \\| A _ { i } - A _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 1 } \\bigwedge \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 0 } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\left( \\xi _ { t } + \\frac { 1 } { \\vert \\mathsf { g o o d } _ { t } \\vert } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\right) ; } \\end{array}", + "type": "inline_equation", + "image_path": "5ad0e7b80cf7f269d2180df8808647b0a565b064ef05822de93f8061eb5d5f3d.jpg" + }, + { + "bbox": [ + 428, + 628, + 433, + 644 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 110, + 646, + 121, + 659 + ], + "spans": [ + { + "bbox": [ + 110, + 646, + 121, + 659 + ], + "score": 1.0, + "content": "9:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 256, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 257, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 257, + 698 + ], + "score": 1.0, + "content": "Our main result is the following theorem.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 730 + ], + "lines": [ + { + "bbox": [ + 104, + 699, + 501, + 718 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 180, + 718 + ], + "score": 1.0, + "content": "Theorem B.1. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 702, + 236, + 715 + ], + "score": 0.92, + "content": "C _ { 3 } = \\alpha ^ { 2 } + { \\textstyle { \\frac { 1 } { m } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 699, + 349, + 718 + ], + "score": 1.0, + "content": ". Suppose we pick parameters", + "type": "text" + }, + { + "bbox": [ + 349, + 703, + 397, + 714 + ], + "score": 0.72, + "content": "p , \\delta \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 699, + 401, + 718 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 401, + 700, + 447, + 716 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\eta \\le \\widetilde { O } ( \\frac { \\delta ^ { 3 } } { C _ { 3 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 699, + 451, + 718 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 452, + 702, + 501, + 714 + ], + "score": 0.76, + "content": "\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 714, + 450, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 714, + 151, + 730 + ], + "score": 0.86, + "content": "\\begin{array} { r } { T _ { 0 } = \\widetilde { \\Theta } \\left( \\frac { 1 } { \\eta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 714, + 155, + 731 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 155, + 715, + 225, + 730 + ], + "score": 0.88, + "content": "\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } \\big ( \\frac { 1 } { \\eta \\delta } \\big ) \\geq T _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 714, + 228, + 731 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 229, + 715, + 283, + 729 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 714, + 303, + 731 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 303, + 715, + 357, + 728 + ], + "score": 0.92, + "content": "{ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 714, + 433, + 731 + ], + "score": 1.0, + "content": ". 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Let us call", + "type": "text" + }, + { + "bbox": [ + 314, + 168, + 400, + 180 + ], + "score": 0.91, + "content": "M = \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 165, + 454, + 181 + ], + "score": 1.0, + "content": ", and focus on", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 151, + 506, + 181 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 183, + 452, + 324 + ], + "lines": [ + { + "bbox": [ + 160, + 183, + 452, + 324 + ], + "spans": [ + { + "bbox": [ + 160, + 183, + 452, + 324 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left| \\displaystyle \\sum _ { i = 0 } ^ { \\ell - 1 } \\left. \\Xi _ { i } ^ { 3 } , \\left( \\mathbf { I } - \\eta \\nabla ^ { 2 } f ( w _ { 0 } ) \\right) ^ { t - 1 - i } g \\right. \\right| } \\\\ & { = \\left| \\displaystyle \\langle \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } , g \\rangle + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\left. \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } , M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\right. \\right| } \\\\ & { \\leq \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { t - 1 } ^ { 3 } \\| \\cdot \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| \\Xi _ { 0 } ^ { 3 } + \\dots + \\Xi _ { i } ^ { 2 } \\| \\cdot \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\left( \\| g \\| + \\displaystyle \\sum _ { i = 0 } ^ { t - 2 } \\| M ^ { t - 1 - i } g - M ^ { t - 2 - i } g \\| \\right) } \\\\ & { \\leq O ( \\sqrt { T C _ { 2 } } \\cdot ( 1 + \\eta \\delta ) ^ { t - 1 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "09b1d9101bd28e7dee275dbbc230f22bc627008d052872758af55e104207bd2b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 160, + 183, + 452, + 230.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 160, + 230.0, + 452, + 277.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 160, + 277.0, + 452, + 324.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 174, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 175, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 175, + 336 + ], + "score": 1.0, + "content": "Together, we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 324, + 175, + 336 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 338, + 431, + 362 + ], + "lines": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "spans": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "score": 0.9, + "content": "\\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| \\leq O ( \\eta \\sqrt { T C _ { 2 } } ) \\cdot ( 1 + \\eta \\delta ) ^ { t } + O ( R \\eta T \\frac { \\sqrt { \\eta C _ { 1 } } } { \\sqrt { \\delta } } \\nu ( 1 + \\eta \\delta ) ^ { t } )", + "type": "interline_equation", + "image_path": "3b7d722923f62180d19941de86c94aa7a00331c3eec18295687b3457afce6f05.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 179, + 338, + 431, + 362 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 101, + 363, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 101, + 363, + 221, + 384 + ], + "score": 1.0, + "content": "Under our assumption, we have", + "type": "text" + }, + { + "bbox": [ + 221, + 365, + 337, + 382 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| \\psi _ { t } - \\overline { { \\psi } } _ { t } \\| < \\frac { 1 } { 2 0 0 0 } \\big ( \\frac { \\sqrt { \\eta } \\nu ( 1 + \\eta \\delta ) ^ { t } } { \\sqrt { \\delta } } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 363, + 388, + 384 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 388, + 367, + 505, + 380 + ], + "score": 0.91, + "content": "\\lVert \\overline { { \\psi } } _ { T } \\rVert \\ge \\lVert \\psi _ { T } \\rVert - \\lVert \\psi _ { T } - \\overline { { \\psi } } _ { T } \\rVert \\ge", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 120, + 379, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 120, + 379, + 456, + 398 + ], + "score": 1.0, + "content": "0 ( √ην(1+ηδ)t√ ). Thus, within T iterations, we have kψtk > R and this gives a contradiction.", + "type": "text" + }, + { + "bbox": [ + 491, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 101, + 363, + 505, + 398 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 284, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 285, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 285, + 424 + ], + "score": 1.0, + "content": "B FINAL: DOUBLE SAFE GUARD", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 506, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "We now come to our final Algorithm 3 which is our perturbed SGD algorithm with two safeguards. The two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 377, + 452 + ], + "score": 1.0, + "content": "safeguard algorithm naturally divides itself into epochs, each consisting of", + "type": "text" + }, + { + "bbox": [ + 378, + 441, + 388, + 450 + ], + "score": 0.88, + "content": "T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "iterations. We will demonstrate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 396, + 462 + ], + "score": 1.0, + "content": "that within most epochs, we make good progress. Thus, consider some iterate", + "type": "text" + }, + { + "bbox": [ + 396, + 452, + 417, + 461 + ], + "score": 0.87, + "content": "x _ { m T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 450, + 457, + 462 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 457, + 450, + 501, + 461 + ], + "score": 0.92, + "content": "m < T / T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 450, + 505, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 395, + 474 + ], + "score": 1.0, + "content": "Our goal will be to argue that we make good function value progress by iterate", + "type": "text" + }, + { + "bbox": [ + 395, + 462, + 431, + 471 + ], + "score": 0.9, + "content": "x _ { ( m + 1 ) T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 459, + 506, + 474 + ], + "score": 1.0, + "content": ", and that we do not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 331, + 482 + ], + "score": 1.0, + "content": "settle into any saddle points. To slightly simplify notation, let", + "type": "text" + }, + { + "bbox": [ + 331, + 471, + 376, + 481 + ], + "score": 0.89, + "content": "w _ { 0 } = x _ { m T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 469, + 506, + 482 + ], + "score": 1.0, + "content": ", and let the sequence of iterates be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 480, + 480, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 482, + 163, + 491 + ], + "score": 0.89, + "content": "w _ { 0 } , \\dots , w _ { T _ { 1 } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 480, + 193, + 492 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + }, + { + "bbox": [ + 193, + 482, + 276, + 492 + ], + "score": 0.9, + "content": "w _ { T _ { 1 } - 1 } = x _ { ( m + 1 ) T _ { 1 } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 480, + 480, + 492 + ], + "score": 1.0, + "content": ". For completeness’ sake we rewrite this as Algorithm 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 430, + 506, + 492 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 407, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 408, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 408, + 528 + ], + "score": 1.0, + "content": "Algorithm 3 Perturbed SGD with double safe guard (for analysis purpose)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "index", + "bbox": [ + 108, + 529, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 491, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 138, + 543 + ], + "score": 1.0, + "content": "Input:", + "type": "text" + }, + { + "bbox": [ + 138, + 529, + 175, + 541 + ], + "score": 0.9, + "content": "w _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 528, + 192, + 543 + ], + "score": 1.0, + "content": ", set", + "type": "text" + }, + { + "bbox": [ + 193, + 531, + 253, + 542 + ], + "score": 0.85, + "content": "\\mathtt { g o o d } _ { 0 } \\supseteq \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 528, + 274, + 543 + ], + "score": 1.0, + "content": ", rate", + "type": "text" + }, + { + "bbox": [ + 274, + 531, + 299, + 542 + ], + "score": 0.87, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 528, + 334, + 543 + ], + "score": 1.0, + "content": ", lengths", + "type": "text" + }, + { + "bbox": [ + 334, + 531, + 387, + 542 + ], + "score": 0.9, + "content": "T _ { 1 } \\ge T _ { 0 } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 528, + 430, + 543 + ], + "score": 1.0, + "content": ", threshold", + "type": "text" + }, + { + "bbox": [ + 431, + 531, + 486, + 542 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 1 } > \\mathfrak { T } _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 528, + 491, + 543 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 540, + 222, + 553 + ], + "spans": [ + { + "bbox": [ + 110, + 540, + 140, + 553 + ], + "score": 1.0, + "content": "1: for", + "type": "text" + }, + { + "bbox": [ + 140, + 542, + 167, + 551 + ], + "score": 0.86, + "content": "t \\gets 0", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 540, + 178, + 553 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 178, + 542, + 207, + 552 + ], + "score": 0.89, + "content": "T _ { 1 } - 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A _ { i } \\right\\| \\leq \\mathfrak { T } _ { 1 } \\right\\} \\right| > m / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 603, + 505, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 615, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 110, + 617, + 121, + 629 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 138, + 617, + 187, + 628 + ], + "score": 0.87, + "content": "B _ { \\sf m e d } B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 615, + 215, + 630 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 215, + 618, + 256, + 629 + ], + "score": 0.9, + "content": "i \\in \\mathtt { g o o d } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 615, + 335, + 630 + ], + "score": 1.0, + "content": "is any machine s.t.", + "type": "text" + }, + { + "bbox": [ + 335, + 617, + 501, + 630 + ], + "score": 0.84, + "content": "\\left| \\left\\{ j \\in \\mathbf { g 0 0 d } _ { t } \\colon \\left\\| B _ { j } - B _ { i } \\right\\| \\le \\mathfrak { T } _ { 0 } \\right\\} \\right| > m / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 615, + 505, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 628, + 433, + 663 + ], + "spans": [ + { + "bbox": [ + 110, + 630, + 121, + 642 + ], + "score": 1.0, + "content": "8:", + "type": "text" + }, + { + "bbox": [ + 137, + 629, + 426, + 663 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\mathsf { g o o d } _ { t + 1 } \\gets \\big \\{ i \\in \\mathsf { g o o d } _ { t } : \\| A _ { i } - A _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 1 } \\bigwedge \\| B _ { i } - B _ { \\mathsf { m e d } } \\| \\le 2 \\mathfrak { T } _ { 0 } \\big \\} } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta \\left( \\xi _ { t } + \\frac { 1 } { \\vert \\mathsf { g o o d } _ { t } \\vert } \\sum _ { i \\in \\mathsf { g o o d } _ { t } } \\nabla _ { t , i } \\right) ; } \\end{array}", + "type": "inline_equation", + "image_path": "5ad0e7b80cf7f269d2180df8808647b0a565b064ef05822de93f8061eb5d5f3d.jpg" + }, + { + "bbox": [ + 428, + 628, + 433, + 644 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 646, + 121, + 659 + ], + "spans": [ + { + "bbox": [ + 110, + 646, + 121, + 659 + ], + "score": 1.0, + "content": "9:", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 528, + 505, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 256, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 257, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 257, + 698 + ], + "score": 1.0, + "content": "Our main result is the following theorem.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 684, + 257, + 698 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 730 + ], + "lines": [ + { + "bbox": [ + 104, + 699, + 501, + 718 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 180, + 718 + ], + "score": 1.0, + "content": "Theorem B.1. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 702, + 236, + 715 + ], + "score": 0.92, + "content": "C _ { 3 } = \\alpha ^ { 2 } + { \\textstyle { \\frac { 1 } { m } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 699, + 349, + 718 + ], + "score": 1.0, + "content": ". Suppose we pick parameters", + "type": "text" + }, + { + "bbox": [ + 349, + 703, + 397, + 714 + ], + "score": 0.72, + "content": "p , \\delta \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 699, + 401, + 718 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 401, + 700, + 447, + 716 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\eta \\le \\widetilde { O } ( \\frac { \\delta ^ { 3 } } { C _ { 3 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 699, + 451, + 718 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 452, + 702, + 501, + 714 + ], + "score": 0.76, + "content": "\\nu ^ { 2 } = \\widetilde { \\Theta } ( C _ { 3 } )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 714, + 450, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 714, + 151, + 730 + ], + "score": 0.86, + "content": "\\begin{array} { r } { T _ { 0 } = \\widetilde { \\Theta } \\left( \\frac { 1 } { \\eta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 714, + 155, + 731 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 155, + 715, + 225, + 730 + ], + "score": 0.88, + "content": "\\begin{array} { r } { T _ { 1 } = \\widetilde { \\Theta } \\big ( \\frac { 1 } { \\eta \\delta } \\big ) \\geq T _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 714, + 228, + 731 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 229, + 715, + 283, + 729 + ], + "score": 0.91, + "content": "\\mathfrak { T } _ { 1 } = \\widetilde { \\Theta } ( \\sqrt { T _ { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 714, + 303, + 731 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 303, + 715, + 357, + 728 + ], + "score": 0.92, + "content": "{ \\mathfrak { T } } _ { 0 } = { \\widetilde { \\Theta } } ( { \\sqrt { T _ { 0 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 714, + 433, + 731 + ], + "score": 1.0, + "content": ". 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f ( w _ { T _ { 1 } } ) \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - \\widetilde { O } ( \\eta C _ { 3 } d ) \\right) \\ .", + "type": "interline_equation", + "image_path": "f5ce478307ac4462e58ec1c9150577d414608ca2b228c375ee251c03bc716873.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 96, + 425, + 111.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 200, + 111.5, + 425, + 127.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 504, + 156 + ], + "lines": [ + { + "bbox": [ + 103, + 132, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 103, + 132, + 161, + 148 + ], + "score": 1.0, + "content": "(b) As long as", + "type": "text" + }, + { + "bbox": [ + 161, + 135, + 223, + 146 + ], + "score": 0.89, + "content": "\\| w _ { t } - w _ { 0 } \\| \\ge R", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 132, + 258, + 148 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 258, + 134, + 326, + 146 + ], + "score": 0.92, + "content": "t \\in \\{ 1 , 2 , \\dots , T _ { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 132, + 356, + 148 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 357, + 133, + 414, + 147 + ], + "score": 0.94, + "content": "\\begin{array} { r } { R = \\widetilde { \\Theta } ( \\delta ) \\leq \\frac { \\delta } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 132, + 506, + 148 + ], + "score": 1.0, + "content": ", then with probability at", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 120, + 145, + 213, + 156 + ], + "spans": [ + { + "bbox": [ + 120, + 145, + 140, + 156 + ], + "score": 1.0, + "content": "least", + "type": "text" + }, + { + "bbox": [ + 140, + 146, + 162, + 155 + ], + "score": 0.87, + "content": "1 - 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\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 194, + 373, + 208 + ], + "score": 1.0, + "content": ", we also have with probability at least 0.45,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 208, + 413, + 240 + ], + "lines": [ + { + "bbox": [ + 212, + 208, + 413, + 240 + ], + "spans": [ + { + "bbox": [ + 212, + 208, + 413, + 240 + ], + "score": 0.93, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\Big ( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\Big ) + \\widetilde { \\Omega } ( \\delta ^ { 3 } )", + "type": "interline_equation", + "image_path": "5c4a89bd641d5f843ca361f5283db612d80b5f7444d48b02dcfbd3d86ffd0e3f.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 208, + 413, + 224.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 212, + 224.0, + 413, + 240.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 247, + 320, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 320, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 320, + 260 + ], + "score": 1.0, + "content": "B.1 WHY THEOREM B.1 IMPLIES THEOREM 2.3", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 265, + 504, + 290 + ], + "lines": [ + { + "bbox": [ + 104, + 263, + 507, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 210, + 283 + ], + "score": 1.0, + "content": "Using the parameter choice", + "type": "text" + }, + { + "bbox": [ + 211, + 265, + 262, + 281 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\eta = \\widetilde { \\Theta } ( \\frac { \\varepsilon ^ { 2 } } { C _ { 3 } d } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 263, + 373, + 283 + ], + "score": 1.0, + "content": "from Theorem 2.3, we know", + "type": "text" + }, + { + "bbox": [ + 373, + 266, + 444, + 279 + ], + "score": 0.92, + "content": "\\widetilde { \\cal O } ( \\eta C _ { 3 } d ) \\le 0 . 1 \\varepsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 263, + 507, + 283 + ], + "score": 1.0, + "content": ". We claim two", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 135, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 135, + 293 + ], + "score": 1.0, + "content": "things:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 111, + 292, + 503, + 313 + ], + "lines": [ + { + "bbox": [ + 110, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 110, + 291, + 162, + 305 + ], + "score": 1.0, + "content": "• For at least", + "type": "text" + }, + { + "bbox": [ + 162, + 293, + 180, + 302 + ], + "score": 0.88, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 291, + 343, + 305 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy (denoting by", + "type": "text" + }, + { + "bbox": [ + 343, + 294, + 354, + 303 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 291, + 371, + 305 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 294, + 387, + 303 + ], + "score": 0.88, + "content": "w _ { T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "the beginning and ending points", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 119, + 301, + 173, + 315 + ], + "spans": [ + { + "bbox": [ + 119, + 301, + 173, + 315 + ], + "score": 1.0, + "content": "of this epoch)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 313, + 404, + 337 + ], + "lines": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "spans": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "score": 0.93, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\leq 2 0 \\frac { f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } \\leq \\varepsilon ^ { 1 . 5 }", + "type": "interline_equation", + "image_path": "552df0d7326fc1432351a7449bcc1c89b32ce7dcbd3a5278a619a999fedd3633.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 121, + 340, + 328, + 351 + ], + "lines": [ + { + "bbox": [ + 119, + 337, + 329, + 353 + ], + "spans": [ + { + "bbox": [ + 119, + 337, + 257, + 353 + ], + "score": 1.0, + "content": "The last inequality uses our choice of", + "type": "text" + }, + { + "bbox": [ + 257, + 340, + 265, + 349 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 337, + 281, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 338, + 325, + 351 + ], + "score": 0.93, + "content": "\\delta = \\widetilde { \\Theta } ( \\sqrt { \\varepsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 337, + 329, + 353 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 120, + 352, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 119, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 119, + 351, + 388, + 363 + ], + "score": 1.0, + "content": "The reason for this is by way of contradiction. Suppose for at least", + "type": "text" + }, + { + "bbox": [ + 389, + 352, + 407, + 362 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "of the epochs it satisfies", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 121, + 358, + 507, + 378 + ], + "spans": [ + { + "bbox": [ + 121, + 362, + 264, + 376 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) \\ : - \\ : f ( w _ { T _ { 1 } } ) > 2 0 \\frac { \\dot { f } ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 358, + 507, + 378 + ], + "score": 1.0, + "content": ", then, for the remainder of the epochs, they must at least satisfy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 121, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 121, + 377, + 260, + 388 + ], + "score": 0.82, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq - 0 . 7 \\eta T _ { 1 } \\cdot 0 . 1 \\varepsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 376, + 434, + 389 + ], + "score": 1.0, + "content": ". Summing over all the epochs, we shall obtain", + "type": "text" + }, + { + "bbox": [ + 435, + 376, + 505, + 388 + ], + "score": 0.91, + "content": "f ( x _ { 0 } ) - f ( x _ { T } ) >", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 121, + 386, + 299, + 399 + ], + "spans": [ + { + "bbox": [ + 121, + 388, + 190, + 398 + ], + "score": 0.86, + "content": "f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 386, + 299, + 399 + ], + "score": 1.0, + "content": "but this gives a contradiction.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 113, + 398, + 439, + 410 + ], + "lines": [ + { + "bbox": [ + 110, + 397, + 441, + 411 + ], + "spans": [ + { + "bbox": [ + 110, + 397, + 162, + 411 + ], + "score": 1.0, + "content": "• For at least", + "type": "text" + }, + { + "bbox": [ + 163, + 398, + 181, + 408 + ], + "score": 0.87, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 397, + 441, + 411 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy the three properties from Theorem B.1.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 105, + 411, + 444, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 446, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 195, + 425 + ], + "score": 1.0, + "content": "In particular, for at least", + "type": "text" + }, + { + "bbox": [ + 195, + 412, + 213, + 422 + ], + "score": 0.88, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 410, + 372, + 425 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy both. Since", + "type": "text" + }, + { + "bbox": [ + 372, + 411, + 388, + 421 + ], + "score": 0.88, + "content": "\\varepsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 410, + 446, + 425 + ], + "score": 1.0, + "content": "is so small that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 425, + 462, + 456 + ], + "lines": [ + { + "bbox": [ + 148, + 425, + 462, + 456 + ], + "spans": [ + { + "bbox": [ + 148, + 425, + 462, + 456 + ], + "score": 0.92, + "content": "\\varepsilon ^ { 1 . 5 } \\ge f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\right) + \\widetilde { \\Omega } ( \\delta ^ { 3 } ) \\ge \\widetilde { \\Omega } ( \\delta ^ { 3 } ) - 0 . 0 5 \\eta T _ { 1 } \\varepsilon ^ { 2 }", + "type": "interline_equation", + "image_path": "4582c309c5cb7a8c589c4f9111e26a8cd1c1496ae52ed095ec5a0cbce50b38b9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 148, + 425, + 462, + 435.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 148, + 435.3333333333333, + 462, + 445.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 148, + 445.66666666666663, + 462, + 455.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 313, + 471 + ], + "score": 1.0, + "content": "would give a contradiction (for instance, one can choose", + "type": "text" + }, + { + "bbox": [ + 313, + 459, + 319, + 468 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 458, + 411, + 471 + ], + "score": 1.0, + "content": "to be slightly larger than", + "type": "text" + }, + { + "bbox": [ + 412, + 459, + 425, + 470 + ], + "score": 0.89, + "content": "\\sqrt { \\varepsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "by some log factors),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 183, + 480 + ], + "score": 1.0, + "content": "this means, for those", + "type": "text" + }, + { + "bbox": [ + 183, + 469, + 201, + 479 + ], + "score": 0.88, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 468, + 320, + 480 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 112, + 481, + 296, + 522 + ], + "lines": [ + { + "bbox": [ + 109, + 477, + 292, + 500 + ], + "spans": [ + { + "bbox": [ + 109, + 477, + 119, + 500 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 119, + 481, + 287, + 496 + ], + "score": 0.37, + "content": "\\begin{array} { r } { \\varepsilon ^ { 1 . 5 } \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - 0 . 1 \\varepsilon ^ { 2 } \\right) \\ , } \\end{array}", + "type": "inline_equation", + "image_path": "91dae17177064d90d91028c9fc6c48f9fc34604af51a16bc4b4113da1c461e8e.jpg" + }, + { + "bbox": [ + 287, + 477, + 292, + 500 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 111, + 496, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 111, + 496, + 120, + 509 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 496, + 181, + 509 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| w _ { t } - w _ { 0 } \\| \\leq \\frac { \\delta } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 496, + 217, + 509 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 217, + 498, + 277, + 508 + ], + "score": 0.9, + "content": "t = 1 , 2 , \\dots , T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 496, + 297, + 509 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 112, + 508, + 189, + 523 + ], + "spans": [ + { + "bbox": [ + 112, + 508, + 120, + 523 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 510, + 186, + 522 + ], + "score": 0.78, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } ) \\succeq - \\delta \\mathbf { I } .", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 508, + 189, + 523 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 260, + 538 + ], + "score": 1.0, + "content": "The latter two properties together implies", + "type": "text" + }, + { + "bbox": [ + 261, + 524, + 327, + 536 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } f ( w _ { t } ) \\succeq - \\frac { \\delta } { 2 } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 524, + 364, + 538 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 364, + 526, + 426, + 536 + ], + "score": 0.92, + "content": "t = 1 , 2 , \\ldots , T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "(by the second-order", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 299, + 547 + ], + "score": 1.0, + "content": "smoothness). The first property implies for at least", + "type": "text" + }, + { + "bbox": [ + 299, + 536, + 317, + 546 + ], + "score": 0.85, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 535, + 380, + 547 + ], + "score": 1.0, + "content": "of the iterations", + "type": "text" + }, + { + "bbox": [ + 380, + 537, + 385, + 545 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "in this epoch, they must satisfy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 548, + 501, + 560 + ], + "spans": [ + { + "bbox": [ + 107, + 548, + 159, + 560 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 548, + 306, + 559 + ], + "score": 1.0, + "content": ". This finishes the proof of Theorem 2.3.", + "type": "text" + }, + { + "bbox": [ + 495, + 549, + 501, + 552 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 568, + 240, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 240, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 240, + 581 + ], + "score": 1.0, + "content": "B.2 PROOF OF THEOREM B.1", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 232, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 233, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 233, + 596 + ], + "score": 1.0, + "content": "We first have the following lemma", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 343, + 613 + ], + "score": 1.0, + "content": "Lemma B.2 (double safe guard). In Algorithm 3, suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 599, + 456, + 612 + ], + "score": 0.92, + "content": "\\mathfrak { T } _ { 1 } ~ = ~ 8 \\sqrt { T _ { 1 } \\log ( 1 6 m T _ { 1 } / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 599, + 478, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 600, + 505, + 611 + ], + "score": 0.84, + "content": "\\mathfrak { T } _ { 0 } ~ =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 611, + 441, + 625 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 187, + 624 + ], + "score": 0.9, + "content": "8 \\sqrt { T _ { 0 } \\log ( 1 6 m T _ { 1 } / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 612, + 302, + 625 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 302, + 612, + 333, + 623 + ], + "score": 0.93, + "content": "1 - p / 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 612, + 370, + 625 + ], + "score": 1.0, + "content": ", for every", + "type": "text" + }, + { + "bbox": [ + 370, + 613, + 437, + 623 + ], + "score": 0.9, + "content": "t = 0 , \\ldots , T _ { 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 612, + 441, + 625 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 111, + 626, + 498, + 705 + ], + "lines": [ + { + "bbox": [ + 111, + 624, + 179, + 638 + ], + "spans": [ + { + "bbox": [ + 111, + 624, + 119, + 638 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 120, + 626, + 176, + 637 + ], + "score": 0.59, + "content": "\\mathbf { g } _ { 0 0 } \\mathbf { d } _ { t } \\supseteq \\mathbf { g } _ { 0 0 } \\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 624, + 179, + 638 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 110, + 636, + 496, + 705 + ], + "spans": [ + { + "bbox": [ + 110, + 636, + 120, + 655 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 110, + 654, + 121, + 672 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 124, + 637, + 496, + 705 + ], + "score": 0.32, + "content": "\\begin{array} { r l } & { | \\sigma _ { t } | | ^ { 2 } \\leq O ( \\frac { \\log ( T _ { 1 } / p ) } { m } ) , \\| \\Delta _ { t } \\| ^ { 2 } \\leq \\alpha ^ { 2 } , \\| \\xi _ { t } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T _ { 1 } / p ) ) , } \\\\ & { | \\sigma _ { 0 } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 1 } \\log ( T _ { 1 } / p ) } { m } ) , \\| \\sigma _ { t a s t } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 0 } \\log ( T _ { 1 } / p ) } { m } ) } \\\\ & { | \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 1 } \\log ( m T _ { 1 } / p ) ) a n d \\| \\Delta _ { l a s t } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 0 } \\log ( m T _ { 1 } / p ) ) } \\\\ & { | \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 1 } \\log ( T _ { 1 } / p ) ) a n d \\| \\xi _ { l a s t } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 0 } \\log ( T _ { 1 } / p ) ) . } \\\\ & { \\quad \\cdot \\quad \\cdot \\quad \\cdot \\quad \\cdot } \\end{array}", + "type": "inline_equation", + "image_path": "71fefc8670a3238f94647c230dab755fe971d3971c7fd148e056309d0e3cf889.jpg" + } + ], + "index": 40 + }, + { + "bbox": [ + 110, + 670, + 119, + 686 + ], + "spans": [ + { + "bbox": [ + 110, + 670, + 119, + 686 + ], + "score": 1.0, + "content": "•", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 110, + 685, + 120, + 699 + ], + "spans": [ + { + "bbox": [ + 110, + 685, + 120, + 699 + ], + "score": 1.0, + "content": "•", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 417, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 418, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 241, + 717 + ], + "score": 1.0, + "content": "We call this probabilistic event Event", + "type": "text" + }, + { + "bbox": [ + 241, + 701, + 279, + 713 + ], + "score": 0.72, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 698, + 296, + 717 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 701, + 414, + 713 + ], + "score": 0.71, + "content": "\\operatorname* { P r } [ \\mathsf { E v e n t } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ) ] \\geq 1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 698, + 418, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 502, + 733 + ], + "lines": [ + { + "bbox": [ + 103, + 713, + 501, + 738 + ], + "spans": [ + { + "bbox": [ + 103, + 713, + 403, + 738 + ], + "score": 1.0, + "content": "The proof is a direct corollary of Lemma 3.2, by combining events Even", + "type": "text" + }, + { + "bbox": [ + 404, + 720, + 438, + 734 + ], + "score": 0.83, + "content": "\\mathbf { \\cdot } _ { T _ { 1 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 713, + 467, + 738 + ], + "score": 1.0, + "content": ", Event", + "type": "text" + }, + { + "bbox": [ + 467, + 720, + 501, + 734 + ], + "score": 0.65, + "content": "\\mathbf { \\cdot } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + } + ], + "index": 43 + } + ], + "index": 43 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 83, + 263, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 82, + 263, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 208, + 96 + ], + "score": 1.0, + "content": "(a) with probability at least", + "type": "text" + }, + { + "bbox": [ + 209, + 84, + 230, + 94 + ], + "score": 0.83, + "content": "1 - 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We claim two", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 135, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 135, + 293 + ], + "score": 1.0, + "content": "things:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 263, + 507, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 292, + 503, + 313 + ], + "lines": [ + { + "bbox": [ + 110, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 110, + 291, + 162, + 305 + ], + "score": 1.0, + "content": "• For at least", + "type": "text" + }, + { + "bbox": [ + 162, + 293, + 180, + 302 + ], + "score": 0.88, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 291, + 343, + 305 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy (denoting by", + "type": "text" + }, + { + "bbox": [ + 343, + 294, + 354, + 303 + ], + "score": 0.84, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 291, + 371, + 305 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 371, + 294, + 387, + 303 + ], + "score": 0.88, + "content": "w _ { T _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "the beginning and ending points", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 119, + 301, + 173, + 315 + ], + "spans": [ + { + "bbox": [ + 119, + 301, + 173, + 315 + ], + "score": 1.0, + "content": "of this epoch)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 110, + 291, + 505, + 315 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 313, + 404, + 337 + ], + "lines": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "spans": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "score": 0.93, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\leq 2 0 \\frac { f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } \\leq \\varepsilon ^ { 1 . 5 }", + "type": "interline_equation", + "image_path": "552df0d7326fc1432351a7449bcc1c89b32ce7dcbd3a5278a619a999fedd3633.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 221, + 313, + 404, + 337 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 121, + 340, + 328, + 351 + ], + "lines": [ + { + "bbox": [ + 119, + 337, + 329, + 353 + ], + "spans": [ + { + "bbox": [ + 119, + 337, + 257, + 353 + ], + "score": 1.0, + "content": "The last inequality uses our choice of", + "type": "text" + }, + { + "bbox": [ + 257, + 340, + 265, + 349 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 337, + 281, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 338, + 325, + 351 + ], + "score": 0.93, + "content": "\\delta = \\widetilde { \\Theta } ( \\sqrt { \\varepsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 337, + 329, + 353 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 119, + 337, + 329, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 120, + 352, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 119, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 119, + 351, + 388, + 363 + ], + "score": 1.0, + "content": "The reason for this is by way of contradiction. Suppose for at least", + "type": "text" + }, + { + "bbox": [ + 389, + 352, + 407, + 362 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "of the epochs it satisfies", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 121, + 358, + 507, + 378 + ], + "spans": [ + { + "bbox": [ + 121, + 362, + 264, + 376 + ], + "score": 0.88, + "content": "\\begin{array} { r } { f ( w _ { 0 } ) \\ : - \\ : f ( w _ { T _ { 1 } } ) > 2 0 \\frac { \\dot { f } ( x _ { 0 } ) - \\operatorname* { m i n } f ( x ) } { T / T _ { 1 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 358, + 507, + 378 + ], + "score": 1.0, + "content": ", then, for the remainder of the epochs, they must at least satisfy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 121, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 121, + 377, + 260, + 388 + ], + "score": 0.82, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\geq - 0 . 7 \\eta T _ { 1 } \\cdot 0 . 1 \\varepsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 376, + 434, + 389 + ], + "score": 1.0, + "content": ". Summing over all the epochs, we shall obtain", + "type": "text" + }, + { + "bbox": [ + 435, + 376, + 505, + 388 + ], + "score": 0.91, + "content": "f ( x _ { 0 } ) - f ( x _ { T } ) >", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 121, + 386, + 299, + 399 + ], + "spans": [ + { + "bbox": [ + 121, + 388, + 190, + 398 + ], + "score": 0.86, + "content": "f ( x _ { 0 } ) - \\operatorname* { m i n } f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 386, + 299, + 399 + ], + "score": 1.0, + "content": "but this gives a contradiction.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 119, + 351, + 507, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 113, + 398, + 439, + 410 + ], + "lines": [ + { + "bbox": [ + 110, + 397, + 441, + 411 + ], + "spans": [ + { + "bbox": [ + 110, + 397, + 162, + 411 + ], + "score": 1.0, + "content": "• For at least", + "type": "text" + }, + { + "bbox": [ + 163, + 398, + 181, + 408 + ], + "score": 0.87, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 397, + 441, + 411 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy the three properties from Theorem B.1.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 110, + 397, + 441, + 411 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 411, + 444, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 446, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 195, + 425 + ], + "score": 1.0, + "content": "In particular, for at least", + "type": "text" + }, + { + "bbox": [ + 195, + 412, + 213, + 422 + ], + "score": 0.88, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 410, + 372, + 425 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy both. Since", + "type": "text" + }, + { + "bbox": [ + 372, + 411, + 388, + 421 + ], + "score": 0.88, + "content": "\\varepsilon ^ { 1 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 410, + 446, + 425 + ], + "score": 1.0, + "content": "is so small that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 410, + 446, + 425 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 425, + 462, + 456 + ], + "lines": [ + { + "bbox": [ + 148, + 425, + 462, + 456 + ], + "spans": [ + { + "bbox": [ + 148, + 425, + 462, + 456 + ], + "score": 0.92, + "content": "\\varepsilon ^ { 1 . 5 } \\ge f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( - \\widetilde { O } ( \\eta C _ { 3 } d ) \\right) + \\widetilde { \\Omega } ( \\delta ^ { 3 } ) \\ge \\widetilde { \\Omega } ( \\delta ^ { 3 } ) - 0 . 0 5 \\eta T _ { 1 } \\varepsilon ^ { 2 }", + "type": "interline_equation", + "image_path": "4582c309c5cb7a8c589c4f9111e26a8cd1c1496ae52ed095ec5a0cbce50b38b9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 148, + 425, + 462, + 435.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 148, + 435.3333333333333, + 462, + 445.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 148, + 445.66666666666663, + 462, + 455.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 313, + 471 + ], + "score": 1.0, + "content": "would give a contradiction (for instance, one can choose", + "type": "text" + }, + { + "bbox": [ + 313, + 459, + 319, + 468 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 458, + 411, + 471 + ], + "score": 1.0, + "content": "to be slightly larger than", + "type": "text" + }, + { + "bbox": [ + 412, + 459, + 425, + 470 + ], + "score": 0.89, + "content": "\\sqrt { \\varepsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "by some log factors),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 183, + 480 + ], + "score": 1.0, + "content": "this means, for those", + "type": "text" + }, + { + "bbox": [ + 183, + 469, + 201, + 479 + ], + "score": 0.88, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 468, + 320, + 480 + ], + "score": 1.0, + "content": "of the epochs, they must satisfy:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 458, + 505, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 112, + 481, + 296, + 522 + ], + "lines": [ + { + "bbox": [ + 109, + 477, + 292, + 500 + ], + "spans": [ + { + "bbox": [ + 109, + 477, + 119, + 500 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 119, + 481, + 287, + 496 + ], + "score": 0.37, + "content": "\\begin{array} { r } { \\varepsilon ^ { 1 . 5 } \\geq 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\Vert \\nabla f ( w _ { t } ) \\Vert ^ { 2 } - 0 . 1 \\varepsilon ^ { 2 } \\right) \\ , } \\end{array}", + "type": "inline_equation", + "image_path": "91dae17177064d90d91028c9fc6c48f9fc34604af51a16bc4b4113da1c461e8e.jpg" + }, + { + "bbox": [ + 287, + 477, + 292, + 500 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 111, + 496, + 297, + 509 + ], + "spans": [ + { + "bbox": [ + 111, + 496, + 120, + 509 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 496, + 181, + 509 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| w _ { t } - w _ { 0 } \\| \\leq \\frac { \\delta } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 496, + 217, + 509 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 217, + 498, + 277, + 508 + ], + "score": 0.9, + "content": "t = 1 , 2 , \\dots , T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 496, + 297, + 509 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 112, + 508, + 189, + 523 + ], + "spans": [ + { + "bbox": [ + 112, + 508, + 120, + 523 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 121, + 510, + 186, + 522 + ], + "score": 0.78, + "content": "\\nabla ^ { 2 } f ( w _ { 0 } ) \\succeq - \\delta \\mathbf { I } .", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 508, + 189, + 523 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 109, + 477, + 297, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 260, + 538 + ], + "score": 1.0, + "content": "The latter two properties together implies", + "type": "text" + }, + { + "bbox": [ + 261, + 524, + 327, + 536 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } f ( w _ { t } ) \\succeq - \\frac { \\delta } { 2 } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 524, + 364, + 538 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 364, + 526, + 426, + 536 + ], + "score": 0.92, + "content": "t = 1 , 2 , \\ldots , T _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "(by the second-order", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 299, + 547 + ], + "score": 1.0, + "content": "smoothness). The first property implies for at least", + "type": "text" + }, + { + "bbox": [ + 299, + 536, + 317, + 546 + ], + "score": 0.85, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 535, + 380, + 547 + ], + "score": 1.0, + "content": "of the iterations", + "type": "text" + }, + { + "bbox": [ + 380, + 537, + 385, + 545 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "in this epoch, they must satisfy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 548, + 501, + 560 + ], + "spans": [ + { + "bbox": [ + 107, + 548, + 159, + 560 + ], + "score": 0.91, + "content": "\\| \\nabla f ( x ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 548, + 306, + 559 + ], + "score": 1.0, + "content": ". This finishes the proof of Theorem 2.3.", + "type": "text" + }, + { + "bbox": [ + 495, + 549, + 501, + 552 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 524, + 506, + 560 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 568, + 240, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 240, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 240, + 581 + ], + "score": 1.0, + "content": "B.2 PROOF OF THEOREM B.1", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 232, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 233, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 233, + 596 + ], + "score": 1.0, + "content": "We first have the following lemma", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 582, + 233, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 343, + 613 + ], + "score": 1.0, + "content": "Lemma B.2 (double safe guard). In Algorithm 3, suppose", + "type": "text" + }, + { + "bbox": [ + 344, + 599, + 456, + 612 + ], + "score": 0.92, + "content": "\\mathfrak { T } _ { 1 } ~ = ~ 8 \\sqrt { T _ { 1 } \\log ( 1 6 m T _ { 1 } / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 599, + 478, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 478, + 600, + 505, + 611 + ], + "score": 0.84, + "content": "\\mathfrak { T } _ { 0 } ~ =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 611, + 441, + 625 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 187, + 624 + ], + "score": 0.9, + "content": "8 \\sqrt { T _ { 0 } \\log ( 1 6 m T _ { 1 } / p ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 612, + 302, + 625 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 302, + 612, + 333, + 623 + ], + "score": 0.93, + "content": "1 - p / 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 612, + 370, + 625 + ], + "score": 1.0, + "content": ", for every", + "type": "text" + }, + { + "bbox": [ + 370, + 613, + 437, + 623 + ], + "score": 0.9, + "content": "t = 0 , \\ldots , T _ { 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 612, + 441, + 625 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 599, + 505, + 625 + ] + }, + { + "type": "list", + "bbox": [ + 111, + 626, + 498, + 705 + ], + "lines": [ + { + "bbox": [ + 111, + 624, + 179, + 638 + ], + "spans": [ + { + "bbox": [ + 111, + 624, + 119, + 638 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 120, + 626, + 176, + 637 + ], + "score": 0.59, + "content": "\\mathbf { g } _ { 0 0 } \\mathbf { d } _ { t } \\supseteq \\mathbf { g } _ { 0 0 } \\mathbf { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 624, + 179, + 638 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 636, + 496, + 705 + ], + "spans": [ + { + "bbox": [ + 110, + 636, + 120, + 655 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 110, + 654, + 121, + 672 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 124, + 637, + 496, + 705 + ], + "score": 0.32, + "content": "\\begin{array} { r l } & { | \\sigma _ { t } | | ^ { 2 } \\leq O ( \\frac { \\log ( T _ { 1 } / p ) } { m } ) , \\| \\Delta _ { t } \\| ^ { 2 } \\leq \\alpha ^ { 2 } , \\| \\xi _ { t } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d \\log ( T _ { 1 } / p ) ) , } \\\\ & { | \\sigma _ { 0 } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 1 } \\log ( T _ { 1 } / p ) } { m } ) , \\| \\sigma _ { t a s t } + \\cdot \\cdot + \\sigma _ { t - 1 } \\| ^ { 2 } \\leq O ( \\frac { T _ { 0 } \\log ( T _ { 1 } / p ) } { m } ) } \\\\ & { | \\Delta _ { 0 } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 1 } \\log ( m T _ { 1 } / p ) ) a n d \\| \\Delta _ { l a s t } + \\cdot \\cdot \\cdot + \\Delta _ { t - 1 } \\| ^ { 2 } \\leq O ( \\alpha ^ { 2 } T _ { 0 } \\log ( m T _ { 1 } / p ) ) } \\\\ & { | \\xi _ { 0 } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 1 } \\log ( T _ { 1 } / p ) ) a n d \\| \\xi _ { l a s t } + \\cdot \\cdot \\cdot + \\xi _ { t - 1 } \\| ^ { 2 } \\leq O ( \\nu ^ { 2 } d T _ { 0 } \\log ( T _ { 1 } / p ) ) . } \\\\ & { \\quad \\cdot \\quad \\cdot \\quad \\cdot \\quad \\cdot } \\end{array}", + "type": "inline_equation", + "image_path": "71fefc8670a3238f94647c230dab755fe971d3971c7fd148e056309d0e3cf889.jpg" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 670, + 119, + 686 + ], + "spans": [ + { + "bbox": [ + 110, + 670, + 119, + 686 + ], + "score": 1.0, + "content": "•", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 685, + 120, + 699 + ], + "spans": [ + { + "bbox": [ + 110, + 685, + 120, + 699 + ], + "score": 1.0, + "content": "•", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 39.5, + "bbox_fs": [ + 110, + 624, + 496, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 417, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 418, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 241, + 717 + ], + "score": 1.0, + "content": "We call this probabilistic event Event", + "type": "text" + }, + { + "bbox": [ + 241, + 701, + 279, + 713 + ], + "score": 0.72, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 698, + 296, + 717 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 701, + 414, + 713 + ], + "score": 0.71, + "content": "\\operatorname* { P r } [ \\mathsf { E v e n t } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ) ] \\geq 1 - p / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 698, + 418, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 698, + 418, + 717 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 502, + 733 + ], + "lines": [ + { + "bbox": [ + 103, + 713, + 501, + 738 + ], + "spans": [ + { + "bbox": [ + 103, + 713, + 403, + 738 + ], + "score": 1.0, + "content": "The proof is a direct corollary of Lemma 3.2, by combining events Even", + "type": "text" + }, + { + "bbox": [ + 404, + 720, + 438, + 734 + ], + "score": 0.83, + "content": "\\mathbf { \\cdot } _ { T _ { 1 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 713, + 467, + 738 + ], + "score": 1.0, + "content": ", Event", + "type": "text" + }, + { + "bbox": [ + 467, + 720, + 501, + 734 + ], + "score": 0.65, + "content": "\\mathbf { \\cdot } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { 0 } )", + "type": "inline_equation" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 103, + 713, + 501, + 738 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 102, + 75, + 510, + 105 + ], + "spans": [ + { + "bbox": [ + 102, + 75, + 128, + 105 + ], + "score": 1.0, + "content": "EvenLemm", + "type": "text" + }, + { + "bbox": [ + 128, + 82, + 167, + 95 + ], + "score": 0.79, + "content": "\\mathbf { \\cdot } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { T _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 75, + 173, + 105 + ], + "score": 1.0, + "content": ", ro", + "type": "text" + }, + { + "bbox": [ + 173, + 82, + 236, + 95 + ], + "score": 0.78, + "content": "\\mathsf { E v e n t } _ { T _ { 0 } } ^ { \\mathsf { s i n g l e } } ( w _ { 2 T _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 75, + 510, + 105 + ], + "score": 1.0, + "content": "and so on. The next lemma is a simple corollary by repeatedly applyinga.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 103, + 107, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 103, + 107, + 139, + 140 + ], + "score": 1.0, + "content": "LemmaSuppose", + "type": "text" + }, + { + "bbox": [ + 221, + 107, + 225, + 140 + ], + "score": 1.0, + "content": "e,", + "type": "text" + }, + { + "bbox": [ + 297, + 107, + 305, + 140 + ], + "score": 1.0, + "content": "by a", + "type": "text" + }, + { + "bbox": [ + 305, + 110, + 369, + 121 + ], + "score": 0.92, + "content": "C _ { 1 } = \\log ( T _ { 1 } / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 107, + 387, + 140 + ], + "score": 1.0, + "content": "and start", + "type": "text" + }, + { + "bbox": [ + 387, + 108, + 501, + 123 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C _ { 2 } = \\alpha ^ { 2 } \\log \\frac { m T _ { 1 } } { p } + \\frac { \\log ( T _ { 1 } / p ) } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 107, + 507, + 140 + ], + "score": 1.0, + "content": "..", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 139, + 122, + 417, + 136 + ], + "spans": [ + { + "bbox": [ + 139, + 122, + 221, + 135 + ], + "score": 0.9, + "content": "\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 122, + 297, + 136 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 122, + 348, + 133 + ], + "score": 0.91, + "content": "T _ { 1 } \\geq T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 124, + 417, + 133 + ], + "score": 0.81, + "content": "w _ { 0 }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 132, + 257, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 174, + 151 + ], + "score": 1.0, + "content": "Under event Even", + "type": "text" + }, + { + "bbox": [ + 175, + 136, + 213, + 148 + ], + "score": 0.87, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 132, + 257, + 151 + ], + "score": 1.0, + "content": ", it satisfies", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 150, + 479, + 182 + ], + "lines": [ + { + "bbox": [ + 131, + 150, + 479, + 182 + ], + "spans": [ + { + "bbox": [ + 131, + 150, + 479, + 182 + ], + "score": 0.92, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right)", + "type": "interline_equation", + "image_path": "d60869e06ea14288f79f8940a2cbea5df5826aa6f03ccfb49c175ded6b3f1f0f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 131, + 150, + 479, + 160.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 131, + 160.66666666666666, + 479, + 171.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 131, + 171.33333333333331, + 479, + 181.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 308, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 309, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 309, + 198 + ], + "score": 1.0, + "content": "The next lemma can be easily derived from Lemma 3.5.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 293, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 294, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 294, + 214 + ], + "score": 1.0, + "content": "Lemma B.4 (modified from Lemma 3.5). Suppose", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 215, + 415, + 240 + ], + "lines": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "spans": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "score": 0.9, + "content": "R = \\Theta ( \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( \\delta ^ { 3 } / \\eta C _ { 2 } ) } ) a n d \\nu ^ { 2 } = \\Theta ( C _ { 2 } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } )", + "type": "interline_equation", + "image_path": "31b7245e824074578fbf95553f57a5df19d563db350646d4ccc23cab8452b24a.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 100, + 237, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 100, + 237, + 139, + 264 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 139, + 244, + 222, + 258 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\lbrace 1 , \\frac { \\delta ^ { 3 } } { C _ { 2 } } \\rbrace } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 237, + 226, + 264 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 227, + 244, + 299, + 259 + ], + "score": 0.94, + "content": "\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 237, + 317, + 264 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 243, + 420, + 259 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T _ { 1 } = \\Theta ( \\frac { 1 } { \\eta \\delta } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } ) \\geq T _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 237, + 441, + 264 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 441, + 245, + 477, + 256 + ], + "score": 0.9, + "content": "\\boldsymbol { w _ { 0 } } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 237, + 510, + 264 + ], + "score": 1.0, + "content": "be any", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 220, + 272 + ], + "score": 1.0, + "content": "point in the space and suppose", + "type": "text" + }, + { + "bbox": [ + 220, + 259, + 307, + 270 + ], + "score": 0.83, + "content": "\\lambda _ { \\operatorname* { m i n } } \\bigl ( \\nabla ^ { 2 } f ( w _ { 0 } ) \\bigr ) \\leq - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 257, + 340, + 272 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 341, + 259, + 364, + 270 + ], + "score": 0.89, + "content": "\\delta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 257, + 506, + 272 + ], + "score": 1.0, + "content": ". Given two coupled sequences defined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 264, + 473, + 286 + ], + "spans": [ + { + "bbox": [ + 102, + 264, + 214, + 286 + ], + "score": 1.0, + "content": "as before, under events Event", + "type": "text" + }, + { + "bbox": [ + 214, + 270, + 251, + 282 + ], + "score": 0.87, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ^ { \\mathsf { a } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 264, + 291, + 286 + ], + "score": 1.0, + "content": "and Event", + "type": "text" + }, + { + "bbox": [ + 291, + 270, + 328, + 282 + ], + "score": 0.85, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } ^ { \\mathsf { b } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 264, + 473, + 286 + ], + "score": 1.0, + "content": ", we have with probability at least 0.98", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 284, + 462, + 332 + ], + "lines": [ + { + "bbox": [ + 150, + 284, + 462, + 332 + ], + "spans": [ + { + "bbox": [ + 150, + 284, + 462, + 332 + ], + "score": 0.92, + "content": "\\begin{array} { l } { { \\displaystyle \\operatorname* { m a x } \\left\\{ f ( w _ { 0 } ^ { \\mathsf { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathsf { a } } ) , f ( w _ { 0 } ^ { \\mathsf { b } } ) - f ( w _ { T _ { 1 } } ^ { \\mathsf { b } } ) \\right\\} } \\ ~ } \\\\ { \\displaystyle \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\Omega ( \\frac { \\delta ^ { 3 } } { C _ { 1 } \\log ^ { 3 } \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } } ) } \\end{array}", + "type": "interline_equation", + "image_path": "9f1aab65008332c4ba73404ea5369c81fb947e771b72169aa7dbadb4c75b3616.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 150, + 284, + 462, + 300.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 150, + 300.0, + 462, + 316.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 150, + 316.0, + 462, + 332.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 336, + 506, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Lemma B.4 directly proves the second half of Theorem B.1c, because given two coupled sequences with the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 239, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 239, + 358 + ], + "score": 1.0, + "content": "same marginal distribution, we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 360, + 464, + 381 + ], + "lines": [ + { + "bbox": [ + 147, + 360, + 464, + 381 + ], + "spans": [ + { + "bbox": [ + 147, + 360, + 464, + 381 + ], + "score": 0.91, + "content": "\\operatorname* { P r } [ f ( w _ { 0 } ^ { \\mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { a } } ) \\ge X ] \\ge \\frac { 1 } { 2 } \\operatorname* { P r } [ \\operatorname* { m a x } \\left\\{ f ( w _ { 0 } ^ { \\mathrm { a } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { a } } ) , f ( w _ { 0 } ^ { \\mathrm { b } } ) - f ( w _ { T _ { 1 } } ^ { \\mathrm { b } } ) \\right\\} \\ge X ]", + "type": "interline_equation", + "image_path": "ac1ca7536fd15a004e1bffc1afa24097042bc9aad4fcf8cc39bdaaec7d19a0cf.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 147, + 360, + 464, + 381 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 403, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 240, + 416 + ], + "score": 1.0, + "content": "Proof of Lemma B.4. Our choice on", + "type": "text" + }, + { + "bbox": [ + 240, + 407, + 246, + 413 + ], + "score": 0.78, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 403, + 263, + 416 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 405, + 271, + 414 + ], + "score": 0.84, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "satisfy the requirement of Lemma 3.5. Suppose without loss of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 176, + 425 + ], + "score": 1.0, + "content": "generality that the", + "type": "text" + }, + { + "bbox": [ + 177, + 415, + 188, + 425 + ], + "score": 0.88, + "content": "w _ { t } ^ { \\mathsf { a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 414, + 252, + 425 + ], + "score": 1.0, + "content": "sequence leaves", + "type": "text" + }, + { + "bbox": [ + 252, + 416, + 263, + 424 + ], + "score": 0.86, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 414, + 318, + 425 + ], + "score": 1.0, + "content": "by more than", + "type": "text" + }, + { + "bbox": [ + 318, + 415, + 326, + 424 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 414, + 348, + 425 + ], + "score": 1.0, + "content": ". 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w _ { 0 } ^ { \\mathtt { a } } \\| ^ { 2 } = \\eta ^ { 2 } \\| \\nabla f ( w _ { 0 } ^ { \\mathtt { a } } ) + \\cdot \\cdot \\cdot + \\nabla f ( w _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } ) + \\Xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\Xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } + \\xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } \\| ^ { 2 } } } \\\\ { { \\displaystyle \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } } \\\\ { { \\displaystyle \\le O ( \\eta ^ { 2 } T _ { 1 } ) \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\mathtt { a } } - 1 } \\| \\nabla f ( w _ { t } ^ { \\mathtt { a } } ) \\| ^ { 2 } + O ( C _ { 2 } \\eta ^ { 2 } T _ { 1 } ) + O ( C _ { 1 } \\eta ^ { 2 } \\nu ^ { 2 } T _ { 1 } d ) } } \\end{array}", + "type": "interline_equation", + "image_path": "01a014c680c9853ae0eb278f04660b7558cc3bfa95b5d84e0c92d220132950aa.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 129, + 436, + 482, + 453.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 129, + 453.0, + 482, + 470.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 129, + 470.0, + 482, + 487.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 261, + 499 + ], + "lines": [ + { + "bbox": [ + 107, + 488, + 261, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 488, + 261, + 500 + ], + "score": 1.0, + "content": "Combining this with Lemma B.3, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 501, + 514, + 568 + ], + "lines": [ + { + "bbox": [ + 111, + 501, + 514, + 568 + ], + "spans": [ + { + "bbox": [ + 111, + 501, + 514, + 568 + ], + "score": 0.89, + "content": "\\begin{array} { l } { \\displaystyle \\lvert \\left( w _ { 0 } ^ { 3 } \\right) - f \\left( w _ { T _ { 1 } ^ { 3 } } ^ { \\mathfrak { a } } \\right) \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { \\lVert w _ { T _ { 1 } ^ { \\flat } } ^ { \\mathfrak { a } } - w _ { 0 } ^ { \\mathfrak { a } } \\rVert } { 1 0 0 \\eta T _ { 1 } } } \\\\ { \\displaystyle \\qquad \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "9a3c48a8ab46594dc021223bd4ac9c34aa958f9ae2c84d06fd0eb4f6694271c4.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 111, + 501, + 514, + 523.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 111, + 523.3333333333334, + 514, + 545.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 111, + 545.6666666666667, + 514, + 568.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 397, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 396, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 396, + 581 + ], + "score": 1.0, + "content": "Combining this with Lemma B.3 again but for the remainder iterations, we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 581, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 111, + 581, + 504, + 614 + ], + "spans": [ + { + "bbox": [ + 111, + 581, + 504, + 614 + ], + "score": 0.93, + "content": "f ( w _ { 0 } ^ { 3 } ) - f ( w _ { T _ { 1 } } ^ { 3 } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ^ { 3 } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } }", + "type": "interline_equation", + "image_path": "deb8ffcd9383de8a3eda38953948f33f4d103572a2b900e77ba78203442ccbfa.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 111, + 581, + 504, + 592.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 111, + 592.0, + 504, + 603.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 603.0, + 504, + 614.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 507, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "In fact, the above same proof of Lemma B.4 also implies Theorem B.1b. These together finish the proof of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 159, + 677 + ], + "score": 1.0, + "content": "Theorem B.1.", + "type": "text" + }, + { + "bbox": [ + 496, + 666, + 501, + 672 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 255, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 257, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 257, + 704 + ], + "score": 1.0, + "content": "C MORE ON EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "score": 1.0, + "content": "We conduct experiments on training a residual network ResNet-20 He et al. (2016) on the CIFAR-10/100 image", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 267, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 267, + 732 + ], + "score": 1.0, + "content": "classification tasks Krizhevsky et al. 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The next lemma is a simple corollary by repeatedly applyinga.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 102, + 75, + 510, + 105 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 108, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 103, + 107, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 103, + 107, + 139, + 140 + ], + "score": 1.0, + "content": "LemmaSuppose", + "type": "text" + }, + { + "bbox": [ + 221, + 107, + 225, + 140 + ], + "score": 1.0, + "content": "e,", + "type": "text" + }, + { + "bbox": [ + 297, + 107, + 305, + 140 + ], + "score": 1.0, + "content": "by a", + "type": "text" + }, + { + "bbox": [ + 305, + 110, + 369, + 121 + ], + "score": 0.92, + "content": "C _ { 1 } = \\log ( T _ { 1 } / p )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 107, + 387, + 140 + ], + "score": 1.0, + "content": "and start", + "type": "text" + }, + { + "bbox": [ + 387, + 108, + 501, + 123 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C _ { 2 } = \\alpha ^ { 2 } \\log \\frac { m T _ { 1 } } { p } + \\frac { \\log ( T _ { 1 } / p ) } { m } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 107, + 507, + 140 + ], + "score": 1.0, + "content": "..", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 139, + 122, + 417, + 136 + ], + "spans": [ + { + "bbox": [ + 139, + 122, + 221, + 135 + ], + "score": 0.9, + "content": "\\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\{ 1 , \\frac { 1 } { C _ { 2 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 122, + 297, + 136 + ], + "score": 0.92, + "content": "\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 122, + 348, + 133 + ], + "score": 0.91, + "content": "T _ { 1 } \\geq T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 124, + 417, + 133 + ], + "score": 0.81, + "content": "w _ { 0 }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 132, + 257, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 174, + 151 + ], + "score": 1.0, + "content": "Under event Even", + "type": "text" + }, + { + "bbox": [ + 175, + 136, + 213, + 148 + ], + "score": 0.87, + "content": "\\mathsf { \\Pi } _ { T _ { 1 } , T _ { 0 } } ^ { \\mathsf { d o u b l e } } ( w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 132, + 257, + 151 + ], + "score": 1.0, + "content": ", it satisfies", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 103, + 107, + 507, + 151 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 150, + 479, + 182 + ], + "lines": [ + { + "bbox": [ + 131, + 150, + 479, + 182 + ], + "spans": [ + { + "bbox": [ + 131, + 150, + 479, + 182 + ], + "score": 0.92, + "content": "f ( w _ { 0 } ) - f ( w _ { T _ { 1 } } ) \\ge 0 . 7 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right)", + "type": "interline_equation", + "image_path": "d60869e06ea14288f79f8940a2cbea5df5826aa6f03ccfb49c175ded6b3f1f0f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 131, + 150, + 479, + 160.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 131, + 160.66666666666666, + 479, + 171.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 131, + 171.33333333333331, + 479, + 181.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 308, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 309, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 309, + 198 + ], + "score": 1.0, + "content": "The next lemma can be easily derived from Lemma 3.5.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 187, + 309, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 293, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 294, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 294, + 214 + ], + "score": 1.0, + "content": "Lemma B.4 (modified from Lemma 3.5). Suppose", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 200, + 294, + 214 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 215, + 415, + 240 + ], + "lines": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "spans": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "score": 0.9, + "content": "R = \\Theta ( \\frac { \\delta } { \\sqrt { C _ { 1 } } \\log ( \\delta ^ { 3 } / \\eta C _ { 2 } ) } ) a n d \\nu ^ { 2 } = \\Theta ( C _ { 2 } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } )", + "type": "interline_equation", + "image_path": "31b7245e824074578fbf95553f57a5df19d563db350646d4ccc23cab8452b24a.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 196, + 215, + 415, + 240 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 282 + ], + "lines": [ + { + "bbox": [ + 100, + 237, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 100, + 237, + 139, + 264 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 139, + 244, + 222, + 258 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\eta \\leq 0 . 0 1 \\operatorname* { m i n } \\lbrace 1 , \\frac { \\delta ^ { 3 } } { C _ { 2 } } \\rbrace } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 237, + 226, + 264 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 227, + 244, + 299, + 259 + ], + "score": 0.94, + "content": "\\begin{array} { r } { T _ { 0 } = \\frac { 1 } { 1 0 0 \\eta ( 1 + \\sqrt { C _ { 2 } } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 237, + 317, + 264 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 243, + 420, + 259 + ], + "score": 0.93, + "content": "\\begin{array} { r } { T _ { 1 } = \\Theta ( \\frac { 1 } { \\eta \\delta } \\log \\frac { \\delta ^ { 3 } } { \\eta C _ { 2 } } ) \\geq T _ { 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 237, + 441, + 264 + ], + "score": 1.0, + "content": ". 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Our choice on", + "type": "text" + }, + { + "bbox": [ + 240, + 407, + 246, + 413 + ], + "score": 0.78, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 403, + 263, + 416 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 405, + 271, + 414 + ], + "score": 0.84, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "satisfy the requirement of Lemma 3.5. Suppose without loss of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 176, + 425 + ], + "score": 1.0, + "content": "generality that the", + "type": "text" + }, + { + "bbox": [ + 177, + 415, + 188, + 425 + ], + "score": 0.88, + "content": "w _ { t } ^ { \\mathsf { a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 414, + 252, + 425 + ], + "score": 1.0, + "content": "sequence leaves", + "type": "text" + }, + { + "bbox": [ + 252, + 416, + 263, + 424 + ], + "score": 0.86, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 414, + 318, + 425 + ], + "score": 1.0, + "content": "by more than", + "type": "text" + }, + { + "bbox": [ + 318, + 415, + 326, + 424 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 414, + 348, + 425 + ], + "score": 1.0, + "content": ". 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w _ { 0 } ^ { \\mathtt { a } } \\| ^ { 2 } = \\eta ^ { 2 } \\| \\nabla f ( w _ { 0 } ^ { \\mathtt { a } } ) + \\cdot \\cdot \\cdot + \\nabla f ( w _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } ) + \\Xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\Xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } + \\xi _ { 0 } ^ { \\mathtt { a } } + \\cdot \\cdot \\cdot + \\xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } \\| ^ { 2 } } } \\\\ { { \\displaystyle \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } } \\\\ { { \\displaystyle \\le O ( \\eta ^ { 2 } T _ { 1 } ) \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\mathtt { a } } - 1 } \\| \\nabla f ( w _ { t } ^ { \\mathtt { a } } ) \\| ^ { 2 } + O ( C _ { 2 } \\eta ^ { 2 } T _ { 1 } ) + O ( C _ { 1 } \\eta ^ { 2 } \\nu ^ { 2 } T _ { 1 } d ) } } \\end{array}", + "type": "interline_equation", + "image_path": "01a014c680c9853ae0eb278f04660b7558cc3bfa95b5d84e0c92d220132950aa.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 129, + 436, + 482, + 453.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 129, + 453.0, + 482, + 470.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 129, + 470.0, + 482, + 487.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 261, + 499 + ], + "lines": [ + { + "bbox": [ + 107, + 488, + 261, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 488, + 261, + 500 + ], + "score": 1.0, + "content": "Combining this with Lemma B.3, we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 107, + 488, + 261, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 501, + 514, + 568 + ], + "lines": [ + { + "bbox": [ + 111, + 501, + 514, + 568 + ], + "spans": [ + { + "bbox": [ + 111, + 501, + 514, + 568 + ], + "score": 0.89, + "content": "\\begin{array} { l } { \\displaystyle \\lvert \\left( w _ { 0 } ^ { 3 } \\right) - f \\left( w _ { T _ { 1 } ^ { 3 } } ^ { \\mathfrak { a } } \\right) \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { \\lVert w _ { T _ { 1 } ^ { \\flat } } ^ { \\mathfrak { a } } - w _ { 0 } ^ { \\mathfrak { a } } \\rVert } { 1 0 0 \\eta T _ { 1 } } } \\\\ { \\displaystyle \\qquad \\geq 0 . 5 \\eta \\displaystyle \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\sharp } - 1 } \\left( \\lVert \\nabla f ( w _ { t } ^ { \\mathfrak { a } } ) \\rVert ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "9a3c48a8ab46594dc021223bd4ac9c34aa958f9ae2c84d06fd0eb4f6694271c4.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 111, + 501, + 514, + 523.3333333333334 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 111, + 523.3333333333334, + 514, + 545.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 111, + 545.6666666666667, + 514, + 568.0000000000001 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 397, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 396, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 396, + 581 + ], + "score": 1.0, + "content": "Combining this with Lemma B.3 again but for the remainder iterations, we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 568, + 396, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 581, + 504, + 614 + ], + "lines": [ + { + "bbox": [ + 111, + 581, + 504, + 614 + ], + "spans": [ + { + "bbox": [ + 111, + 581, + 504, + 614 + ], + "score": 0.93, + "content": "f ( w _ { 0 } ^ { 3 } ) - f ( w _ { T _ { 1 } } ^ { 3 } ) \\geq 0 . 5 \\eta \\sum _ { t = 0 } ^ { T _ { 1 } - 1 } \\left( \\| \\nabla f ( w _ { t } ^ { 3 } ) \\| ^ { 2 } - \\eta \\cdot O ( C _ { 2 } + ( C _ { 2 } ) ^ { 1 . 5 } ) - O ( C _ { 1 } \\nu ^ { 2 } \\eta ( d + \\sqrt { C _ { 2 } } ) ) \\right) + \\frac { R ^ { 2 } } { 1 0 0 \\eta T _ { 1 } }", + "type": "interline_equation", + "image_path": "deb8ffcd9383de8a3eda38953948f33f4d103572a2b900e77ba78203442ccbfa.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 111, + 581, + 504, + 592.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 111, + 592.0, + 504, + 603.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 603.0, + 504, + 614.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 507, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "In fact, the above same proof of Lemma B.4 also implies Theorem B.1b. These together finish the proof of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 159, + 677 + ], + "score": 1.0, + "content": "Theorem B.1.", + "type": "text" + }, + { + "bbox": [ + 496, + 666, + 501, + 672 + ], + "score": 1.0, + "content": "\u0004", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 654, + 505, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 255, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 257, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 257, + 704 + ], + "score": 1.0, + "content": "C MORE ON EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 725 + ], + "score": 1.0, + "content": "We conduct experiments on training a residual network ResNet-20 He et al. (2016) on the CIFAR-10/100 image", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 267, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 267, + 732 + ], + "score": 1.0, + "content": "classification tasks Krizhevsky et al. (2014).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 709, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 304, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 306, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 306, + 95 + ], + "score": 1.0, + "content": "C.1 SETTING AND IMPLEMENTED METHODS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 98, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 505, + 111 + ], + "score": 1.0, + "content": "In all of our experiments, we use 10 workers and mini-batch size 10 per worker. Given any attacker and any", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 109, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 505, + 120 + ], + "score": 1.0, + "content": "defender algorithm, we run SGD three times for 140 epochs, each time with a different initial learning rate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 178, + 130 + ], + "score": 0.89, + "content": "\\eta \\in \\{ 0 . 1 , 0 . 2 , 0 . 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 117, + 506, + 131 + ], + "score": 1.0, + "content": ".7 We let the learning rate decrease by a factor of 10 on epochs 80 and 110, and present", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 490, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 490, + 141 + ], + "score": 1.0, + "content": "present the best testing accuracies in the three runs (each corresponding to a different initial learning rate).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 109, + 142, + 443, + 153 + ], + "lines": [ + { + "bbox": [ + 107, + 141, + 444, + 154 + ], + "spans": [ + { + "bbox": [ + 107, + 141, + 444, + 154 + ], + "score": 1.0, + "content": "We use standard data augmentation (random crops, random flips, and channel normalization).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "We compare against Geometric Median Chen et al. (2017), Coordinate-wise Median Yin et al. (2018; 2019),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 412, + 176 + ], + "score": 1.0, + "content": "Krum Blanchard et al. (2017), and Zeno Xie et al. (2018b) with attacks. We set", + "type": "text" + }, + { + "bbox": [ + 413, + 166, + 449, + 175 + ], + "score": 0.88, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "so there are 4", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "score": 1.0, + "content": "Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 360, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 360, + 198 + ], + "score": 1.0, + "content": "Byzantine workers.) We formally define those prior works as follows.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 504, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 406, + 213 + ], + "score": 1.0, + "content": "Definition C.1 (GeoMed Chen et al. (2017)). The geometric median of", + "type": "text" + }, + { + "bbox": [ + 407, + 200, + 452, + 212 + ], + "score": 0.93, + "content": "\\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 198, + 506, + 213 + ], + "score": 1.0, + "content": ", denoted by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 200, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 187, + 221 + ], + "score": 0.84, + "content": "g e o \\_ m e d \\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 210, + 200, + 222 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 223, + 407, + 237 + ], + "lines": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "spans": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "score": 0.89, + "content": "\\begin{array} { r } { g e o \\ L _ { - } m e d \\{ y _ { 1 } , . . . , y _ { m } \\} : = \\arg \\operatorname* { m i n } _ { y \\in \\mathbb { R } ^ { d } } \\sum _ { i = 1 } ^ { m } \\| y - y _ { i } \\| } \\end{array}", + "type": "interline_equation", + "image_path": "b7976376997ce922fcbac75cb86e0a9ae225dbb6a492f3ccd9fb8747facd20cb.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 239, + 379, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 379, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 329, + 253 + ], + "score": 1.0, + "content": "In our experiments, we choose the geometric median from set", + "type": "text" + }, + { + "bbox": [ + 329, + 239, + 375, + 251 + ], + "score": 0.92, + "content": "\\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 237, + 379, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 506, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 481, + 266 + ], + "score": 1.0, + "content": "Definition C.2 (coordinate-wise median Yin et al. (2018; 2019)). Coordinate-wise median", + "type": "text" + }, + { + "bbox": [ + 481, + 255, + 504, + 265 + ], + "score": 0.83, + "content": "g \\quad =", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 170, + 275 + ], + "score": 0.9, + "content": "m e d \\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 263, + 280, + 276 + ], + "score": 1.0, + "content": "is defined as a vector with its", + "type": "text" + }, + { + "bbox": [ + 280, + 265, + 285, + 273 + ], + "score": 0.35, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 263, + 360, + 276 + ], + "score": 1.0, + "content": "-th coordinate being", + "type": "text" + }, + { + "bbox": [ + 361, + 264, + 471, + 275 + ], + "score": 0.91, + "content": "g [ k ] = m e d \\{ y _ { 1 } [ k ] , . . . , y _ { m } [ k ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 274, + 321, + 286 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 133, + 285 + ], + "score": 0.9, + "content": "k \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 274, + 321, + 286 + ], + "score": 1.0, + "content": ", where med is the usual (one-dimensional) median.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 278, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 278, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 278, + 301 + ], + "score": 1.0, + "content": "Definition C.3 (Krum Blanchard et al. 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Given any attacker and any", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 109, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 505, + 120 + ], + "score": 1.0, + "content": "defender algorithm, we run SGD three times for 140 epochs, each time with a different initial learning rate", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 178, + 130 + ], + "score": 0.89, + "content": "\\eta \\in \\{ 0 . 1 , 0 . 2 , 0 . 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 117, + 506, + 131 + ], + "score": 1.0, + "content": ".7 We let the learning rate decrease by a factor of 10 on epochs 80 and 110, and present", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 490, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 490, + 141 + ], + "score": 1.0, + "content": "present the best testing accuracies in the three runs (each corresponding to a different initial learning rate).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 98, + 506, + 141 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 142, + 443, + 153 + ], + "lines": [ + { + "bbox": [ + 107, + 141, + 444, + 154 + ], + "spans": [ + { + "bbox": [ + 107, + 141, + 444, + 154 + ], + "score": 1.0, + "content": "We use standard data augmentation (random crops, random flips, and channel normalization).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 107, + 141, + 444, + 154 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "We compare against Geometric Median Chen et al. (2017), Coordinate-wise Median Yin et al. (2018; 2019),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 412, + 176 + ], + "score": 1.0, + "content": "Krum Blanchard et al. (2017), and Zeno Xie et al. (2018b) with attacks. We set", + "type": "text" + }, + { + "bbox": [ + 413, + 166, + 449, + 175 + ], + "score": 0.88, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "so there are 4", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "score": 1.0, + "content": "Byzantine workers. (This exceeds the fault-tolerance of Krum, and so we also tested Krum with only 3", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 360, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 360, + 198 + ], + "score": 1.0, + "content": "Byzantine workers.) We formally define those prior works as follows.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 156, + 505, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 200, + 504, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 406, + 213 + ], + "score": 1.0, + "content": "Definition C.1 (GeoMed Chen et al. (2017)). The geometric median of", + "type": "text" + }, + { + "bbox": [ + 407, + 200, + 452, + 212 + ], + "score": 0.93, + "content": "\\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 198, + 506, + 213 + ], + "score": 1.0, + "content": ", denoted by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 200, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 187, + 221 + ], + "score": 0.84, + "content": "g e o \\_ m e d \\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 210, + 200, + 222 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 198, + 506, + 222 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 223, + 407, + 237 + ], + "lines": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "spans": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "score": 0.89, + "content": "\\begin{array} { r } { g e o \\ L _ { - } m e d \\{ y _ { 1 } , . . . , y _ { m } \\} : = \\arg \\operatorname* { m i n } _ { y \\in \\mathbb { R } ^ { d } } \\sum _ { i = 1 } ^ { m } \\| y - y _ { i } \\| } \\end{array}", + "type": "interline_equation", + "image_path": "b7976376997ce922fcbac75cb86e0a9ae225dbb6a492f3ccd9fb8747facd20cb.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 204, + 223, + 407, + 237 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 239, + 379, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 379, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 329, + 253 + ], + "score": 1.0, + "content": "In our experiments, we choose the geometric median from set", + "type": "text" + }, + { + "bbox": [ + 329, + 239, + 375, + 251 + ], + "score": 0.92, + "content": "\\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 237, + 379, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 237, + 379, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 506, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 481, + 266 + ], + "score": 1.0, + "content": "Definition C.2 (coordinate-wise median Yin et al. (2018; 2019)). Coordinate-wise median", + "type": "text" + }, + { + "bbox": [ + 481, + 255, + 504, + 265 + ], + "score": 0.83, + "content": "g \\quad =", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 170, + 275 + ], + "score": 0.9, + "content": "m e d \\{ y _ { 1 } , . . . , y _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 263, + 280, + 276 + ], + "score": 1.0, + "content": "is defined as a vector with its", + "type": "text" + }, + { + "bbox": [ + 280, + 265, + 285, + 273 + ], + "score": 0.35, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 263, + 360, + 276 + ], + "score": 1.0, + "content": "-th coordinate being", + "type": "text" + }, + { + "bbox": [ + 361, + 264, + 471, + 275 + ], + "score": 0.91, + "content": "g [ k ] = m e d \\{ y _ { 1 } [ k ] , . . . , y _ { m } [ k ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 274, + 321, + 286 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 133, + 285 + ], + "score": 0.9, + "content": "k \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 274, + 321, + 286 + ], + "score": 1.0, + "content": ", where med is the usual (one-dimensional) median.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 253, + 506, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 278, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 278, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 278, + 301 + ], + "score": 1.0, + "content": "Definition C.3 (Krum Blanchard et al. 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f _ { r } ( x - \\eta u ) - \\rho \\| u \\| ^ { 2 }", + "type": "interline_equation", + "image_path": "f2957468c08b0590e3d0fce50ba76472b832024fc6f3e7083ef8697ba2b43195.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 216, + 450, + 395, + 464 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 502, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 181, + 477 + ], + "score": 0.88, + "content": "f _ { r } ( x ) - f _ { r } ( x - \\eta u )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 464, + 367, + 478 + ], + "score": 1.0, + "content": "is the estimated descendant of the loss function and", + "type": "text" + }, + { + "bbox": [ + 368, + 465, + 392, + 477 + ], + "score": 0.91, + "content": "\\rho \\| u \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 464, + 504, + 478 + ], + "score": 1.0, + "content": "is the magnitude of the update.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 464, + 504, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 482, + 503, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 204, + 495 + ], + "score": 1.0, + "content": "In our experiments, we let", + "type": "text" + }, + { + "bbox": [ + 204, + 483, + 226, + 494 + ], + "score": 0.92, + "content": "f _ { r } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 482, + 415, + 495 + ], + "score": 1.0, + "content": "be the estimated objective over a mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 416, + 483, + 449, + 493 + ], + "score": 0.89, + "content": "n _ { r } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "(so the time to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 504 + ], + "score": 1.0, + "content": "perform this estimation is on the same magnitude as the gradient evaluation for each individual worker). We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 503, + 427, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 146, + 514 + ], + "score": 1.0, + "content": "also chose", + "type": "text" + }, + { + "bbox": [ + 146, + 503, + 190, + 514 + ], + "score": 0.9, + "content": "\\rho = 0 . 0 0 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 504, + 427, + 514 + ], + "score": 1.0, + "content": "(and this value does not affect our experimental results by much).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 482, + 506, + 514 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 516, + 504, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "Safeguard SGD. Our Algorithm 1 is stated in a way to make our theoretical proofs as clean as possible. Here,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 526, + 298, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 298, + 538 + ], + "score": 1.0, + "content": "we discuss how we actually implement it in practice.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 515, + 505, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 540, + 503, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 358, + 552 + ], + "score": 1.0, + "content": "First of all, as common in the literature, we omit the Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 359, + 541, + 367, + 551 + ], + "score": 0.86, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "that is added for theoretical purpose,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 549, + 407, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 407, + 561 + ], + "score": 1.0, + "content": "and instead rely on the natural noise in the training process to escape saddle points.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 539, + 505, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "Also, we make universal choices for our safeguard window sizes (across all attackers): for our algorithm with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 318, + 585 + ], + "score": 1.0, + "content": "a single safeguard we have used a universal window size", + "type": "text" + }, + { + "bbox": [ + 318, + 574, + 346, + 583 + ], + "score": 0.89, + "content": "T = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "epochs, and for our algorithm with double", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 583, + 374, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 248, + 594 + ], + "score": 1.0, + "content": "safeguards we have used window sizes", + "type": "text" + }, + { + "bbox": [ + 248, + 583, + 275, + 593 + ], + "score": 0.91, + "content": "T _ { 0 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 584, + 315, + 594 + ], + "score": 1.0, + "content": "epoch and", + "type": "text" + }, + { + "bbox": [ + 315, + 584, + 343, + 593 + ], + "score": 0.9, + "content": "T _ { 1 } = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 584, + 374, + 594 + ], + "score": 1.0, + "content": "epochs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 563, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 597, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "We also provide an automatic empirical process to select safeguard thresholds and eliminate bad workers.8 The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 183, + 619 + ], + "score": 1.0, + "content": "process to determine", + "type": "text" + }, + { + "bbox": [ + 184, + 607, + 203, + 617 + ], + "score": 0.87, + "content": "A _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 606, + 268, + 619 + ], + "score": 1.0, + "content": "(and likewise for", + "type": "text" + }, + { + "bbox": [ + 268, + 607, + 288, + 617 + ], + "score": 0.88, + "content": "B _ { \\mathrm { m e d . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 606, + 506, + 619 + ], + "score": 1.0, + "content": ") is described as follows. In each iteration, for every worker", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 613, + 508, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 138, + 628 + ], + "score": 0.91, + "content": "\\hat { i } \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 613, + 172, + 634 + ], + "score": 1.0, + "content": ", we sort", + "type": "text" + }, + { + "bbox": [ + 172, + 617, + 246, + 631 + ], + "score": 0.94, + "content": "\\left\\{ \\Vert A _ { i } - A _ { j } \\Vert \\right\\} _ { j \\in [ m ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 613, + 328, + 634 + ], + "score": 1.0, + "content": "and pick the smallest", + "type": "text" + }, + { + "bbox": [ + 329, + 617, + 371, + 628 + ], + "score": 0.93, + "content": "\\lceil m / 2 + 1 \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 613, + 508, + 634 + ], + "score": 1.0, + "content": "-th entry, and let this number be the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 176, + 641 + ], + "score": 1.0, + "content": "“score” for worker", + "type": "text" + }, + { + "bbox": [ + 176, + 630, + 181, + 638 + ], + "score": 0.64, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 628, + 365, + 641 + ], + "score": 1.0, + "content": ". We select the worker with the smallest “score” as", + "type": "text" + }, + { + "bbox": [ + 366, + 629, + 385, + 639 + ], + "score": 0.89, + "content": "A _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 628, + 457, + 641 + ], + "score": 1.0, + "content": "and call its “score”", + "type": "text" + }, + { + "bbox": [ + 457, + 629, + 465, + 638 + ], + "score": 0.6, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 628, + 505, + 641 + ], + "score": 1.0, + "content": ". Then, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 637, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 121, + 653 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 122, + 639, + 175, + 650 + ], + "score": 0.93, + "content": "1 . 5 \\mathrm { m i n } \\{ S , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 637, + 460, + 653 + ], + "score": 1.0, + "content": "as the safeguard threshold for this iteration. Namely, we declare any worker", + "type": "text" + }, + { + "bbox": [ + 460, + 640, + 466, + 650 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 637, + 506, + 653 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 596, + 508, + 653 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 102, + 287, + 114 + ], + "lines": [ + { + "bbox": [ + 107, + 101, + 289, + 115 + ], + "spans": [ + { + "bbox": [ + 107, + 102, + 224, + 114 + ], + "score": 0.9, + "content": "\\| A _ { j } - A _ { \\sf m e d } \\| \\geq 1 . 5 \\operatorname* { m a x } \\{ S , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 101, + 289, + 115 + ], + "score": 1.0, + "content": "as a bad worker.9", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 123, + 288, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 289, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 289, + 136 + ], + "score": 1.0, + "content": "C.2 EXPERIMENT RESULTS BY ATTACKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 139, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 196, + 151 + ], + "score": 1.0, + "content": "The ideal accuracies are", + "type": "text" + }, + { + "bbox": [ + 197, + 139, + 251, + 149 + ], + "score": 0.83, + "content": "9 1 . 7 \\% / 6 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "for CIFAR-10/100, which correspond to applying SGD using only the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 150, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 160 + ], + "score": 1.0, + "content": "stochastic gradients from the honest workers. Below we discuss about the experimental results one attack at a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 158, + 128, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 128, + 171 + ], + "score": 1.0, + "content": "time.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 179, + 224, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 226, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 226, + 192 + ], + "score": 1.0, + "content": "C.2.1 VARIANCE ATTACK", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "We begin by looking at the hardest proposed attack from prior works. The Variance attack follows the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "strategy prescribed by Baruch et al. (2019), by which Byzantine workers collude in order to shift the mean", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 220, + 227 + ], + "score": 1.0, + "content": "among all gradients by a factor", + "type": "text" + }, + { + "bbox": [ + 221, + 216, + 227, + 225 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "times the standard deviation of the gradient, while staying within population", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "variance. More precisely, the maximal change to the mean that can be applied by an attacker without the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "fear of being detected, by using the properties of the Normal distribution, specifically the cumulative standard", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "normal function, and compute the maximal possible shift so that the attackers’s values stay within population", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 390, + 267 + ], + "score": 1.0, + "content": "variance. (See (Baruch et al., 2019, Algorithm 3) for a precise description. Our", + "type": "text" + }, + { + "bbox": [ + 390, + 255, + 397, + 266 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 254, + 405, + 267 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 406, + 256, + 425, + 265 + ], + "score": 0.88, + "content": "z _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "in their notation.) We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 471, + 277 + ], + "score": 1.0, + "content": "implement this strategy coordinate-wise, the same way as they did. Their work observes that the shift", + "type": "text" + }, + { + "bbox": [ + 471, + 266, + 478, + 276 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "non-trivial in practice, since stochastic gradients tend to have large variance in neural network training (which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "we also observed in our setup). Critically, the attack cannot be defended against by historyless algorithms, as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 295, + 464, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 464, + 307 + ], + "score": 1.0, + "content": "the attacker’s values are statistically indistinguishable from a regular execution in a single iteration.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 241, + 320 + ], + "score": 1.0, + "content": "In our setting, for 10 total nodes and", + "type": "text" + }, + { + "bbox": [ + 241, + 308, + 273, + 318 + ], + "score": 0.86, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 308, + 276, + 320 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 277, + 308, + 284, + 319 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "is upper bounded by 0.3 (following the classic tables for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "cumulative normal). We also ran the same attack in the setup from their paper (50 nodes total, of which 24 are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 198, + 340 + ], + "score": 1.0, + "content": "Byzantine, which allows", + "type": "text" + }, + { + "bbox": [ + 199, + 329, + 234, + 339 + ], + "score": 0.89, + "content": "\\beta \\sim 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "and observed a similar outcome. Results for this experiment are given in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 142, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 142, + 350 + ], + "score": 1.0, + "content": "Figure 3.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "image", + "bbox": [ + 104, + 363, + 505, + 495 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 363, + 505, + 495 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 363, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 505, + 495 + ], + "score": 0.973, + "type": "image", + "image_path": "64aff80ab494c66dd3b961892259a7e29da2e3c1faa1ace4ae36ced5717be94e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 104, + 363, + 505, + 407.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 104, + 407.0, + 505, + 451.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 104, + 451.0, + 505, + 495.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 193, + 502, + 414, + 513 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 501, + 415, + 515 + ], + "spans": [ + { + "bbox": [ + 192, + 501, + 415, + 515 + ], + "score": 1.0, + "content": "Figure 3: Performance comparison under the variance attack.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 23.0 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "As shown by the results, our algorithm provably circumvents this attack, and recovers full accuracy. This is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "score": 1.0, + "content": "explained by the fact that the algorithm has memory: in particular, Byzantine nodes following this strategy will", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 284, + 558 + ], + "score": 1.0, + "content": "progressively diverge from the (honest) “median”", + "type": "text" + }, + { + "bbox": [ + 285, + 547, + 304, + 557 + ], + "score": 0.9, + "content": "A _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "(at a “linear” rate, recall Figure 2(a)), and therefore will", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 557, + 387, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 387, + 568 + ], + "score": 1.0, + "content": "eventually exceed the threshold and be marked as malicious by the algorithm.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 581 + ], + "score": 1.0, + "content": "Specifically, both variants of the algorithm successfully catch all the bad nodes after at most 150 iterations.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 299, + 592 + ], + "score": 1.0, + "content": "Indeed, at the 100-th iteration, the pair-wise distances", + "type": "text" + }, + { + "bbox": [ + 299, + 581, + 338, + 591 + ], + "score": 0.93, + "content": "\\| A _ { i } - A _ { j } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 580, + 416, + 592 + ], + "score": 1.0, + "content": "among good workers", + "type": "text" + }, + { + "bbox": [ + 416, + 581, + 438, + 591 + ], + "score": 0.8, + "content": "i , j \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "good are between", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 424, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 424, + 602 + ], + "score": 1.0, + "content": "5.3 and 6.3, but the pair-wise distance between a good and a bad worker is at least 12.5.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 610, + 244, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 246, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 246, + 623 + ], + "score": 1.0, + "content": "C.2.2 SIGN-FLIPPING ATTACK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 626, + 503, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "We next move onto the sign-flipping attack. Recall that, in a sign-flipping attack, each Byzantine worker sends", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 636, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 647 + ], + "score": 1.0, + "content": "the negative gradient to the master. This is still a strong attack since if one does not avoid any bad workers, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 646, + 381, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 381, + 658 + ], + "score": 1.0, + "content": "test accuracy will suffer from a significant drop. The results are in Figure 4.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "From the plots, one can see that again our single and double safe-guard algorithms both outperform prior", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "works. They also successfully catch all the bad workers within 150 iterations. (For instance, at iteration 150", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 680, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 239, + 691 + ], + "score": 1.0, + "content": "for CIFAR-10 training, the distance", + "type": "text" + }, + { + "bbox": [ + 240, + 680, + 291, + 691 + ], + "score": 0.92, + "content": "\\| A _ { \\sf m e d } - A _ { j } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 680, + 361, + 691 + ], + "score": 1.0, + "content": "for a good worker", + "type": "text" + }, + { + "bbox": [ + 361, + 680, + 399, + 690 + ], + "score": 0.3, + "content": "j \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 680, + 506, + 691 + ], + "score": 1.0, + "content": "is at most 6.9, but for a bad", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "score": 1.0, + "content": "9This process appears a little different from Algorithm 1, but a similar proof also holds for this new empirical", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 712, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 506, + 722 + ], + "score": 1.0, + "content": "process. The constant factor 1.5 requires no tuning, and the constant threshold 5 is chosen so that the stochastic", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 420, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 420, + 732 + ], + "score": 1.0, + "content": "gradient of batch size 500 at random initialization has Euclidean norm no more than 5.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 102, + 287, + 114 + ], + "lines": [ + { + "bbox": [ + 107, + 101, + 289, + 115 + ], + "spans": [ + { + "bbox": [ + 107, + 102, + 224, + 114 + ], + "score": 0.9, + "content": "\\| A _ { j } - A _ { \\sf m e d } \\| \\geq 1 . 5 \\operatorname* { m a x } \\{ S , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 101, + 289, + 115 + ], + "score": 1.0, + "content": "as a bad worker.9", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 107, + 101, + 289, + 115 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 123, + 288, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 289, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 289, + 136 + ], + "score": 1.0, + "content": "C.2 EXPERIMENT RESULTS BY ATTACKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 139, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 196, + 151 + ], + "score": 1.0, + "content": "The ideal accuracies are", + "type": "text" + }, + { + "bbox": [ + 197, + 139, + 251, + 149 + ], + "score": 0.83, + "content": "9 1 . 7 \\% / 6 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "for CIFAR-10/100, which correspond to applying SGD using only the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 150, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 160 + ], + "score": 1.0, + "content": "stochastic gradients from the honest workers. Below we discuss about the experimental results one attack at a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 158, + 128, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 128, + 171 + ], + "score": 1.0, + "content": "time.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 138, + 506, + 171 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 179, + 224, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 226, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 226, + 192 + ], + "score": 1.0, + "content": "C.2.1 VARIANCE ATTACK", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "We begin by looking at the hardest proposed attack from prior works. The Variance attack follows the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "strategy prescribed by Baruch et al. (2019), by which Byzantine workers collude in order to shift the mean", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 220, + 227 + ], + "score": 1.0, + "content": "among all gradients by a factor", + "type": "text" + }, + { + "bbox": [ + 221, + 216, + 227, + 225 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "times the standard deviation of the gradient, while staying within population", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "variance. More precisely, the maximal change to the mean that can be applied by an attacker without the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "fear of being detected, by using the properties of the Normal distribution, specifically the cumulative standard", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "normal function, and compute the maximal possible shift so that the attackers’s values stay within population", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 390, + 267 + ], + "score": 1.0, + "content": "variance. (See (Baruch et al., 2019, Algorithm 3) for a precise description. Our", + "type": "text" + }, + { + "bbox": [ + 390, + 255, + 397, + 266 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 254, + 405, + 267 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 406, + 256, + 425, + 265 + ], + "score": 0.88, + "content": "z _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "in their notation.) We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 471, + 277 + ], + "score": 1.0, + "content": "implement this strategy coordinate-wise, the same way as they did. Their work observes that the shift", + "type": "text" + }, + { + "bbox": [ + 471, + 266, + 478, + 276 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "non-trivial in practice, since stochastic gradients tend to have large variance in neural network training (which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "we also observed in our setup). Critically, the attack cannot be defended against by historyless algorithms, as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 295, + 464, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 464, + 307 + ], + "score": 1.0, + "content": "the attacker’s values are statistically indistinguishable from a regular execution in a single iteration.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 195, + 506, + 307 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 241, + 320 + ], + "score": 1.0, + "content": "In our setting, for 10 total nodes and", + "type": "text" + }, + { + "bbox": [ + 241, + 308, + 273, + 318 + ], + "score": 0.86, + "content": "\\alpha = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 308, + 276, + 320 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 277, + 308, + 284, + 319 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "is upper bounded by 0.3 (following the classic tables for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "cumulative normal). We also ran the same attack in the setup from their paper (50 nodes total, of which 24 are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 198, + 340 + ], + "score": 1.0, + "content": "Byzantine, which allows", + "type": "text" + }, + { + "bbox": [ + 199, + 329, + 234, + 339 + ], + "score": 0.89, + "content": "\\beta \\sim 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "and observed a similar outcome. Results for this experiment are given in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 142, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 142, + 350 + ], + "score": 1.0, + "content": "Figure 3.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 308, + 505, + 350 + ] + }, + { + "type": "image", + "bbox": [ + 104, + 363, + 505, + 495 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 363, + 505, + 495 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 363, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 505, + 495 + ], + "score": 0.973, + "type": "image", + "image_path": "64aff80ab494c66dd3b961892259a7e29da2e3c1faa1ace4ae36ced5717be94e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 104, + 363, + 505, + 407.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 104, + 407.0, + 505, + 451.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 104, + 451.0, + 505, + 495.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 193, + 502, + 414, + 513 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 501, + 415, + 515 + ], + "spans": [ + { + "bbox": [ + 192, + 501, + 415, + 515 + ], + "score": 1.0, + "content": "Figure 3: Performance comparison under the variance attack.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 23.0 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "As shown by the results, our algorithm provably circumvents this attack, and recovers full accuracy. This is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "score": 1.0, + "content": "explained by the fact that the algorithm has memory: in particular, Byzantine nodes following this strategy will", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 284, + 558 + ], + "score": 1.0, + "content": "progressively diverge from the (honest) “median”", + "type": "text" + }, + { + "bbox": [ + 285, + 547, + 304, + 557 + ], + "score": 0.9, + "content": "A _ { \\mathrm { m e d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "(at a “linear” rate, recall Figure 2(a)), and therefore will", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 557, + 387, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 387, + 568 + ], + "score": 1.0, + "content": "eventually exceed the threshold and be marked as malicious by the algorithm.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 527, + 505, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 581 + ], + "score": 1.0, + "content": "Specifically, both variants of the algorithm successfully catch all the bad nodes after at most 150 iterations.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 299, + 592 + ], + "score": 1.0, + "content": "Indeed, at the 100-th iteration, the pair-wise distances", + "type": "text" + }, + { + "bbox": [ + 299, + 581, + 338, + 591 + ], + "score": 0.93, + "content": "\\| A _ { i } - A _ { j } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 580, + 416, + 592 + ], + "score": 1.0, + "content": "among good workers", + "type": "text" + }, + { + "bbox": [ + 416, + 581, + 438, + 591 + ], + "score": 0.8, + "content": "i , j \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "good are between", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 424, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 424, + 602 + ], + "score": 1.0, + "content": "5.3 and 6.3, but the pair-wise distance between a good and a bad worker is at least 12.5.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 570, + 505, + 602 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 610, + 244, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 246, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 246, + 623 + ], + "score": 1.0, + "content": "C.2.2 SIGN-FLIPPING ATTACK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 626, + 503, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "We next move onto the sign-flipping attack. Recall that, in a sign-flipping attack, each Byzantine worker sends", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 636, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 647 + ], + "score": 1.0, + "content": "the negative gradient to the master. This is still a strong attack since if one does not avoid any bad workers, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 646, + 381, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 381, + 658 + ], + "score": 1.0, + "content": "test accuracy will suffer from a significant drop. The results are in Figure 4.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 626, + 505, + 658 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "From the plots, one can see that again our single and double safe-guard algorithms both outperform prior", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "works. They also successfully catch all the bad workers within 150 iterations. (For instance, at iteration 150", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 680, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 239, + 691 + ], + "score": 1.0, + "content": "for CIFAR-10 training, the distance", + "type": "text" + }, + { + "bbox": [ + 240, + 680, + 291, + 691 + ], + "score": 0.92, + "content": "\\| A _ { \\sf m e d } - A _ { j } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 680, + 361, + 691 + ], + "score": 1.0, + "content": "for a good worker", + "type": "text" + }, + { + "bbox": [ + 361, + 680, + 399, + 690 + ], + "score": 0.3, + "content": "j \\in \\mathtt { g o o d }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 680, + 506, + 691 + ], + "score": 1.0, + "content": "is at most 6.9, but for a bad", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 660, + 506, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 81, + 506, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 81, + 506, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 506, + 214 + ], + "score": 0.964, + "type": "image", + "image_path": "0830d7ad719c3952f22957fe4c980b78a111cbf63eee160f2e9a694efee17c31.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 81, + 506, + 125.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 125.33333333333334, + 506, + 169.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 169.66666666666669, + 506, + 214.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 185, + 222, + 422, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 220, + 423, + 236 + ], + "spans": [ + { + "bbox": [ + 184, + 220, + 423, + 236 + ], + "score": 1.0, + "content": "Figure 4: Performance comparison under the sign-flipping attack.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 108, + 258, + 258, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 259, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 133, + 270 + ], + "score": 1.0, + "content": "worker", + "type": "text" + }, + { + "bbox": [ + 134, + 258, + 149, + 269 + ], + "score": 0.75, + "content": "j \\not \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 257, + 259, + 270 + ], + "score": 1.0, + "content": "good it can be more than 11.)", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 440, + 283 + ], + "score": 1.0, + "content": "In contrast, prior work Zeno completely fails because locally at a training step, using merely", + "type": "text" + }, + { + "bbox": [ + 440, + 272, + 473, + 281 + ], + "score": 0.89, + "content": "n _ { r } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "samples", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "to evaluate the objective, it is statistically not possible to even distinguish if the sign of the stochastic gradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "is flipped or not. For prior works Krum and GeoMedian, although they appear to have some non-negligible", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 302, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 504, + 312 + ], + "score": 1.0, + "content": "performances, but they are actually no better than simply applying SGD with the naive mean of gradients from", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "all the workers (including those from bad workers).10 Therefore, we conclude that prior works all fail to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 321, + 258, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 258, + 332 + ], + "score": 1.0, + "content": "Byzantine fault tolerant under this attack.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 341, + 270, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 272, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 272, + 354 + ], + "score": 1.0, + "content": "C.2.3 DELAYED-GRADIENT ATTACK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "score": 1.0, + "content": "Recall that, in a delayed-gradient attack, each Byzantine worker sends an old gradient to the master. In our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 211, + 379 + ], + "score": 1.0, + "content": "experiments, the delay is of", + "type": "text" + }, + { + "bbox": [ + 212, + 367, + 254, + 377 + ], + "score": 0.9, + "content": "D = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 367, + 294, + 379 + ], + "score": 1.0, + "content": "iterations", + "type": "text" + }, + { + "bbox": [ + 294, + 368, + 309, + 377 + ], + "score": 0.79, + "content": "{ \\it \\Delta \\phi } = 2 { \\it \\Delta \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "epochs). We believe this is not a very strong attack,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "because delayed gradients are not sufficiently malicious: they are still “correct” to certain extent albeit being", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 387, + 264, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 264, + 399 + ], + "score": 1.0, + "content": "delayed. The results are shown in Figure 5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "image", + "bbox": [ + 104, + 411, + 506, + 543 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 411, + 506, + 543 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 411, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 411, + 506, + 543 + ], + "score": 0.972, + "type": "image", + "image_path": "f098289e873c796a5efe146a5bd454d7949f3336d83227ba94172db45a7fec07.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 104, + 411, + 506, + 455.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 104, + 455.0, + 506, + 499.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 104, + 499.0, + 506, + 543.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 178, + 551, + 429, + 562 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 177, + 549, + 430, + 563 + ], + "spans": [ + { + "bbox": [ + 177, + 549, + 430, + 563 + ], + "score": 1.0, + "content": "Figure 5: Performance comparison under the delayed-gradient attack.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 504, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "From the plots, one can see that our single and double safe-guard algorithms again both match the ideal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 584, + 436, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 436, + 595 + ], + "score": 1.0, + "content": "accuracies. All the prior works suffer from a significant performance loss under this attack.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "It is worth noting that our single and double safe-guard algorithms do not catch any bad worker under this attack,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "so they simply use the “naive mean” of gradients from all the workers (including those delayed gradients from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "bad workers). However, there is no performance loss even if we use those delayed gradients. That is why we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 627, + 467, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 467, + 639 + ], + "score": 1.0, + "content": "believe the delayed-gradient attack is not very strong, as the gradients are not sufficiently malicious.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 108, + 641, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "score": 1.0, + "content": "Prior work Zeno suffers from some performance loss, because it only uses 6 workers out of 10, in which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 169, + 664 + ], + "score": 1.0, + "content": "statistically only", + "type": "text" + }, + { + "bbox": [ + 169, + 651, + 237, + 661 + ], + "score": 0.9, + "content": "6 \\times 0 . 6 \\approx 3 \\sim 4", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "gradients are correct.11 Other prior works suffer from performance loss,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 670, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 115, + 668, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 115, + 668, + 506, + 684 + ], + "score": 1.0, + "content": "10We did not include this “naive mean” algorithm in the plots for cleanness, but under the sign-flipping", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 679, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 162, + 694 + ], + "score": 1.0, + "content": "attack, it gives", + "type": "text" + }, + { + "bbox": [ + 162, + 681, + 186, + 690 + ], + "score": 0.87, + "content": "8 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 679, + 304, + 694 + ], + "score": 1.0, + "content": "test accuracy on CIFAR-10 and", + "type": "text" + }, + { + "bbox": [ + 305, + 681, + 329, + 690 + ], + "score": 0.88, + "content": "3 8 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 679, + 505, + 694 + ], + "score": 1.0, + "content": "on CIFAR-100. (This should not be surprising,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 151, + 703 + ], + "score": 1.0, + "content": "since using", + "type": "text" + }, + { + "bbox": [ + 152, + 691, + 214, + 702 + ], + "score": 0.92, + "content": "( 1 - \\alpha ) m = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 690, + 302, + 703 + ], + "score": 1.0, + "content": "positive gradients plus", + "type": "text" + }, + { + "bbox": [ + 302, + 691, + 339, + 700 + ], + "score": 0.87, + "content": "\\alpha m \\ = \\ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "negative gradients still gives non-negligible", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 240, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 240, + 712 + ], + "score": 1.0, + "content": "information about the true gradient.)", + "type": "text" + } + ] + }, + { + "bbox": [ + 116, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 116, + 710, + 132, + 721 + ], + "score": 0.39, + "content": "^ { 1 1 } \\mathrm { { I n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "fact, we observed Zeno slightly favors delayed gradients, where each delay gradient is chosen with", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 436, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 148, + 733 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 148, + 721, + 166, + 731 + ], + "score": 0.87, + "content": "6 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 721, + 414, + 733 + ], + "score": 1.0, + "content": ", comparing to true stochastic gradients each chosen with probability", + "type": "text" + }, + { + "bbox": [ + 414, + 721, + 432, + 731 + ], + "score": 0.88, + "content": "5 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 721, + 436, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 81, + 506, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 81, + 506, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 506, + 214 + ], + "score": 0.964, + "type": "image", + "image_path": "0830d7ad719c3952f22957fe4c980b78a111cbf63eee160f2e9a694efee17c31.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 81, + 506, + 125.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 125.33333333333334, + 506, + 169.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 169.66666666666669, + 506, + 214.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 185, + 222, + 422, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 220, + 423, + 236 + ], + "spans": [ + { + "bbox": [ + 184, + 220, + 423, + 236 + ], + "score": 1.0, + "content": "Figure 4: Performance comparison under the sign-flipping attack.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 108, + 258, + 258, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 259, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 133, + 270 + ], + "score": 1.0, + "content": "worker", + "type": "text" + }, + { + "bbox": [ + 134, + 258, + 149, + 269 + ], + "score": 0.75, + "content": "j \\not \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 257, + 259, + 270 + ], + "score": 1.0, + "content": "good it can be more than 11.)", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 106, + 257, + 259, + 270 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 440, + 283 + ], + "score": 1.0, + "content": "In contrast, prior work Zeno completely fails because locally at a training step, using merely", + "type": "text" + }, + { + "bbox": [ + 440, + 272, + 473, + 281 + ], + "score": 0.89, + "content": "n _ { r } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "samples", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "to evaluate the objective, it is statistically not possible to even distinguish if the sign of the stochastic gradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "is flipped or not. For prior works Krum and GeoMedian, although they appear to have some non-negligible", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 302, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 504, + 312 + ], + "score": 1.0, + "content": "performances, but they are actually no better than simply applying SGD with the naive mean of gradients from", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "all the workers (including those from bad workers).10 Therefore, we conclude that prior works all fail to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 321, + 258, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 258, + 332 + ], + "score": 1.0, + "content": "Byzantine fault tolerant under this attack.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 271, + 506, + 332 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 341, + 270, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 272, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 272, + 354 + ], + "score": 1.0, + "content": "C.2.3 DELAYED-GRADIENT ATTACK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "score": 1.0, + "content": "Recall that, in a delayed-gradient attack, each Byzantine worker sends an old gradient to the master. In our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 211, + 379 + ], + "score": 1.0, + "content": "experiments, the delay is of", + "type": "text" + }, + { + "bbox": [ + 212, + 367, + 254, + 377 + ], + "score": 0.9, + "content": "D = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 367, + 294, + 379 + ], + "score": 1.0, + "content": "iterations", + "type": "text" + }, + { + "bbox": [ + 294, + 368, + 309, + 377 + ], + "score": 0.79, + "content": "{ \\it \\Delta \\phi } = 2 { \\it \\Delta \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "epochs). We believe this is not a very strong attack,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "because delayed gradients are not sufficiently malicious: they are still “correct” to certain extent albeit being", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 387, + 264, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 264, + 399 + ], + "score": 1.0, + "content": "delayed. The results are shown in Figure 5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 357, + 506, + 399 + ] + }, + { + "type": "image", + "bbox": [ + 104, + 411, + 506, + 543 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 411, + 506, + 543 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 411, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 411, + 506, + 543 + ], + "score": 0.972, + "type": "image", + "image_path": "f098289e873c796a5efe146a5bd454d7949f3336d83227ba94172db45a7fec07.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 104, + 411, + 506, + 455.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 104, + 455.0, + 506, + 499.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 104, + 499.0, + 506, + 543.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 178, + 551, + 429, + 562 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 177, + 549, + 430, + 563 + ], + "spans": [ + { + "bbox": [ + 177, + 549, + 430, + 563 + ], + "score": 1.0, + "content": "Figure 5: Performance comparison under the delayed-gradient attack.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 504, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "From the plots, one can see that our single and double safe-guard algorithms again both match the ideal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 584, + 436, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 436, + 595 + ], + "score": 1.0, + "content": "accuracies. All the prior works suffer from a significant performance loss under this attack.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 574, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "It is worth noting that our single and double safe-guard algorithms do not catch any bad worker under this attack,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "so they simply use the “naive mean” of gradients from all the workers (including those delayed gradients from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "bad workers). However, there is no performance loss even if we use those delayed gradients. That is why we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 627, + 467, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 467, + 639 + ], + "score": 1.0, + "content": "believe the delayed-gradient attack is not very strong, as the gradients are not sufficiently malicious.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 597, + 505, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 641, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "score": 1.0, + "content": "Prior work Zeno suffers from some performance loss, because it only uses 6 workers out of 10, in which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 169, + 664 + ], + "score": 1.0, + "content": "statistically only", + "type": "text" + }, + { + "bbox": [ + 169, + 651, + 237, + 661 + ], + "score": 0.9, + "content": "6 \\times 0 . 6 \\approx 3 \\sim 4", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "gradients are correct.11 Other prior works suffer from performance loss,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "score": 1.0, + "content": "because they only pick one single stochastic gradient from the 10 workers, and it is sometimes even from the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 152, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 152, + 114 + ], + "score": 1.0, + "content": "bad worker.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 641, + 506, + 664 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 93, + 504, + 113 + ], + "lines": [ + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "score": 1.0, + "content": "because they only pick one single stochastic gradient from the 10 workers, and it is sometimes even from the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 102, + 152, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 152, + 114 + ], + "score": 1.0, + "content": "bad worker.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 123, + 252, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 254, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 254, + 136 + ], + "score": 1.0, + "content": "C.2.4 LABEL-FLIPPING ATTACK", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "Recall that, in the label-flipping attack, each Byzantine worker computes its gradient based on the cross-entropy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 271, + 160 + ], + "score": 1.0, + "content": "loss with flipped labels: for CIFAR-10, label", + "type": "text" + }, + { + "bbox": [ + 272, + 149, + 323, + 160 + ], + "score": 0.93, + "content": "\\ell \\in \\{ 0 , . . . , 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 149, + 369, + 160 + ], + "score": 1.0, + "content": "is flipped to", + "type": "text" + }, + { + "bbox": [ + 370, + 149, + 391, + 159 + ], + "score": 0.83, + "content": "9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 149, + 490, + 160 + ], + "score": 1.0, + "content": ", and for CIFAR-100, label", + "type": "text" + }, + { + "bbox": [ + 490, + 150, + 495, + 158 + ], + "score": 0.71, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 149, + 506, + 160 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 297, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 143, + 171 + ], + "score": 1.0, + "content": "flipped to", + "type": "text" + }, + { + "bbox": [ + 143, + 159, + 168, + 169 + ], + "score": 0.73, + "content": "9 9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 159, + 297, + 171 + ], + "score": 1.0, + "content": ". The results are shown in Figure 6.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 104, + 185, + 506, + 316 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 185, + 506, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 185, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 185, + 506, + 316 + ], + "score": 0.972, + "type": "image", + "image_path": "7db635d9c4932b3527dd2bd95505447e076401213199725b8306f27dfeae79d7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 104, + 185, + 506, + 228.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 104, + 228.66666666666666, + 506, + 272.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 104, + 272.3333333333333, + 506, + 316.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 183, + 324, + 423, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 324, + 424, + 336 + ], + "spans": [ + { + "bbox": [ + 184, + 324, + 424, + 336 + ], + "score": 1.0, + "content": "Figure 6: Performance comparison under the label-flipping attack.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "From the plots, one can see that our single and double safe-guard algorithms even outperform the “ideal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 360, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 154, + 371 + ], + "score": 1.0, + "content": "accuracies.”", + "type": "text" + }, + { + "bbox": [ + 155, + 360, + 180, + 369 + ], + "score": 0.85, + "content": "9 2 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 360, + 284, + 371 + ], + "score": 1.0, + "content": "accuracy vs “ideal accuracy”", + "type": "text" + }, + { + "bbox": [ + 285, + 360, + 309, + 369 + ], + "score": 0.86, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 360, + 374, + 371 + ], + "score": 1.0, + "content": "under CIFAR-10;", + "type": "text" + }, + { + "bbox": [ + 375, + 360, + 399, + 369 + ], + "score": 0.85, + "content": "6 9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 360, + 504, + 371 + ], + "score": 1.0, + "content": "accuracy vs “ideal accuracy”", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 368, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 382 + ], + "score": 1.0, + "content": "68.0 under CIFAR-100.) In addition, we have found out that the safeguard algorithms did not catch any", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "bad worker. This should not be surprising, since label-flipping (a.k.a. label smoothing) is known to be a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 389, + 435, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 435, + 401 + ], + "score": 1.0, + "content": "regularization technique to actually improve test accuracy, as opposed to hurt performance.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 414 + ], + "score": 1.0, + "content": "Zeno also performs well under this attack (but it does not outperform the ideal accuracy). We have investigated", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "into Zeno, and found out that it cannot distinguish good workers from bad workers under label-flipping attack;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "and therefore Zeno effectively always runs under 6 random workers as opposed to using the full power of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 433, + 139, + 442 + ], + "score": 0.89, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 433, + 505, + 444 + ], + "score": 1.0, + "content": "workers (recall Zeno picks 6 workers with the topmost scores, see Definition C.4). This explains its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 442, + 297, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 297, + 455 + ], + "score": 1.0, + "content": "(minor) under-performance comparing to safeguard.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 107, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "Other prior works perform significantly worse, and this should be alarming since label-flipping is one type of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 466, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 469, + 478 + ], + "score": 1.0, + "content": "smoothing technique to improve test accuracy, as opposed to an actual “attack” to hurt performance.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 486, + 237, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 239, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 239, + 499 + ], + "score": 1.0, + "content": "C.2.5 SAFEGUARD ATTACKS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "Finally, in the safeguard attack that we design, Byzantine workers send negative but re-scaled gradient to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "master. We choose the re-scale factor so that it hardly triggers the safe-guard conditions at the master. From our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "experiment, choosing the re-scale factor as 0.6 in all the cases do not trigger the safe-guard conditions, while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "choosing a re-scale factor as 0.7 enables the algorithm to catch Byzantine workers occasionally. Our results are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 542, + 395, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 395, + 554 + ], + "score": 1.0, + "content": "shown in Figure 7 (for re-scale factor 0.6) and Figure 8 (for re-scale factor 0.7).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "image", + "bbox": [ + 104, + 568, + 506, + 700 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 568, + 506, + 700 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 568, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 506, + 700 + ], + "score": 0.972, + "type": "image", + "image_path": "a4d787756f71e5c67aa9104e5787baf0489820e638d83faedff07126f19b5336.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 104, + 568, + 506, + 612.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 104, + 612.0, + 506, + 656.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 104, + 656.0, + 506, + 700.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 102, + 708, + 506, + 729 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 719 + ], + "score": 1.0, + "content": "Figure 7: Performance comparison under the safeguard attack with re-scale factor 0.6. (Recall this attack is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 718, + 380, + 729 + ], + "spans": [ + { + "bbox": [ + 142, + 718, + 380, + 729 + ], + "score": 1.0, + "content": "designed to maximally impact the performance of our algorithm.)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "index": 30.25 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "22", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 93, + 504, + 113 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 93, + 505, + 114 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 123, + 252, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 254, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 254, + 136 + ], + "score": 1.0, + "content": "C.2.4 LABEL-FLIPPING ATTACK", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "Recall that, in the label-flipping attack, each Byzantine worker computes its gradient based on the cross-entropy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 271, + 160 + ], + "score": 1.0, + "content": "loss with flipped labels: for CIFAR-10, label", + "type": "text" + }, + { + "bbox": [ + 272, + 149, + 323, + 160 + ], + "score": 0.93, + "content": "\\ell \\in \\{ 0 , . . . , 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 149, + 369, + 160 + ], + "score": 1.0, + "content": "is flipped to", + "type": "text" + }, + { + "bbox": [ + 370, + 149, + 391, + 159 + ], + "score": 0.83, + "content": "9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 149, + 490, + 160 + ], + "score": 1.0, + "content": ", and for CIFAR-100, label", + "type": "text" + }, + { + "bbox": [ + 490, + 150, + 495, + 158 + ], + "score": 0.71, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 149, + 506, + 160 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 297, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 143, + 171 + ], + "score": 1.0, + "content": "flipped to", + "type": "text" + }, + { + "bbox": [ + 143, + 159, + 168, + 169 + ], + "score": 0.73, + "content": "9 9 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 159, + 297, + 171 + ], + "score": 1.0, + "content": ". 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In addition, we have found out that the safeguard algorithms did not catch any", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 391 + ], + "score": 1.0, + "content": "bad worker. This should not be surprising, since label-flipping (a.k.a. label smoothing) is known to be a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 389, + 435, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 435, + 401 + ], + "score": 1.0, + "content": "regularization technique to actually improve test accuracy, as opposed to hurt performance.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 349, + 506, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 414 + ], + "score": 1.0, + "content": "Zeno also performs well under this attack (but it does not outperform the ideal accuracy). We have investigated", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "into Zeno, and found out that it cannot distinguish good workers from bad workers under label-flipping attack;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "and therefore Zeno effectively always runs under 6 random workers as opposed to using the full power of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 433, + 139, + 442 + ], + "score": 0.89, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 433, + 505, + 444 + ], + "score": 1.0, + "content": "workers (recall Zeno picks 6 workers with the topmost scores, see Definition C.4). This explains its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 442, + 297, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 297, + 455 + ], + "score": 1.0, + "content": "(minor) under-performance comparing to safeguard.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 403, + 505, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 107, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 107, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "Other prior works perform significantly worse, and this should be alarming since label-flipping is one type of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 466, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 469, + 478 + ], + "score": 1.0, + "content": "smoothing technique to improve test accuracy, as opposed to an actual “attack” to hurt performance.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 456, + 505, + 478 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 486, + 237, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 239, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 239, + 499 + ], + "score": 1.0, + "content": "C.2.5 SAFEGUARD ATTACKS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "Finally, in the safeguard attack that we design, Byzantine workers send negative but re-scaled gradient to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "master. We choose the re-scale factor so that it hardly triggers the safe-guard conditions at the master. From our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "experiment, choosing the re-scale factor as 0.6 in all the cases do not trigger the safe-guard conditions, while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "choosing a re-scale factor as 0.7 enables the algorithm to catch Byzantine workers occasionally. Our results are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 542, + 395, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 395, + 554 + ], + "score": 1.0, + "content": "shown in Figure 7 (for re-scale factor 0.6) and Figure 8 (for re-scale factor 0.7).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 502, + 505, + 554 + ] + }, + { + "type": "image", + "bbox": [ + 104, + 568, + 506, + 700 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 568, + 506, + 700 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 568, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 506, + 700 + ], + "score": 0.972, + "type": "image", + "image_path": "a4d787756f71e5c67aa9104e5787baf0489820e638d83faedff07126f19b5336.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 104, + 568, + 506, + 612.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 104, + 612.0, + 506, + 656.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 104, + 656.0, + 506, + 700.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 102, + 708, + 506, + 729 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 719 + ], + "score": 1.0, + "content": "Figure 7: Performance comparison under the safeguard attack with re-scale factor 0.6. (Recall this attack is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 718, + 380, + 729 + ], + "spans": [ + { + "bbox": [ + 142, + 718, + 380, + 729 + ], + "score": 1.0, + "content": "designed to maximally impact the performance of our algorithm.)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "index": 30.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 505, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Re-scale factor 0.6. In Figure 7, the performance of our (single and double) safeguard algorithms indeed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "get hurt a bit. Recall in Figure 7 the re-scale factor 0.6 is chosen to maximally impact our algorithm. The test", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 181, + 115 + ], + "score": 1.0, + "content": "accuracy drops from", + "type": "text" + }, + { + "bbox": [ + 181, + 103, + 206, + 113 + ], + "score": 0.92, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 102, + 215, + 115 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 216, + 103, + 240, + 113 + ], + "score": 0.88, + "content": "8 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 102, + 361, + 115 + ], + "score": 1.0, + "content": "under CIFAR-10; and drops from", + "type": "text" + }, + { + "bbox": [ + 362, + 103, + 387, + 113 + ], + "score": 0.9, + "content": "6 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 102, + 396, + 115 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 397, + 103, + 422, + 113 + ], + "score": 0.89, + "content": "6 \\mathrm { { 0 . 0 \\% } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "under CIFAR-100 (for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 124 + ], + "score": 1.0, + "content": "both single and double safeguards). In these cases, we confirm that both versions of the safeguard algorithms", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 123, + 432, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 432, + 135 + ], + "score": 1.0, + "content": "did not catch any bad worker. However, this still significantly outperforms all prior works.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "image", + "bbox": [ + 104, + 156, + 505, + 287 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 156, + 505, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 156, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 505, + 287 + ], + "score": 0.971, + "type": "image", + "image_path": "49b2f75c69622e5f6fd3ef4971d15626ae1482d4b1c61087e1ae4910caf31307.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 104, + 156, + 505, + 199.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 104, + 199.66666666666666, + 505, + 243.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 104, + 243.33333333333331, + 505, + 287.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 295, + 505, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "Figure 8: Performance comparison under the safeguard attack with re-scale factor 0.7. (In this case, our our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 304, + 415, + 317 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 415, + 317 + ], + "score": 1.0, + "content": "algorithm can catch some bad workers, and thus perform nearly optimally.)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "Re-scale factor 0.7. In Figure 8, we present the scenario when the re-scale factor is 0.7, so that the safeguard", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "algorithms can occasionally catch some bad workers (depending on the randomness and learning rate). We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 465, + 374 + ], + "score": 1.0, + "content": "confirm that in the three runs of single safeguard, it catches 1, 2, 3 bad workers for CIFAR-10, and", + "type": "text" + }, + { + "bbox": [ + 465, + 362, + 488, + 372 + ], + "score": 0.78, + "content": "1 , 0 , 0", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "bad", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 420, + 384 + ], + "score": 1.0, + "content": "workers for CIFAR-100 respectively; in the three runs of double safeguard, it catches", + "type": "text" + }, + { + "bbox": [ + 420, + 373, + 444, + 383 + ], + "score": 0.48, + "content": "1 , 2 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "bad workers for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 380, + 339, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 339, + 394 + ], + "score": 1.0, + "content": "CIFAR-10, and 2, 2, 2 bad workers for CIFAR-100 respectively.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Since there is a significant performance gain when our safeguard algorithms catch bad workers, this explains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "why safeguard algorithms in Figure 8 outperform their counterparts in Figure 7 with rescale factor 0.6. At the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "same time, we notice that the double safeguard algorithm has the ability to catch bad workers more easily. This", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 395, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 395, + 437 + ], + "score": 1.0, + "content": "is why double safeguard significantly outperforms single sageguard in Figure 8.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "In contrast, all other prior algorithms perform extremely bad under this attack. To some extent, safeguard attack", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "is even stronger than the previously proposed variance attack, since it can drag the 100-class test accuracy on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 303, + 471 + ], + "score": 1.0, + "content": "CIFAR-100 for all prior defense algorithms to nearly", + "type": "text" + }, + { + "bbox": [ + 303, + 459, + 316, + 469 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 458, + 505, + 471 + ], + "score": 1.0, + "content": ", while variance attack can only drag them down to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 469, + 155, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 134, + 480 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 134, + 469, + 151, + 479 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 469, + 155, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 489, + 248, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 249, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 249, + 501 + ], + "score": 1.0, + "content": "C.3 FULL COMPARISON TABLE", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 345, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 346, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 346, + 517 + ], + "score": 1.0, + "content": "We also include the full test accuracy comparison table in Table 1.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "table", + "bbox": [ + 144, + 534, + 466, + 714 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 144, + 534, + 466, + 714 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 534, + 466, + 714 + ], + "spans": [ + { + "bbox": [ + 144, + 534, + 466, + 714 + ], + "score": 0.875, + "html": "
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack92.0291.7521.4322.0121.4722.433.42
sign-flipping attack91.9392.082259.6557.0370.8722.4
label-flipping attack92.3392.4431.9383.0783.1883.5291.66
delayed-gradient attack91.5891.4229.4374.3661.8179.2977.27
safeguard(x0.6) attack89.0189.2621.4423.1212.4828.6674.46
safeguard(x0.7) attack91.2492.0821.6119.9515.1724.5233.36
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack68.2767.956.65.815.056.110.87
sign-flipping attack68.0268.082.1310.1910.9328.342.59
label-flipping attack69.4368.85.3451.8552.1351.6667.86
delayed-gradient attack67.0366.424.0436.3431.4344.8536.6
safeguard(x0.6) attack59.87602.011.91.251.725.02
safeguard(x0.7) attack59.8464.912.071.971.321.553.31
", + "type": "table", + "image_path": "13e28cf13d108a2ccad363e9bf3180cb61e26e5dd371738eb440bbe3697cfe15.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 144, + 534, + 466, + 594.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 144, + 594.0, + 466, + 654.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 144, + 654.0, + 466, + 714.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 135, + 718, + 471, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 717, + 473, + 730 + ], + "spans": [ + { + "bbox": [ + 135, + 717, + 473, + 730 + ], + "score": 1.0, + "content": "Table 1: Table of test accuracy performances for CIFAR-10 (above) and CIFAR-100 (below).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 505, + 134 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Re-scale factor 0.6. In Figure 7, the performance of our (single and double) safeguard algorithms indeed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "get hurt a bit. Recall in Figure 7 the re-scale factor 0.6 is chosen to maximally impact our algorithm. The test", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 181, + 115 + ], + "score": 1.0, + "content": "accuracy drops from", + "type": "text" + }, + { + "bbox": [ + 181, + 103, + 206, + 113 + ], + "score": 0.92, + "content": "9 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 102, + 215, + 115 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 216, + 103, + 240, + 113 + ], + "score": 0.88, + "content": "8 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 102, + 361, + 115 + ], + "score": 1.0, + "content": "under CIFAR-10; and drops from", + "type": "text" + }, + { + "bbox": [ + 362, + 103, + 387, + 113 + ], + "score": 0.9, + "content": "6 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 102, + 396, + 115 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 397, + 103, + 422, + 113 + ], + "score": 0.89, + "content": "6 \\mathrm { { 0 . 0 \\% } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "under CIFAR-100 (for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 124 + ], + "score": 1.0, + "content": "both single and double safeguards). In these cases, we confirm that both versions of the safeguard algorithms", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 123, + 432, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 432, + 135 + ], + "score": 1.0, + "content": "did not catch any bad worker. However, this still significantly outperforms all prior works.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 135 + ] + }, + { + "type": "image", + "bbox": [ + 104, + 156, + 505, + 287 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 156, + 505, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 156, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 505, + 287 + ], + "score": 0.971, + "type": "image", + "image_path": "49b2f75c69622e5f6fd3ef4971d15626ae1482d4b1c61087e1ae4910caf31307.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 104, + 156, + 505, + 199.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 104, + 199.66666666666666, + 505, + 243.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 104, + 243.33333333333331, + 505, + 287.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 295, + 505, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "Figure 8: Performance comparison under the safeguard attack with re-scale factor 0.7. (In this case, our our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 304, + 415, + 317 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 415, + 317 + ], + "score": 1.0, + "content": "algorithm can catch some bad workers, and thus perform nearly optimally.)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "Re-scale factor 0.7. In Figure 8, we present the scenario when the re-scale factor is 0.7, so that the safeguard", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "algorithms can occasionally catch some bad workers (depending on the randomness and learning rate). We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 465, + 374 + ], + "score": 1.0, + "content": "confirm that in the three runs of single safeguard, it catches 1, 2, 3 bad workers for CIFAR-10, and", + "type": "text" + }, + { + "bbox": [ + 465, + 362, + 488, + 372 + ], + "score": 0.78, + "content": "1 , 0 , 0", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "bad", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 420, + 384 + ], + "score": 1.0, + "content": "workers for CIFAR-100 respectively; in the three runs of double safeguard, it catches", + "type": "text" + }, + { + "bbox": [ + 420, + 373, + 444, + 383 + ], + "score": 0.48, + "content": "1 , 2 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 372, + 505, + 384 + ], + "score": 1.0, + "content": "bad workers for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 380, + 339, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 339, + 394 + ], + "score": 1.0, + "content": "CIFAR-10, and 2, 2, 2 bad workers for CIFAR-100 respectively.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 342, + 505, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "Since there is a significant performance gain when our safeguard algorithms catch bad workers, this explains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "why safeguard algorithms in Figure 8 outperform their counterparts in Figure 7 with rescale factor 0.6. At the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "same time, we notice that the double safeguard algorithm has the ability to catch bad workers more easily. This", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 395, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 395, + 437 + ], + "score": 1.0, + "content": "is why double safeguard significantly outperforms single sageguard in Figure 8.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 395, + 505, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "In contrast, all other prior algorithms perform extremely bad under this attack. To some extent, safeguard attack", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "is even stronger than the previously proposed variance attack, since it can drag the 100-class test accuracy on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 303, + 471 + ], + "score": 1.0, + "content": "CIFAR-100 for all prior defense algorithms to nearly", + "type": "text" + }, + { + "bbox": [ + 303, + 459, + 316, + 469 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 458, + 505, + 471 + ], + "score": 1.0, + "content": ", while variance attack can only drag them down to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 469, + 155, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 134, + 480 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 134, + 469, + 151, + 479 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 469, + 155, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 439, + 505, + 480 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 489, + 248, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 249, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 249, + 501 + ], + "score": 1.0, + "content": "C.3 FULL COMPARISON TABLE", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 345, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 346, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 346, + 517 + ], + "score": 1.0, + "content": "We also include the full test accuracy comparison table in Table 1.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 505, + 346, + 517 + ] + }, + { + "type": "table", + "bbox": [ + 144, + 534, + 466, + 714 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 144, + 534, + 466, + 714 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 534, + 466, + 714 + ], + "spans": [ + { + "bbox": [ + 144, + 534, + 466, + 714 + ], + "score": 0.875, + "html": "
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack92.0291.7521.4322.0121.4722.433.42
sign-flipping attack91.9392.082259.6557.0370.8722.4
label-flipping attack92.3392.4431.9383.0783.1883.5291.66
delayed-gradient attack91.5891.4229.4374.3661.8179.2977.27
safeguard(x0.6) attack89.0189.2621.4423.1212.4828.6674.46
safeguard(x0.7) attack91.2492.0821.6119.9515.1724.5233.36
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack68.2767.956.65.815.056.110.87
sign-flipping attack68.0268.082.1310.1910.9328.342.59
label-flipping attack69.4368.85.3451.8552.1351.6667.86
delayed-gradient attack67.0366.424.0436.3431.4344.8536.6
safeguard(x0.6) attack59.87602.011.91.251.725.02
safeguard(x0.7) attack59.8464.912.071.971.321.553.31
", + "type": "table", + "image_path": "13e28cf13d108a2ccad363e9bf3180cb61e26e5dd371738eb440bbe3697cfe15.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 144, + 534, + 466, + 594.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 144, + 594.0, + 466, + 654.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 144, + 654.0, + 466, + 714.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 135, + 718, + 471, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 717, + 473, + 730 + ], + "spans": [ + { + "bbox": [ + 135, + 717, + 473, + 730 + ], + "score": 1.0, + "content": "Table 1: Table of test accuracy performances for CIFAR-10 (above) and CIFAR-100 (below).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 449, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 450, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 450, + 96 + ], + "score": 1.0, + "content": "C.4 ATTACK AGAINST THE CONVEX ALGORITHM OF ALISTARH ET AL. (2018)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 106, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "We now briefly describe an attack against this algorithm. The attack specifically leverages the fact that the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 254, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 254, + 120 + ], + "score": 1.0, + "content": "algorithm does not use sliding windows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "One can first run the vanilla SGD to compute “the maximum deviation per good worker” for the accumulation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 102, + 128, + 509, + 151 + ], + "spans": [ + { + "bbox": [ + 102, + 128, + 212, + 151 + ], + "score": 1.0, + "content": "vector used by the algorithm", + "type": "text" + }, + { + "bbox": [ + 213, + 132, + 248, + 145 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 0 } ^ { T } \\nabla _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 128, + 509, + 151 + ], + "score": 1.0, + "content": ". This maximum deviation is therefore a lower bound for the threshold", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "used in their algorithm. Next, we design an attacker who evenly distributes this total allowed deviation to e.g.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "5 consecutive epochs, and behaves honestly for the remaining epochs. Such an attacker cannot be identified by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "this algorithm, because its total deviation across all the iterations is identical to that of a good worker. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 245, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 245, + 185 + ], + "score": 1.0, + "content": "this leads the algorithm to divergence.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Specifically, suppose 4 Byzantine workers all maliciously report their stochastic gradients multiplied by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 129, + 209 + ], + "score": 1.0, + "content": "scalar", + "type": "text" + }, + { + "bbox": [ + 130, + 198, + 142, + 207 + ], + "score": 0.67, + "content": "^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 197, + 505, + 209 + ], + "score": 1.0, + "content": ", and the remaining 6 good workers report their true stochastic gradients. One can verify numerically", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 339, + 220 + ], + "score": 1.0, + "content": "that this attacker can run for 5 consecutive epochs (say, epochs", + "type": "text" + }, + { + "bbox": [ + 340, + 207, + 451, + 218 + ], + "score": 0.92, + "content": "a , a + 1 , a + 2 , a + 3 , a + 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 205, + 505, + 220 + ], + "score": 1.0, + "content": "without being", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 219, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 219, + 229 + ], + "score": 1.0, + "content": "caught by the algorithm. Now,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 112, + 230, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 110, + 229, + 427, + 243 + ], + "spans": [ + { + "bbox": [ + 110, + 229, + 128, + 243 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 128, + 230, + 156, + 241 + ], + "score": 0.88, + "content": "a \\leq 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 229, + 427, + 243 + ], + "score": 1.0, + "content": ", within just 1 epoch of attack, the neural net weights diverge (value NAN).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 110, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 110, + 241, + 128, + 255 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 128, + 243, + 182, + 253 + ], + "score": 0.9, + "content": "8 0 \\leq a \\leq 1 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 241, + 505, + 255 + ], + "score": 1.0, + "content": ", this attack is applied after the first learning rate decay. Within just 1 epoch of the attack,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 251, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 118, + 251, + 288, + 266 + ], + "score": 1.0, + "content": "the objective explodes and accuracy becomes", + "type": "text" + }, + { + "bbox": [ + 288, + 252, + 306, + 262 + ], + "score": 0.87, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 251, + 505, + 266 + ], + "score": 1.0, + "content": "(random), and within 3 epochs the algorithm diverges", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 119, + 263, + 164, + 274 + ], + "spans": [ + { + "bbox": [ + 119, + 263, + 164, + 274 + ], + "score": 1.0, + "content": "completely.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 111, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 111, + 273, + 129, + 286 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 129, + 275, + 190, + 285 + ], + "score": 0.89, + "content": "1 2 0 \\leq a \\leq 1 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ", this attack is after the second learning rate decay. Within just 2 epochs of attack, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 119, + 284, + 378, + 295 + ], + "spans": [ + { + "bbox": [ + 119, + 284, + 186, + 295 + ], + "score": 1.0, + "content": "accuracy drops to", + "type": "text" + }, + { + "bbox": [ + 186, + 284, + 204, + 294 + ], + "score": 0.89, + "content": "1 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 284, + 356, + 295 + ], + "score": 1.0, + "content": ". Later, the accuracy never recovers above", + "type": "text" + }, + { + "bbox": [ + 356, + 284, + 374, + 294 + ], + "score": 0.89, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 284, + 378, + 295 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 449, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 450, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 450, + 96 + ], + "score": 1.0, + "content": "C.4 ATTACK AGAINST THE CONVEX ALGORITHM OF ALISTARH ET AL. (2018)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 106, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "We now briefly describe an attack against this algorithm. The attack specifically leverages the fact that the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 254, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 254, + 120 + ], + "score": 1.0, + "content": "algorithm does not use sliding windows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 97, + 505, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "One can first run the vanilla SGD to compute “the maximum deviation per good worker” for the accumulation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 102, + 128, + 509, + 151 + ], + "spans": [ + { + "bbox": [ + 102, + 128, + 212, + 151 + ], + "score": 1.0, + "content": "vector used by the algorithm", + "type": "text" + }, + { + "bbox": [ + 213, + 132, + 248, + 145 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 0 } ^ { T } \\nabla _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 128, + 509, + 151 + ], + "score": 1.0, + "content": ". This maximum deviation is therefore a lower bound for the threshold", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "used in their algorithm. Next, we design an attacker who evenly distributes this total allowed deviation to e.g.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "5 consecutive epochs, and behaves honestly for the remaining epochs. Such an attacker cannot be identified by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "this algorithm, because its total deviation across all the iterations is identical to that of a good worker. However,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 245, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 245, + 185 + ], + "score": 1.0, + "content": "this leads the algorithm to divergence.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 102, + 122, + 509, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Specifically, suppose 4 Byzantine workers all maliciously report their stochastic gradients multiplied by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 129, + 209 + ], + "score": 1.0, + "content": "scalar", + "type": "text" + }, + { + "bbox": [ + 130, + 198, + 142, + 207 + ], + "score": 0.67, + "content": "^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 197, + 505, + 209 + ], + "score": 1.0, + "content": ", and the remaining 6 good workers report their true stochastic gradients. One can verify numerically", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 339, + 220 + ], + "score": 1.0, + "content": "that this attacker can run for 5 consecutive epochs (say, epochs", + "type": "text" + }, + { + "bbox": [ + 340, + 207, + 451, + 218 + ], + "score": 0.92, + "content": "a , a + 1 , a + 2 , a + 3 , a + 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 205, + 505, + 220 + ], + "score": 1.0, + "content": "without being", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 219, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 219, + 229 + ], + "score": 1.0, + "content": "caught by the algorithm. Now,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 187, + 505, + 229 + ] + }, + { + "type": "list", + "bbox": [ + 112, + 230, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 110, + 229, + 427, + 243 + ], + "spans": [ + { + "bbox": [ + 110, + 229, + 128, + 243 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 128, + 230, + 156, + 241 + ], + "score": 0.88, + "content": "a \\leq 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 229, + 427, + 243 + ], + "score": 1.0, + "content": ", within just 1 epoch of attack, the neural net weights diverge (value NAN).", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 110, + 241, + 128, + 255 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 128, + 243, + 182, + 253 + ], + "score": 0.9, + "content": "8 0 \\leq a \\leq 1 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 241, + 505, + 255 + ], + "score": 1.0, + "content": ", this attack is applied after the first learning rate decay. Within just 1 epoch of the attack,", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 251, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 118, + 251, + 288, + 266 + ], + "score": 1.0, + "content": "the objective explodes and accuracy becomes", + "type": "text" + }, + { + "bbox": [ + 288, + 252, + 306, + 262 + ], + "score": 0.87, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 251, + 505, + 266 + ], + "score": 1.0, + "content": "(random), and within 3 epochs the algorithm diverges", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 119, + 263, + 164, + 274 + ], + "spans": [ + { + "bbox": [ + 119, + 263, + 164, + 274 + ], + "score": 1.0, + "content": "completely.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 111, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 111, + 273, + 129, + 286 + ], + "score": 1.0, + "content": "• if", + "type": "text" + }, + { + "bbox": [ + 129, + 275, + 190, + 285 + ], + "score": 0.89, + "content": "1 2 0 \\leq a \\leq 1 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ", this attack is after the second learning rate decay. Within just 2 epochs of attack, the", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 119, + 284, + 378, + 295 + ], + "spans": [ + { + "bbox": [ + 119, + 284, + 186, + 295 + ], + "score": 1.0, + "content": "accuracy drops to", + "type": "text" + }, + { + "bbox": [ + 186, + 284, + 204, + 294 + ], + "score": 0.89, + "content": "1 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 284, + 356, + 295 + ], + "score": 1.0, + "content": ". 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1:good←[m];Input:point xo ∈Rd,rate η >0,lengths T≥T1 ≥T ≥1,threshold ∑1 >o >O;
2:fort←0toT-1do
3:last1 ←max{t1 ∈[t]: t1 is a multiple ofTi};
4:lasto ←max{to ∈[t]: to is amultiple ofTo}
5:for each i ∈ goodt do
6:receive Vt,i ∈Rd from machine i;Vki
7:Ai←∑=last1 Igoodl ki tlgood
8:Amed ← Ai where i∈ goodt is any machine s.t.|{j ∈ goodt: ||Aj - Aill≤∑1}| > m/2.
9:Bmed ← Bi where i∈ goodt is any machine s.t.|{j ∈ goodt: |/Bj -Bill ≤ o}| > m/2.
10:g0odt+1←{i∈ goodt: |Ai-Amedll≤2£1∧Bi-Bmedll≤20} ;
11:xt+1 = xt-n(st + [god∑iEgoodVt,i);Gaussian noise t ~ N(O,v²I)
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\\cdot + \\xi _ { T _ { 1 } ^ { \\mathtt { a } } - 1 } ^ { \\mathtt { a } } \\| ^ { 2 } } } \\\\ { { \\displaystyle \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } } \\\\ { { \\displaystyle \\le O ( \\eta ^ { 2 } T _ { 1 } ) \\sum _ { t = 0 } ^ { T _ { 1 } ^ { \\mathtt { a } } - 1 } \\| \\nabla f ( w _ { t } ^ { \\mathtt { a } } ) \\| ^ { 2 } + O ( C _ { 2 } \\eta ^ { 2 } T _ { 1 } ) + O ( C _ { 1 } \\eta ^ { 2 } \\nu ^ { 2 } T _ { 1 } d ) } } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 1077, + 301, + 1394, + 301, + 1394, + 342, + 1077, + 342 + ], + "score": 0.92, + "latex": "\\begin{array} { r } { C _ { 2 } = \\alpha ^ { 2 } \\log \\frac { m T _ { 1 } } { p } + \\frac { \\log ( T _ { 1 } / p ) } { m } } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 627, + 339, + 826, + 339, + 826, + 379, + 627, + 379 + 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single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack92.0291.7521.4322.0121.4722.433.42
sign-flipping attack91.9392.082259.6557.0370.8722.4
label-flipping attack92.3392.4431.9383.0783.1883.5291.66
delayed-gradient attack91.5891.4229.4374.3661.8179.2977.27
safeguard(x0.6) attack89.0189.2621.4423.1212.4828.6674.46
safeguard(x0.7) attack91.2492.0821.6119.9515.1724.5233.36
single safeguarddouble safeguardcoord mediangeo medianKrumKrum (3 faulty nodes)Zeno
variance attack68.2767.956.65.815.056.110.87
sign-flipping attack68.0268.082.1310.1910.9328.342.59
label-flipping attack69.4368.85.3451.8552.1351.6667.86
delayed-gradient attack67.0366.424.0436.3431.4344.8536.6
safeguard(x0.6) attack59.87602.011.91.251.725.02
safeguard(x0.7) attack59.8464.912.071.971.321.553.31
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file mode 100644 index 0000000000000000000000000000000000000000..c45adcc20f3354abffbb45b45a23f2104a0e68b4 --- /dev/null +++ b/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. + +
TaskRegularizationData augmentationNo data augmentation
lift aloneTRAIL-0 TRAIL~165~115
lift distractedTRAIL-0~155~165
TRAIL~30 ~180~5 ~10
+ +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. 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Leveraging demonstrations for deep reinforcement learning on robotics problems with sparse rewards. CoRR, 2017. + +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. + +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. + +# 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 + +
Feature NameDimensions
frontal camera base force and torque sensors64×64×3 6
arm joints position6
arm joints velocity6
wrist force and torque sensors6
hand finger joints position3
hand finger joints velocity
hand fingertip sensors3
3
grip site position3
pinch site position3
+ +Table 2: Observation and dimensions. + +
b) Sawyer
Feature Name frontleft cameraDimensions 64×64×3
front right camera64×64×3
arm joint position7
arm joint velocity7
wrist force sensor3
wrist torque sensor3
hand grasp sensor
1
hand joint position1
tool center point cartesian orientation9
tool center point cartesian position
hand joint velocity3 1
+ +# 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. + +
ParametersValues
Actor/Critic Input Width64
Actor/Critic Input Height64
D4PG Parameters
Vmin-50
Vmax150
Vbins21
N step1
Actor learning rate10-4
Critic learning rate10-4
OptimizerAdam (Kingma & Ba (2014))
Batch size256
Target update period100
Discount factor (y)0.99
Replay capacity106
Number of actors32 or 128
Imitation Parameters
Discriminator learning rate10-4
Discriminator Input Width48
Discriminator Input Height48
+ +# 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. \ No newline at end of file diff --git a/parse/train/S1x2PCNKDB/S1x2PCNKDB_content_list.json b/parse/train/S1x2PCNKDB/S1x2PCNKDB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..28f7192070ac0d136fafa70a67728fe27704f1e9 --- /dev/null +++ b/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, + 825, + 511 + ], + "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, + 299, + 825, + 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": [ + 173, + 406, + 825, + 505 + ], + "page_idx": 2 + }, + { + "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: ", + "bbox": [ + 174, + 511, + 825, + 594 + ], + "page_idx": 2 + }, + { + "type": "equation", + "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": [ + 241, + 609, + 753, + 632 + ], + "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: $R ( s , a ) = - \\log ( 1 - D ( s , a ) )$ . ", + "bbox": [ + 174, + 647, + 825, + 676 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 683, + 825, + 739 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 TASK-RELEVANT ADVERSARIAL IMITATION LEARNING (TRAIL) ", + "text_level": 1, + "bbox": [ + 171, + 767, + 750, + 785 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 804, + 825, + 875 + ], + "page_idx": 2 + }, + { + "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: ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 686, + 118 + ], + "page_idx": 3 + }, + { + "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": [ + 254, + 123, + 743, + 183 + ], + "page_idx": 3 + }, + { + "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: ", + "bbox": [ + 176, + 194, + 823, + 236 + ], + "page_idx": 3 + }, + { + "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": [ + 194, + 241, + 802, + 299 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $a c c u r a c y ( \\cdot , \\cdot )$ is defined as the average of discriminator accuracies: ", + "bbox": [ + 171, + 303, + 661, + 319 + ], + "page_idx": 3 + }, + { + "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": [ + 297, + 324, + 699, + 368 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The scalar $\\lambda \\geq 0$ is a tunable hyperparameter. ", + "bbox": [ + 174, + 373, + 473, + 388 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 395, + 825, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 THE SELECTION OF THE INVARIANT SET ", + "text_level": 1, + "bbox": [ + 174, + 454, + 493, + 469 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 481, + 825, + 536 + ], + "page_idx": 3 + }, + { + "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": [ + 174, + 542, + 825, + 627 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 176, + 633, + 825, + 676 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 696, + 326, + 712 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 728, + 825, + 799 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 804, + 823, + 861 + ], + "page_idx": 3 + }, + { + "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. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 166, + 825, + 236 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 243, + 825, + 285 + ], + "page_idx": 4 + }, + { + "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 $T _ { p a t i e n c e } = 1 0$ ). ", + "bbox": [ + 174, + 292, + 825, + 391 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 397, + 825, + 439 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 BLOCK LIFTING WITH DISTRACTORS ", + "text_level": 1, + "bbox": [ + 176, + 457, + 467, + 470 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 482, + 826, + 525 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 531, + 825, + 587 + ], + "page_idx": 4 + }, + { + "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). ", + "bbox": [ + 173, + 594, + 825, + 707 + ], + "page_idx": 4 + }, + { + "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": [], + "bbox": [ + 191, + 715, + 799, + 835 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 180, + 825, + 263 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 176, + 271, + 825, + 313 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CONSTRUCTING THE INVARIANT SET $\\mathcal { T }$ ", + "text_level": 1, + "bbox": [ + 178, + 330, + 444, + 343 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 354, + 826, + 453 + ], + "page_idx": 5 + }, + { + "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": [ + 204, + 468, + 782, + 578 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 176, + 630, + 821, + 671 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 678, + 825, + 762 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 770, + 825, + 825 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 ABLATION STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 844, + 352, + 858 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "MEASURING DISCRIMINATOR MEMORIZATION OF TASK-IRRELEVANT FEATURES ", + "text_level": 1, + "bbox": [ + 174, + 871, + 723, + 885 + ], + "page_idx": 5 + }, + { + "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 ", + "bbox": [ + 173, + 895, + 821, + 922 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 6 + }, + { + "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": [ + 174, + 194, + 825, + 263 + ], + "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": [ + 173, + 271, + 823, + 300 + ], + "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, + 513 + ], + "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, + 614, + 825, + 684 + ], + "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, + 825, + 819 + ], + "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, + 825, + 825, + 924 + ], + "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": [ + 174, + 292, + 826, + 335 + ], + "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": "
TaskRegularizationData augmentationNo data augmentation
lift aloneTRAIL-0 TRAIL~165~115
lift distractedTRAIL-0~155~165
TRAIL~30 ~180~5 ~10
", + "bbox": [ + 253, + 354, + 741, + 440 + ], + "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": [ + 173, + 481, + 825, + 508 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "LEARNING WITH A FIXED, PERFECT DISCRIMINATOR ", + "text_level": 1, + "bbox": [ + 173, + 532, + 539, + 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": [ + 174, + 558, + 825, + 643 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 LEARNING FROM OTHER EMBODIMENTS AND PROPS ", + "text_level": 1, + "bbox": [ + 173, + 667, + 571, + 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": [ + 174, + 695, + 826, + 751 + ], + "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, + 552, + 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": [ + 173, + 130, + 826, + 213 + ], + "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, + 227, + 802, + 343 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 174, + 393, + 328, + 409 + ], + "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": [ + 173, + 425, + 826, + 494 + ], + "page_idx": 8 + }, + { + "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": [ + 176, + 501, + 823, + 544 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 571, + 285, + 587 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Pieter Abbeel and Andrew Y Ng. Apprenticeship learning via inverse reinforcement learning. 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", + "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, + 517, + 146 + ], + "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, + 157, + 825, + 200 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/6bccf9982d118787a6c2979868292057dd3b8d908253da25a031063cb76c20dc.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 205, + 212, + 789, + 324 + ], + "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, + 825, + 367 + ], + "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, + 825, + 436 + ], + "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, + 784, + 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, + 825, + 790 + ], + "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, + 796, + 823, + 839 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/07b4e70c00c60a1551189151f9ee412f3336bc9f65211675aee4c92bca3c9287.jpg", + "table_caption": [ + "a) Jaco " + ], + "table_footnote": [], + "table_body": "
Feature NameDimensions
frontal camera base force and torque sensors64×64×3 6
arm joints position6
arm joints velocity6
wrist force and torque sensors6
hand finger joints position3
hand finger joints velocity
hand fingertip sensors3
3
grip site position3
pinch site position3
", + "bbox": [ + 169, + 133, + 460, + 281 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/35828ddf5a5369eb1bb42ba00067e3cdf43e97fe28c656aea544c66d4b4046ca.jpg", + "table_caption": [ + "Table 2: Observation and dimensions. " + ], + "table_footnote": [], + "table_body": "
b) Sawyer
Feature Name frontleft cameraDimensions 64×64×3
front right camera64×64×3
arm joint position7
arm joint velocity7
wrist force sensor3
wrist torque sensor3
hand grasp sensor
1
hand joint position1
tool center point cartesian orientation9
tool center point cartesian position
hand joint velocity3 1
", + "bbox": [ + 480, + 126, + 816, + 294 + ], + "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, + 825, + 390 + ], + "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, + 790, + 541 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.3 TRAIL-0 WITH random ", + "text_level": 1, + "bbox": [ + 176, + 582, + 382, + 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": [ + 174, + 867, + 826, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "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, + 823, + 443 + ], + "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, + 826, + 492 + ], + "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, + 818, + 655 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.6 D4PG ", + "text_level": 1, + "bbox": [ + 176, + 737, + 261, + 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": [ + 173, + 762, + 825, + 875 + ], + "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, + 881, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Given the $Q$ function, we can update the policy using DPG: ", + "bbox": [ + 173, + 103, + 565, + 119 + ], + "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, + 629, + 150 + ], + "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, + 170, + 825, + 233 + ], + "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, + 237, + 825, + 311 + ], + "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, + 376, + 826, + 462 + ], + "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": [ + 174, + 468, + 825, + 511 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Algorithm 1 Actor ", + "text_level": 1, + "bbox": [ + 174, + 534, + 302, + 547 + ], + "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": [ + 178, + 551, + 620, + 681 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Algorithm 2 Learner ", + "text_level": 1, + "bbox": [ + 174, + 729, + 315, + 743 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Given: ", + "text_level": 1, + "bbox": [ + 191, + 747, + 238, + 760 + ], + "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, + 542, + 805 + ], + "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": "
ParametersValues
Actor/Critic Input Width64
Actor/Critic Input Height64
D4PG Parameters
Vmin-50
Vmax150
Vbins21
N step1
Actor learning rate10-4
Critic learning rate10-4
OptimizerAdam (Kingma & Ba (2014))
Batch size256
Target update period100
Discount factor (y)0.99
Replay capacity106
Number of actors32 or 128
Imitation Parameters
Discriminator learning rate10-4
Discriminator Input Width48
Discriminator Input Height48
", + "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. 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Our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 251, + 470, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 251, + 470, + 264 + ], + "score": 1.0, + "content": "proposed solution, Task-Relevant Adversarial Imitation Learning (TRAIL), uses a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 263, + 470, + 274 + ], + "spans": [ + { + "bbox": [ + 142, + 263, + 470, + 274 + ], + "score": 1.0, + "content": "constrained optimization objective to overcome task-irrelevant features. Compre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 273, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 273, + 470, + 286 + ], + "score": 1.0, + "content": "hensive experiments show that TRAIL can solve challenging manipulation tasks", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 284, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 470, + 297 + ], + "score": 1.0, + "content": "from pixels by imitating human operators, where other agents such as behaviour", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "score": 1.0, + "content": "cloning (BC), standard GAIL, improved GAIL variants including our newly pro-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "score": 1.0, + "content": "posed baselines, and Deterministic Policy Gradients from Demonstrations (DPGfD)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 317, + 447, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 447, + 329 + ], + "score": 1.0, + "content": "fail to find solutions, even when the other agents have access to task reward.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 141, + 196, + 470, + 329 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 361, + 205, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 208, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 208, + 377 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "Generative Adversarial Networks (GANs) have produced breath-taking conditional image synthesis", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "results (Goodfellow et al., 2014; Brock et al., 2019), and have inspired adversarial learning approaches", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "to imitating behavior. In Generative Adversarial Imitation Learning (GAIL) (Ho & Ermon, 2016), a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 421, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 507, + 437 + ], + "score": 1.0, + "content": "discriminator network is trained to distinguish agent and expert behaviour through its observations,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "and is then used as a reward function. GAIL agents can overcome the exploration challenge by taking", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "advantage of expert demonstrations, while also achieving high asymptotic performance by learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 201, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 201, + 470 + ], + "score": 1.0, + "content": "from agent experience.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 390, + 507, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "Despite the huge promise of GAIL, it has not yet had the same impact as GANs; in particular, robust", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "GAIL from pixels for control applications remains a challenge. Here, we study a key shortcoming", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "of GAIL: the tendency of the discriminator to mainly exploit task-irrelevant features. For example,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "by focusing on slight background differences, a discriminator can achieve perfect generalization,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "assigning zero reward to all held-out agent observations. However, this discriminator does not yield", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 528, + 345, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 345, + 540 + ], + "score": 1.0, + "content": "an informative reward function because it ignores behavior.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 473, + 506, + 540 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 559, + 499, + 679 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 559, + 499, + 679 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 559, + 499, + 679 + ], + "spans": [ + { + "bbox": [ + 110, + 559, + 499, + 679 + ], + "score": 0.972, + "type": "image", + "image_path": "21a1b85e2a28f46c1e51c7d95c791c9593dbf312cc715586c7be17a650a74728.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 110, + 559, + 499, + 599.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 110, + 599.0, + 499, + 639.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 110, + 639.0, + 499, + 679.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 685, + 505, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "Figure 1: GAIL and TRAIL succeed at lifting (a), but when distractor objects are added, GAIL fails", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 697, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 697, + 505, + 708 + ], + "score": 1.0, + "content": "while TRAIL succeeds (b). Due to robustness to initial conditions, TRAIL can stack from pixels while", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 720 + ], + "score": 1.0, + "content": "standard GAIL fails (c). We witness this difference again in insertion with distractors (d). A video", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 718, + 497, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 497, + 731 + ], + "score": 1.0, + "content": "showing agents performing these tasks can be seen at https://youtu.be/Rz5G15rDKcg .", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + } + ], + "index": 32.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 127, + 65, + 483, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 127, + 65, + 483, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 65, + 483, + 169 + ], + "spans": [ + { + "bbox": [ + 127, + 65, + 483, + 169 + ], + "score": 0.963, + "type": "image", + "image_path": "862945eb66675f4de9e3964473bbe1038dacc00ee10aed40f2139ea575ebaafb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 127, + 65, + 483, + 99.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 127, + 99.66666666666666, + 483, + 134.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 134.33333333333331, + 483, + 168.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 177, + 506, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "Figure 2: Illustration of several task-irrelevant changes between the expert demonstrations and the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "distribution of agent observations, for the lift (red cube) task. The naively-trained discriminator", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 199, + 488, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 488, + 212 + ], + "score": 1.0, + "content": "network will use these differences rather than task performance to distinguish agent and expert.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 108, + 225, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 245, + 237 + ], + "score": 1.0, + "content": "Assuming there is an expert policy", + "type": "text" + }, + { + "bbox": [ + 245, + 227, + 259, + 236 + ], + "score": 0.86, + "content": "\\pi _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that is optimal for an unknown reward function, here we refer", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "score": 1.0, + "content": "to a feature as task-irrelevant if it does not affect that reward. For example, if the task is to lift a red", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 247, + 432, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 432, + 259 + ], + "score": 1.0, + "content": "block, the positions of other blocks would be task-irrelevant; see Figures 1 and 2.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 290, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 293, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 293, + 277 + ], + "score": 1.0, + "content": "This paper makes the following contributions:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 129, + 284, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 130, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 130, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "1. It reveals a fundamental limitation of GAIL by showing that discriminators do in practice", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 295, + 458, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 458, + 308 + ], + "score": 1.0, + "content": "exploit task-irrelevant information, thereby resulting in poor task performance.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 129, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 129, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "2. It introduces powerful GAIL baselines. In particular, it shows that standard regularization", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 321, + 455, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 321, + 455, + 333 + ], + "score": 1.0, + "content": "and data augmentation are generally useful and improve upon standard GAIL.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 128, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 128, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "3. It shows that these improvements to GAIL, as well as other improvements proposed by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 505, + 360 + ], + "score": 1.0, + "content": "Reed et al. (2018), do not completely solve the problem, allowing GAIL agents to fail", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 357, + 395, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 395, + 369 + ], + "score": 1.0, + "content": "catastrophically with the addition of task-irrelevant distractors.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 129, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 129, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "4. It introduces Task Relevant Adversarial Imitation Learning (TRAIL), using constrained", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 142, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "optimization to force the discriminator to focus on the relevant aspects of the task, which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 393, + 485, + 407 + ], + "spans": [ + { + "bbox": [ + 141, + 393, + 485, + 407 + ], + "score": 1.0, + "content": "improves performance dramatically on manipulation tasks from pixels (see Figure 1).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 420, + 209, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 211, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 211, + 434 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "The use of demonstrations to help agent training has been studied extensively in robotics (Bakker", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "& Kuniyoshi, 1996; Kawato et al., 1994; Miyamoto et al., 1996) with approaches ranging from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 466, + 401, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 401, + 479 + ], + "score": 1.0, + "content": "Q-learning (Schaal, 1997) to behavioral cloning (BC) (Pomerleau, 1989).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "BC: BC is effective in solving many control problems (Pomerleau, 1989; Finn et al., 2017; Duan", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "et al., 2017; Rahmatizadeh et al., 2018). It has also been successfully applied to initialize RL training", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "(Rajeswaran et al., 2017). It, however, suffers from compounding errors as initial small deviations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "from the expert behaviors tend to cause bigger differences (Ross et al., 2011). This often necessitates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "a large number of demonstrations for satisfactory performance. Furthermore, BC typically does not", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 538, + 323, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 323, + 550 + ], + "score": 1.0, + "content": "lead to agents that are superior to their demonstrators.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 238, + 568 + ], + "score": 1.0, + "content": "Inverse RL: Ziebart et al. (2008);", + "type": "text" + }, + { + "bbox": [ + 239, + 555, + 253, + 567 + ], + "score": 0.43, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 554, + 343, + 568 + ], + "score": 1.0, + "content": "et al. (2000); Abbeel &", + "type": "text" + }, + { + "bbox": [ + 343, + 555, + 385, + 567 + ], + "score": 0.33, + "content": "\\mathrm { N g } \\left( 2 0 0 4 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "propose inverse reinforcement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "learning (IRL) as a way of learning reward functions from demonstrations. Reinforcement learning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "can then be used to optimize that learned reward. Recently, Finn et al. (2016b) approached continuous", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 586, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 603 + ], + "score": 1.0, + "content": "robotic control problems with success by applying Maximum Entropy IRL algorithms which are very", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 599, + 397, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 397, + 612 + ], + "score": 1.0, + "content": "closely related to GAIL (Finn et al., 2016a) and have similar drawbacks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 507, + 629 + ], + "score": 1.0, + "content": "Learning from Demonstrations: Hester et al. (2018) developed deep Q-Learning from demonstra-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "tion (DQfD), in which expert trajectories are added to experience replay and jointly used to train", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "agents along with their own experiences. This was later extended by Vecerik et al. (2017) and Pohlen", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "et al. (2018) to better handle sparse-reward problems in control and Atari games respectively. 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Peng et al. (2018) introduces the Variational Bottleneck", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "to regularize the discriminator. Reed et al. (2018) proposes to not train the vision module of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "discriminator (e.g. use the vision module of the critic network instead) and only train a tiny network", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "on top of the vision module to discriminate. Blondé & Kalousis (2018) follow a similar approach.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "Unstructured regularization, however, cannot stop the discriminator from fitting to features that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "systematically different between the agents’ and the demonstration’s behavior, like those illustrated", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 465, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 465, + 172 + ], + "score": 1.0, + "content": "in Figure 2. TRAIL, on the other hand, is much less prone to overfitting to these features.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Stadie et al. (2017) extend GAIL to the setting of third person imitation, in which the demonstrator and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "agent observations come from different views. To prevent the discriminator from discriminating based", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "on viewpoint domain, they use gradient flipping from an auxiliary classifier to learn domain-invariant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "features. Our approach is not to learn domain-invariant features, but instead learn domain-agnostic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 279, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 279, + 233 + ], + "score": 1.0, + "content": "discriminators that only focus on behavior.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "Several recent works have focused on improving the sample efficiency of GAIL (Blondé & Kalousis,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "2018; Sasaki et al., 2018). Common to these approaches and to this work, is the use of off-policy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 471, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 471, + 272 + ], + "score": 1.0, + "content": "actor critic agents and experience replay, to improve the utilization of available experience.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 434, + 306 + ], + "lines": [ + { + "bbox": [ + 104, + 292, + 437, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 292, + 437, + 308 + ], + "score": 1.0, + "content": "3 REINFORCEMENT LEARNING AND ADVERSARIAL IMITATION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 322, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "Following the notation of Sutton & Barto (2018), a Markov Decision Process (MDP) is a tuple", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 167, + 345 + ], + "score": 0.91, + "content": "( \\mathcal { S } , \\mathcal { A } , R , \\mathbf { \\bar { P } } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 333, + 213, + 347 + ], + "score": 1.0, + "content": "with states", + "type": "text" + }, + { + "bbox": [ + 214, + 334, + 222, + 344 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 333, + 257, + 347 + ], + "score": 1.0, + "content": ", actions", + "type": "text" + }, + { + "bbox": [ + 257, + 334, + 266, + 343 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 333, + 336, + 347 + ], + "score": 1.0, + "content": ", reward function", + "type": "text" + }, + { + "bbox": [ + 337, + 334, + 367, + 345 + ], + "score": 0.93, + "content": "R ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 333, + 461, + 347 + ], + "score": 1.0, + "content": ", transition distribution", + "type": "text" + }, + { + "bbox": [ + 461, + 334, + 502, + 345 + ], + "score": 0.92, + "content": "P ( s ^ { \\prime } | s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 333, + 506, + 347 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 160, + 358 + ], + "score": 1.0, + "content": "and discount", + "type": "text" + }, + { + "bbox": [ + 160, + 347, + 168, + 356 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 344, + 243, + 358 + ], + "score": 1.0, + "content": ". An agent in state", + "type": "text" + }, + { + "bbox": [ + 243, + 345, + 268, + 355 + ], + "score": 0.9, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 344, + 320, + 358 + ], + "score": 1.0, + "content": "takes action", + "type": "text" + }, + { + "bbox": [ + 320, + 345, + 346, + 355 + ], + "score": 0.9, + "content": "a \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 344, + 440, + 358 + ], + "score": 1.0, + "content": "according to its policy", + "type": "text" + }, + { + "bbox": [ + 440, + 347, + 448, + 354 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "and moves to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 128, + 369 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 128, + 355, + 156, + 366 + ], + "score": 0.91, + "content": "s ^ { \\prime } \\in \\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "according to the transition distribution. The goal of RL algorithms is to find a policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "that maximizes the expected sum of discounted rewards, represented by the action value function", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 246, + 390 + ], + "score": 0.91, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 375, + 275, + 394 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 276, + 378, + 289, + 388 + ], + "score": 0.91, + "content": "\\mathbb { E } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 375, + 475, + 394 + ], + "score": 1.0, + "content": "is an expectation over trajectories starting from", + "type": "text" + }, + { + "bbox": [ + 476, + 379, + 504, + 388 + ], + "score": 0.86, + "content": "s _ { 0 } = s", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 353, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 177, + 402 + ], + "score": 1.0, + "content": "and taking action", + "type": "text" + }, + { + "bbox": [ + 178, + 391, + 207, + 399 + ], + "score": 0.89, + "content": "a _ { 0 } = a", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 388, + 341, + 402 + ], + "score": 1.0, + "content": "and thereafter running the policy", + "type": "text" + }, + { + "bbox": [ + 342, + 391, + 348, + 398 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 388, + 353, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "To apply RL, it is essential that we have access to the reward function which is often hard to design", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and evaluate (Singh et al., 2019). In addition, sparse rewards can cause exploration difficulties that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "pose great challenges to RL algorithms. We therefore look to imitation learning and particularly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "GAIL to derive a reward function from expert demonstrations. In GAIL, a reward function is learned", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 246, + 461 + ], + "score": 1.0, + "content": "by training a discriminator network", + "type": "text" + }, + { + "bbox": [ + 246, + 449, + 278, + 461 + ], + "score": 0.93, + "content": "D ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "to distinguish between agent and expert state-action pairs.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 310, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 310, + 473 + ], + "score": 1.0, + "content": "The GAIL objective is thus formulated as follows:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 483, + 461, + 501 + ], + "lines": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "spans": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "score": 0.85, + "content": "\\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 ) ,", + "type": "interline_equation", + "image_path": "108a841f22d52fb14679a7fca453346693df0d23802f2235bfac6b83e1870dfa.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 133, + 527 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 516, + 141, + 523 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 512, + 220, + 527 + ], + "score": 1.0, + "content": "is the agent policy,", + "type": "text" + }, + { + "bbox": [ + 220, + 516, + 234, + 524 + ], + "score": 0.85, + "content": "\\pi _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 512, + 325, + 527 + ], + "score": 1.0, + "content": "the expert policy, and", + "type": "text" + }, + { + "bbox": [ + 325, + 514, + 348, + 525 + ], + "score": 0.91, + "content": "H ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 512, + 505, + 527 + ], + "score": 1.0, + "content": "an (optional) entropy regularizer. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 372, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 246, + 537 + ], + "score": 1.0, + "content": "reward function is defined simply:", + "type": "text" + }, + { + "bbox": [ + 246, + 524, + 367, + 537 + ], + "score": 0.91, + "content": "R ( s , a ) = - \\log ( 1 - D ( s , a ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 523, + 372, + 537 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "GAIL is theoretically appealing and practically simple. The discriminator, however, can focus on any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "features to discriminate, whether these features are task-relevant or not. In the next subsection we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "describe a way to constrain the discriminator network in order to prevent it from using task-irrelevant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 574, + 281, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 281, + 587 + ], + "score": 1.0, + "content": "details to distinguish agent and expert data.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 105, + 608, + 459, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 607, + 460, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 460, + 624 + ], + "score": 1.0, + "content": "4 TASK-RELEVANT ADVERSARIAL IMITATION LEARNING (TRAIL)", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "We want the discriminator to focus on task-relevant features. Our proposed solution, TRAIL, prevents", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "the discriminator from being able to distinguish expert and agent behaviour based on selected aspects", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "of the data. For instance, the discriminator should distinguish agent and expert frames only when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "meaningful behavior is present in those frames. In the absence of behavior useful to solve the task,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 681, + 490, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 490, + 694 + ], + "score": 1.0, + "content": "e.g. in initial frames prior to the execution of the behavior, the discriminator should be agnostic.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "To this end, we propose a constraint for the discriminator, such that its accuracy must not be greater", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 298, + 722 + ], + "score": 1.0, + "content": "than chance on a particular set of observations", + "type": "text" + }, + { + "bbox": [ + 298, + 712, + 305, + 720 + ], + "score": 0.87, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 709, + 444, + 722 + ], + "score": 1.0, + "content": "that we will call the invariant set.", + "type": "text" + }, + { + "bbox": [ + 445, + 710, + 453, + 720 + ], + "score": 0.66, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "will include", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "only observations that can be distinguished as agent or expert in task-irrelevant ways. Precisely, we", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "A few papers (Peng et al., 2018; Reed et al., 2018; Blondé & Kalousis, 2018) seek to address the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "problem of overfitting the discriminator. Peng et al. (2018) introduces the Variational Bottleneck", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "to regularize the discriminator. Reed et al. (2018) proposes to not train the vision module of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "discriminator (e.g. use the vision module of the critic network instead) and only train a tiny network", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "on top of the vision module to discriminate. Blondé & Kalousis (2018) follow a similar approach.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "Unstructured regularization, however, cannot stop the discriminator from fitting to features that are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "systematically different between the agents’ and the demonstration’s behavior, like those illustrated", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 465, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 465, + 172 + ], + "score": 1.0, + "content": "in Figure 2. TRAIL, on the other hand, is much less prone to overfitting to these features.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Stadie et al. (2017) extend GAIL to the setting of third person imitation, in which the demonstrator and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "agent observations come from different views. To prevent the discriminator from discriminating based", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "on viewpoint domain, they use gradient flipping from an auxiliary classifier to learn domain-invariant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "features. Our approach is not to learn domain-invariant features, but instead learn domain-agnostic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 279, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 279, + 233 + ], + "score": 1.0, + "content": "discriminators that only focus on behavior.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 176, + 506, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "Several recent works have focused on improving the sample efficiency of GAIL (Blondé & Kalousis,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 261 + ], + "score": 1.0, + "content": "2018; Sasaki et al., 2018). Common to these approaches and to this work, is the use of off-policy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 471, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 471, + 272 + ], + "score": 1.0, + "content": "actor critic agents and experience replay, to improve the utilization of available experience.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 237, + 506, + 272 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 434, + 306 + ], + "lines": [ + { + "bbox": [ + 104, + 292, + 437, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 292, + 437, + 308 + ], + "score": 1.0, + "content": "3 REINFORCEMENT LEARNING AND ADVERSARIAL IMITATION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 322, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "Following the notation of Sutton & Barto (2018), a Markov Decision Process (MDP) is a tuple", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 167, + 345 + ], + "score": 0.91, + "content": "( \\mathcal { S } , \\mathcal { A } , R , \\mathbf { \\bar { P } } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 333, + 213, + 347 + ], + "score": 1.0, + "content": "with states", + "type": "text" + }, + { + "bbox": [ + 214, + 334, + 222, + 344 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 333, + 257, + 347 + ], + "score": 1.0, + "content": ", actions", + "type": "text" + }, + { + "bbox": [ + 257, + 334, + 266, + 343 + ], + "score": 0.81, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 333, + 336, + 347 + ], + "score": 1.0, + "content": ", reward function", + "type": "text" + }, + { + "bbox": [ + 337, + 334, + 367, + 345 + ], + "score": 0.93, + "content": "R ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 333, + 461, + 347 + ], + "score": 1.0, + "content": ", transition distribution", + "type": "text" + }, + { + "bbox": [ + 461, + 334, + 502, + 345 + ], + "score": 0.92, + "content": "P ( s ^ { \\prime } | s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 333, + 506, + 347 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 160, + 358 + ], + "score": 1.0, + "content": "and discount", + "type": "text" + }, + { + "bbox": [ + 160, + 347, + 168, + 356 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 344, + 243, + 358 + ], + "score": 1.0, + "content": ". An agent in state", + "type": "text" + }, + { + "bbox": [ + 243, + 345, + 268, + 355 + ], + "score": 0.9, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 344, + 320, + 358 + ], + "score": 1.0, + "content": "takes action", + "type": "text" + }, + { + "bbox": [ + 320, + 345, + 346, + 355 + ], + "score": 0.9, + "content": "a \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 344, + 440, + 358 + ], + "score": 1.0, + "content": "according to its policy", + "type": "text" + }, + { + "bbox": [ + 440, + 347, + 448, + 354 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "and moves to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 128, + 369 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 128, + 355, + 156, + 366 + ], + "score": 0.91, + "content": "s ^ { \\prime } \\in \\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "according to the transition distribution. The goal of RL algorithms is to find a policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "that maximizes the expected sum of discounted rewards, represented by the action value function", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 375, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 246, + 390 + ], + "score": 0.91, + "content": "\\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}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 375, + 275, + 394 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 276, + 378, + 289, + 388 + ], + "score": 0.91, + "content": "\\mathbb { E } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 375, + 475, + 394 + ], + "score": 1.0, + "content": "is an expectation over trajectories starting from", + "type": "text" + }, + { + "bbox": [ + 476, + 379, + 504, + 388 + ], + "score": 0.86, + "content": "s _ { 0 } = s", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 353, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 177, + 402 + ], + "score": 1.0, + "content": "and taking action", + "type": "text" + }, + { + "bbox": [ + 178, + 391, + 207, + 399 + ], + "score": 0.89, + "content": "a _ { 0 } = a", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 388, + 341, + 402 + ], + "score": 1.0, + "content": "and thereafter running the policy", + "type": "text" + }, + { + "bbox": [ + 342, + 391, + 348, + 398 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 388, + 353, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 322, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "To apply RL, it is essential that we have access to the reward function which is often hard to design", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and evaluate (Singh et al., 2019). In addition, sparse rewards can cause exploration difficulties that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "pose great challenges to RL algorithms. We therefore look to imitation learning and particularly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "GAIL to derive a reward function from expert demonstrations. In GAIL, a reward function is learned", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 246, + 461 + ], + "score": 1.0, + "content": "by training a discriminator network", + "type": "text" + }, + { + "bbox": [ + 246, + 449, + 278, + 461 + ], + "score": 0.93, + "content": "D ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "to distinguish between agent and expert state-action pairs.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 310, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 310, + 473 + ], + "score": 1.0, + "content": "The GAIL objective is thus formulated as follows:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 405, + 506, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 483, + 461, + 501 + ], + "lines": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "spans": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "score": 0.85, + "content": "\\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 ) ,", + "type": "interline_equation", + "image_path": "108a841f22d52fb14679a7fca453346693df0d23802f2235bfac6b83e1870dfa.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 148, + 483, + 461, + 501 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 133, + 527 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 516, + 141, + 523 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 512, + 220, + 527 + ], + "score": 1.0, + "content": "is the agent policy,", + "type": "text" + }, + { + "bbox": [ + 220, + 516, + 234, + 524 + ], + "score": 0.85, + "content": "\\pi _ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 512, + 325, + 527 + ], + "score": 1.0, + "content": "the expert policy, and", + "type": "text" + }, + { + "bbox": [ + 325, + 514, + 348, + 525 + ], + "score": 0.91, + "content": "H ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 512, + 505, + 527 + ], + "score": 1.0, + "content": "an (optional) entropy regularizer. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 372, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 246, + 537 + ], + "score": 1.0, + "content": "reward function is defined simply:", + "type": "text" + }, + { + "bbox": [ + 246, + 524, + 367, + 537 + ], + "score": 0.91, + "content": "R ( s , a ) = - \\log ( 1 - D ( s , a ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 523, + 372, + 537 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 512, + 505, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "GAIL is theoretically appealing and practically simple. The discriminator, however, can focus on any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "features to discriminate, whether these features are task-relevant or not. In the next subsection we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "describe a way to constrain the discriminator network in order to prevent it from using task-irrelevant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 574, + 281, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 281, + 587 + ], + "score": 1.0, + "content": "details to distinguish agent and expert data.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 540, + 506, + 587 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 608, + 459, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 607, + 460, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 607, + 460, + 624 + ], + "score": 1.0, + "content": "4 TASK-RELEVANT ADVERSARIAL IMITATION LEARNING (TRAIL)", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "We want the discriminator to focus on task-relevant features. Our proposed solution, TRAIL, prevents", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "the discriminator from being able to distinguish expert and agent behaviour based on selected aspects", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "of the data. For instance, the discriminator should distinguish agent and expert frames only when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "meaningful behavior is present in those frames. In the absence of behavior useful to solve the task,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 681, + 490, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 490, + 694 + ], + "score": 1.0, + "content": "e.g. in initial frames prior to the execution of the behavior, the discriminator should be agnostic.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 637, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "To this end, we propose a constraint for the discriminator, such that its accuracy must not be greater", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 298, + 722 + ], + "score": 1.0, + "content": "than chance on a particular set of observations", + "type": "text" + }, + { + "bbox": [ + 298, + 712, + 305, + 720 + ], + "score": 0.87, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 709, + 444, + 722 + ], + "score": 1.0, + "content": "that we will call the invariant set.", + "type": "text" + }, + { + "bbox": [ + 445, + 710, + 453, + 720 + ], + "score": 0.66, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "will include", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "only observations that can be distinguished as agent or expert in task-irrelevant ways. Precisely, we", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 80, + 421, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 421, + 97 + ], + "score": 1.0, + "content": "formulate TRAIL in terms of the following constrained optimization problem:", + "type": "text", + "cross_page": true + } + ], + "index": 0 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 420, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 421, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 421, + 97 + ], + "score": 1.0, + "content": "formulate TRAIL in terms of the following constrained optimization problem:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 98, + 455, + 145 + ], + "lines": [ + { + "bbox": [ + 156, + 98, + 455, + 145 + ], + "spans": [ + { + "bbox": [ + 156, + 98, + 455, + 145 + ], + 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^ { ( 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}", + "type": "interline_equation", + "image_path": "4d4c40cd8a255e2543089e2861ddce6e8fe5af859ebd3aa65fb0b438c9523f44.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 119, + 191, + 491, + 206.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 206.33333333333334, + 491, + 221.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 119, + 221.66666666666669, + 491, + 237.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 405, + 253 + ], + "lines": [ + { + "bbox": [ + 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} { 2 } } + \\mathbf { 1 } _ { \\mathrm { D } _ { \\psi } \\left( \\hat { \\mathbf { s } } _ { \\theta } ^ { ( \\mathrm { i } ) } \\right) < \\frac { 1 } { 2 } } \\right] .", + "type": "interline_equation", + "image_path": "d4fae644884feb2b3543a804ce727543c012afad257cd167eb5487d306ecc872.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 182, + 257, + 428, + 268.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 182, + 268.6666666666667, + 428, + 280.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 182, + 280.33333333333337, + 428, + 292.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 296, + 290, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 291, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 150, + 311 + ], + "score": 1.0, + "content": "The scalar", + "type": "text" + }, + { + "bbox": [ + 150, + 297, + 175, + 308 + ], + "score": 0.9, + "content": "\\lambda \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 295, + 291, + 311 + ], + "score": 1.0, + "content": "is a tunable hyperparameter.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 507, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 327 + ], + "score": 1.0, + "content": "We used single frames as states but sequences can also be used, and we do not require expert actions.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "Not requiring actions enables learning in very off-policy settings, where the action dimensions and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 335, + 490, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 490, + 348 + 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"content": "is a design choice. In general, we could always contrive non-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "score": 1.0, + "content": "stationary and adversarial ways of making this choice difficult. However, we argue that in many", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "situations of great interest, including our robotic manipulation setup, it is easy to propose effective", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 415, + 233, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 233, + 426 + ], + "score": 1.0, + "content": "and very general invariant sets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "A straightforward way to collect robot data is to execute a random policy. We can then use the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 430, + 454 + ], + "score": 1.0, + "content": "resulting random episodes, for both expert and agent, to construct the invariant set", + "type": "text" + }, + { + "bbox": [ + 430, + 442, + 438, + 452 + ], + "score": 0.59, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 442, + 505, + 454 + ], + "score": 1.0, + "content": ". Another way to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 145, + 466 + ], + "score": 1.0, + "content": "construct", + "type": "text" + }, + { + "bbox": [ + 146, + 453, + 154, + 463 + ], + "score": 0.79, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "is to use early frames from both expert and agent episodes. Since in early frames little", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "or no task behavior is apparent, this strategy turns out to be effective and no extra data has to be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "collected. This strategy also improves robustness with respect to variation in the initial conditions of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 330, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 330, + 498 + ], + "score": 1.0, + "content": "the task; see for example block insertion in Figure 1(d).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 502, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 107, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 198, + 515 + ], + "score": 1.0, + "content": "Importantly, if the set", + "type": "text" + }, + { + "bbox": [ + 198, + 503, + 206, + 513 + ], + "score": 0.75, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "captures some forms of irrelevance but not all forms, it will nonetheless", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "always help in improving performance. In this regard, TRAIL will dominate its GAIL predecessor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 468, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 468, + 537 + ], + "score": 1.0, + "content": "whenever the designer has some prior on what aspects of the data might be task irrelevant.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 200, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 201, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 201, + 566 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "We focus on solving robot manipulation tasks. The environment implements two work-spaces: one", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "score": 1.0, + "content": "with a Kinova Jaco arm (Jaco), and the other with a Sawyer arm (Sawyer). See supplementary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "material A.1 for a detailed description. Environment rewards, which are not used by GAIL-based", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 239, + 622 + ], + "score": 1.0, + "content": "methods, are sparse and equal to", + "type": "text" + }, + { + "bbox": [ + 239, + 610, + 251, + 620 + ], + "score": 0.84, + "content": "+ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "for each step when a given task is solved and 0 otherwise. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 483, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 483, + 633 + ], + "score": 1.0, + "content": "maximum reward for the episode is 200, since this is the length of a single evaluation episode.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "Our agent is based on the off-policy D4PG algorithm (Barth-Maron et al., 2018) because of its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "stability and data-efficiency (see supplementary material A.6). Following Vecerik et al. (2017), we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "add expert demonstrations into the agents’ experience replay, and refer to the resulting RL algorithm", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 491, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 491, + 683 + ], + "score": 1.0, + "content": "as D4PG from Demonstrations (D4PGfD). For each task we collect 100 human demonstrations.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Data augmentation Traditional data augmentation has proved beneficial in imitation learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(Berseth & Pal, 2019). Surprisingly, to the best of our knowledge, this has not been explicitly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "studied in prior publications on GAIL. However, we find that data augmentation is a generally useful", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "component to prevent discriminator overfitting. 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In general, we could always contrive non-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 406 + ], + "score": 1.0, + "content": "stationary and adversarial ways of making this choice difficult. However, we argue that in many", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "situations of great interest, including our robotic manipulation setup, it is easy to propose effective", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 415, + 233, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 233, + 426 + ], + "score": 1.0, + "content": "and very general invariant sets.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 380, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "A straightforward way to collect robot data is to execute a random policy. We can then use the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 430, + 454 + ], + "score": 1.0, + "content": "resulting random episodes, for both expert and agent, to construct the invariant set", + "type": "text" + }, + { + "bbox": [ + 430, + 442, + 438, + 452 + ], + "score": 0.59, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 442, + 505, + 454 + ], + "score": 1.0, + "content": ". Another way to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 145, + 466 + ], + "score": 1.0, + "content": "construct", + "type": "text" + }, + { + "bbox": [ + 146, + 453, + 154, + 463 + ], + "score": 0.79, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "is to use early frames from both expert and agent episodes. Since in early frames little", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "or no task behavior is apparent, this strategy turns out to be effective and no extra data has to be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "collected. This strategy also improves robustness with respect to variation in the initial conditions of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 330, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 330, + 498 + ], + "score": 1.0, + "content": "the task; see for example block insertion in Figure 1(d).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 431, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 502, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 107, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 198, + 515 + ], + "score": 1.0, + "content": "Importantly, if the set", + "type": "text" + }, + { + "bbox": [ + 198, + 503, + 206, + 513 + ], + "score": 0.75, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "captures some forms of irrelevance but not all forms, it will nonetheless", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "always help in improving performance. In this regard, TRAIL will dominate its GAIL predecessor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 525, + 468, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 468, + 537 + ], + "score": 1.0, + "content": "whenever the designer has some prior on what aspects of the data might be task irrelevant.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 503, + 506, + 537 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 200, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 201, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 201, + 566 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "We focus on solving robot manipulation tasks. The environment implements two work-spaces: one", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 602 + ], + "score": 1.0, + "content": "with a Kinova Jaco arm (Jaco), and the other with a Sawyer arm (Sawyer). See supplementary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "material A.1 for a detailed description. Environment rewards, which are not used by GAIL-based", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 239, + 622 + ], + "score": 1.0, + "content": "methods, are sparse and equal to", + "type": "text" + }, + { + "bbox": [ + 239, + 610, + 251, + 620 + ], + "score": 0.84, + "content": "+ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "for each step when a given task is solved and 0 otherwise. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 483, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 483, + 633 + ], + "score": 1.0, + "content": "maximum reward for the episode is 200, since this is the length of a single evaluation episode.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 577, + 505, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "Our agent is based on the off-policy D4PG algorithm (Barth-Maron et al., 2018) because of its", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "stability and data-efficiency (see supplementary material A.6). Following Vecerik et al. (2017), we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "add expert demonstrations into the agents’ experience replay, and refer to the resulting RL algorithm", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 491, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 491, + 683 + ], + "score": 1.0, + "content": "as D4PG from Demonstrations (D4PGfD). For each task we collect 100 human demonstrations.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 638, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Data augmentation Traditional data augmentation has proved beneficial in imitation learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(Berseth & Pal, 2019). Surprisingly, to the best of our knowledge, this has not been explicitly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "studied in prior publications on GAIL. However, we find that data augmentation is a generally useful", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "component to prevent discriminator overfitting. It drastically improves the baseline GAIL agent, and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "is necessary to solve any of the harder manipulation tasks. We distort images by randomly changing", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "brightness, contrast and saturation; random cropping and rotation; adding Gaussian noise. When", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "multiple sensor inputs are available (e.g. multiple cameras), we also randomly drop out these inputs,", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 229, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 229, + 128 + ], + "score": 1.0, + "content": "but leaving at least one active.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "is necessary to solve any of the harder manipulation tasks. We distort images by randomly changing", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "brightness, contrast and saturation; random cropping and rotation; adding Gaussian noise. When", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "multiple sensor inputs are available (e.g. multiple cameras), we also randomly drop out these inputs,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 229, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 229, + 128 + ], + "score": 1.0, + "content": "but leaving at least one active.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "All discriminator-based methods in this work (including all baselines) use data augmentation except", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "as indicated in the relevant ablations (see Section 5.2). We also considered regularizing the GAIL", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "discriminator with spectral normalization (Miyato et al., 2018). It performed slightly better than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "GAIL, but still failed in the presence of distractor objects, and we thus omit spectral normalization in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 253, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 253, + 190 + ], + "score": 1.0, + "content": "the main experiments for simplicity.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "Actor early stopping When the agent has learned the desired behavior, and the resulting data is used", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "for training, the discriminator will become unable to distinguish expert and agent observations based", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 448, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 448, + 227 + ], + "score": 1.0, + "content": "only on behavior. This forces the discriminator to rely on task-irrelevant information.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "To avoid this scenario, we propose to restart each actor episode after a certain number of steps", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "such that successful behavior is rarely represented in agent data. This enables the discriminator to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "recognize the goal condition, which appears frequently at the end of demonstration episodes, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "representative of expert behavior. To avoid hand-tuning the stopping step number, we found that the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "discriminator score can be used to derive an adaptive stopping criterion. Concretely, we restart an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "score": 1.0, + "content": "episode if the discriminator score at the current step exceeds the median score of the episode so far", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 375, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 120, + 312 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 298, + 158, + 310 + ], + "score": 0.88, + "content": "T _ { p a t i e n c e }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 297, + 306, + 312 + ], + "score": 1.0, + "content": "consecutive steps (in practice we set", + "type": "text" + }, + { + "bbox": [ + 307, + 298, + 368, + 310 + ], + "score": 0.91, + "content": "T _ { p a t i e n c e } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 297, + 375, + 312 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 250, + 327 + ], + "score": 1.0, + "content": "We set invariant set hyperparameter", + "type": "text" + }, + { + "bbox": [ + 250, + 315, + 276, + 325 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "and use adaptive early stopping for TRAIL. The ablation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 126, + 338 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 326, + 152, + 336 + ], + "score": 0.87, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 325, + 505, + 338 + ], + "score": 1.0, + "content": ", i.e. adaptive early stopping only, is referred to as TRAIL-0. Hence, the only difference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 336, + 411, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 411, + 348 + ], + "score": 1.0, + "content": "between TRAIL-0 and TRAIL is that the later uses invariant set constraints.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 286, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 288, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 288, + 375 + ], + "score": 1.0, + "content": "5.1 BLOCK LIFTING WITH DISTRACTORS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "In this section, we consider two variants of the lift task in the Sawyer work space: a) lift alone, where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "only one red cube is present, b) lift distracted, with two extra blocks (blue and green, see Figure 9).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 404, + 425, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 425, + 417 + ], + "score": 1.0, + "content": "We show how adding these additional distractors affects the training procedure.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We first compare our method to baselines. In doing so, the invariant set is constructed using the first", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "10 frames from every episode, as in the rest of the work. This choice does not require us to collect", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "any extra data, and hence the comparison with baselines is fair. In the next section, we elaborate on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 403, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 403, + 466 + ], + "score": 1.0, + "content": "the choice of the invariant set and provide additional experimental results.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "As baselines, we run BC and GAIL (with data augmentation). Baseline GAIL is the strongest we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 494 + ], + "score": 1.0, + "content": "were able to implement. The only difference between this baseline and TRAIL is the use of invariant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "set constrains and actor early stopping (code, agent configurations like number of actors, and network", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "architecture are the same). We additionally consider the approaches proposed by Reed et al. (2018) as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "GAIL-based baselines; using either a randomly initialized convolutional network, or a convolutional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 527, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 506, + 538 + ], + "score": 1.0, + "content": "critic network, to provide fixed vision features on top of which a tiny discriminator network is trained.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "We call these two baselines random and critic respectively. Finally, to show the importance of actor", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 549, + 273, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 273, + 561 + ], + "score": 1.0, + "content": "early stopping, we run TRAIL-0 (Fig. 3).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "image", + "bbox": [ + 117, + 567, + 489, + 662 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 567, + 489, + 662 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 567, + 489, + 662 + ], + "spans": [ + { + "bbox": [ + 117, + 567, + 489, + 662 + ], + "score": 0.967, + "type": "image", + "image_path": "59bb3bff9b0dfb40d7b0e4e3cc9fbc548f0f04232c09b94486607c7ff7cfd013.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 117, + 567, + 489, + 598.6666666666666 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 117, + 598.6666666666666, + 489, + 630.3333333333333 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 117, + 630.3333333333333, + 489, + 661.9999999999999 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 115, + 668, + 492, + 680 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 666, + 492, + 681 + ], + "spans": [ + { + "bbox": [ + 118, + 666, + 492, + 681 + ], + "score": 1.0, + "content": "Figure 3: Results for lift alone, lift distracted, and lift distracted seeded. Only TRAIL excels.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "index": 40.0 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "All methods perform satisfactorily on lift alone, but the proposed methods TRAIL-0 and TRAIL do", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "best. As expected, the performance of BC on lift distracted is similar to its performance on lift alone,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "despite the two additional blocks. The two additional blocks in lift distracted affect the GAIL-based", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 289, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 289, + 732 + ], + "score": 1.0, + "content": "baselines, despite being irrelevant to the task.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "All discriminator-based methods in this work (including all baselines) use data augmentation except", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "as indicated in the relevant ablations (see Section 5.2). We also considered regularizing the GAIL", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "discriminator with spectral normalization (Miyato et al., 2018). It performed slightly better than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "GAIL, but still failed in the presence of distractor objects, and we thus omit spectral normalization in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 253, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 253, + 190 + ], + "score": 1.0, + "content": "the main experiments for simplicity.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 131, + 505, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "Actor early stopping When the agent has learned the desired behavior, and the resulting data is used", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "for training, the discriminator will become unable to distinguish expert and agent observations based", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 448, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 448, + 227 + ], + "score": 1.0, + "content": "only on behavior. This forces the discriminator to rely on task-irrelevant information.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 192, + 505, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "To avoid this scenario, we propose to restart each actor episode after a certain number of steps", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "such that successful behavior is rarely represented in agent data. This enables the discriminator to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "recognize the goal condition, which appears frequently at the end of demonstration episodes, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "representative of expert behavior. To avoid hand-tuning the stopping step number, we found that the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "discriminator score can be used to derive an adaptive stopping criterion. Concretely, we restart an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "score": 1.0, + "content": "episode if the discriminator score at the current step exceeds the median score of the episode so far", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 375, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 120, + 312 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 298, + 158, + 310 + ], + "score": 0.88, + "content": "T _ { p a t i e n c e }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 297, + 306, + 312 + ], + "score": 1.0, + "content": "consecutive steps (in practice we set", + "type": "text" + }, + { + "bbox": [ + 307, + 298, + 368, + 310 + ], + "score": 0.91, + "content": "T _ { p a t i e n c e } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 297, + 375, + 312 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 232, + 506, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 250, + 327 + ], + "score": 1.0, + "content": "We set invariant set hyperparameter", + "type": "text" + }, + { + "bbox": [ + 250, + 315, + 276, + 325 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "and use adaptive early stopping for TRAIL. The ablation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 126, + 338 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 326, + 152, + 336 + ], + "score": 0.87, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 325, + 505, + 338 + ], + "score": 1.0, + "content": ", i.e. adaptive early stopping only, is referred to as TRAIL-0. Hence, the only difference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 336, + 411, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 411, + 348 + ], + "score": 1.0, + "content": "between TRAIL-0 and TRAIL is that the later uses invariant set constraints.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 315, + 505, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 286, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 288, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 288, + 375 + ], + "score": 1.0, + "content": "5.1 BLOCK LIFTING WITH DISTRACTORS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 394 + ], + "score": 1.0, + "content": "In this section, we consider two variants of the lift task in the Sawyer work space: a) lift alone, where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "only one red cube is present, b) lift distracted, with two extra blocks (blue and green, see Figure 9).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 404, + 425, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 425, + 417 + ], + "score": 1.0, + "content": "We show how adding these additional distractors affects the training procedure.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 383, + 506, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We first compare our method to baselines. In doing so, the invariant set is constructed using the first", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "10 frames from every episode, as in the rest of the work. This choice does not require us to collect", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 455 + ], + "score": 1.0, + "content": "any extra data, and hence the comparison with baselines is fair. In the next section, we elaborate on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 403, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 403, + 466 + ], + "score": 1.0, + "content": "the choice of the invariant set and provide additional experimental results.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 421, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "As baselines, we run BC and GAIL (with data augmentation). Baseline GAIL is the strongest we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 494 + ], + "score": 1.0, + "content": "were able to implement. The only difference between this baseline and TRAIL is the use of invariant", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "set constrains and actor early stopping (code, agent configurations like number of actors, and network", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "architecture are the same). We additionally consider the approaches proposed by Reed et al. (2018) as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "GAIL-based baselines; using either a randomly initialized convolutional network, or a convolutional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 527, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 506, + 538 + ], + "score": 1.0, + "content": "critic network, to provide fixed vision features on top of which a tiny discriminator network is trained.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "We call these two baselines random and critic respectively. Finally, to show the importance of actor", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 549, + 273, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 273, + 561 + ], + "score": 1.0, + "content": "early stopping, we run TRAIL-0 (Fig. 3).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 471, + 506, + 561 + ] + }, + { + "type": "image", + "bbox": [ + 117, + 567, + 489, + 662 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 567, + 489, + 662 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 567, + 489, + 662 + ], + "spans": [ + { + "bbox": [ + 117, + 567, + 489, + 662 + ], + "score": 0.967, + "type": "image", + "image_path": "59bb3bff9b0dfb40d7b0e4e3cc9fbc548f0f04232c09b94486607c7ff7cfd013.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 117, + 567, + 489, + 598.6666666666666 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 117, + 598.6666666666666, + 489, + 630.3333333333333 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 117, + 630.3333333333333, + 489, + 661.9999999999999 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 115, + 668, + 492, + 680 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 666, + 492, + 681 + ], + "spans": [ + { + "bbox": [ + 118, + 666, + 492, + 681 + ], + "score": 1.0, + "content": "Figure 3: Results for lift alone, lift distracted, and lift distracted seeded. Only TRAIL excels.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "index": 40.0 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "All methods perform satisfactorily on lift alone, but the proposed methods TRAIL-0 and TRAIL do", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "best. As expected, the performance of BC on lift distracted is similar to its performance on lift alone,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "despite the two additional blocks. The two additional blocks in lift distracted affect the GAIL-based", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 289, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 289, + 732 + ], + "score": 1.0, + "content": "baselines, despite being irrelevant to the task.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "To understand this effect, we conducted an additional experiment (lift distracted seeded). Here the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "initial block positions are randomly drawn from the expert demonstrations. Therefore, it is impossible", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to discriminate between expert and actor episodes using the first few frames of an episode. Note", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "this initialization procedure is not applied to the evaluation actor, keeping the evaluation scores", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 352, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 352, + 139 + ], + "score": 1.0, + "content": "comparable between lift distracted and lift distracted seeded.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "This experiment exposes one major culprit behind the performance degradation of GAIL: memoriza-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "tion. The discriminator can achieve perfect accuracy by memorizing all 100 initial positions from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "the demonstration set, making the reward function uninformative. By constraining the discriminator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "TRAIL squeezes out this irrelevant information and succeeds in solving the task in the presence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "distractions. TRAIL is the only method that is able to handle the variety of initial cube positions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 408, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 408, + 211 + ], + "score": 1.0, + "content": "during training, achieving better than expert performance on lift distracted.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "Interestingly, random performs reasonably on lift distracted. Given random’s strong performance,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "we conducted additional experiments to evaluate its effectiveness when trained with adaptive early", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 410, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 410, + 249 + ], + "score": 1.0, + "content": "stopping and present the results in Figure 12 of the supplementary material.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 109, + 262, + 272, + 272 + ], + "lines": [ + { + "bbox": [ + 107, + 262, + 273, + 273 + ], + "spans": [ + { + "bbox": [ + 107, + 262, + 264, + 273 + ], + "score": 1.0, + "content": "CONSTRUCTING THE INVARIANT SET", + "type": "text" + }, + { + "bbox": [ + 265, + 262, + 273, + 272 + ], + "score": 0.4, + "content": "\\mathcal { T }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 506, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "In the previous subsection, early frames were used to construct the invariant set (TRAIL-early). Here,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 392, + 305 + ], + "score": 1.0, + "content": "we evaluate another previously mentioned approach for constructing", + "type": "text" + }, + { + "bbox": [ + 392, + 293, + 400, + 302 + ], + "score": 0.69, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "; random policy (TRAIL-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "random). The lift distracted task caused all baselines to fail, but was solved by TRAIL. We introduce", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "a harder version of the task, where the expert appearance is different, to tease out the differences", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "between TRAIL-early and TRAIL-random. The difference in appearance between the expert and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "imitator allows the GAIL discriminator to trivially distinguish them. The results and the differences", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 310, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 310, + 360 + ], + "score": 1.0, + "content": "in the expert appearance are presented in Figure 4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "image", + "bbox": [ + 125, + 371, + 479, + 458 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 371, + 479, + 458 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 371, + 479, + 458 + ], + "spans": [ + { + "bbox": [ + 125, + 371, + 479, + 458 + ], + "score": 0.964, + "type": "image", + "image_path": "6b8cf409f166f495f4aaf1c532472a37e5e94c105788ca3abc316459da5a2a6d.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 125, + 371, + 479, + 400.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 125, + 400.0, + 479, + 429.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 125, + 429.0, + 479, + 458.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 464, + 506, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "score": 1.0, + "content": "Figure 4: Lift red block, where expert has a different body appearance, and with distractor blocks.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 475, + 411, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 411, + 487 + ], + "score": 1.0, + "content": "TRAIL-random outperforms GAIL, and performs on par with TRAIL-early.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + } + ], + "index": 24.25 + }, + { + "type": "text", + "bbox": [ + 108, + 499, + 503, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 512 + ], + "score": 1.0, + "content": "The new task is indeed harder and it takes longer for TRAIL methods to achieve performance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "better than BC baseline, which is not affected by the different body appearance. GAIL is clearly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 521, + 251, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 251, + 532 + ], + "score": 1.0, + "content": "outperformed and does not take off.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "The difference between TRAIL methods is negligible. We also tried to mix them but the differences", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "remain imperceptible. Hence, in the following experiments we simply use early frames. This choice", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "is pragmatic as it does not require that we collect any extra data, and hence the comparison with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "GAIL and other baselines is fair. It is also very simple to apply in practice, even if one does not have", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "access to the expert setup anymore. Finally, it is general and powerful enough to be successfully used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 352, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 352, + 605 + ], + "score": 1.0, + "content": "across all robotic manipulation tasks considered in this work.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 363, + 622 + ], + "score": 1.0, + "content": "To decide how many initial frames should be used to construct", + "type": "text" + }, + { + "bbox": [ + 363, + 610, + 371, + 620 + ], + "score": 0.71, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 610, + 505, + 622 + ], + "score": 1.0, + "content": ", we conducted an ablation study", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "and found out that the method is not very sensitive to this choice (see supplementary material A.2).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Hence, we chose 10 initial frames, and intentionally used the same number for all tasks to further", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 252, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 252, + 656 + ], + "score": 1.0, + "content": "emphasize generality of this choice.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 108, + 669, + 216, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 217, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 217, + 681 + ], + "score": 1.0, + "content": "5.2 ABLATION STUDIES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 443, + 701 + ], + "lines": [ + { + "bbox": [ + 107, + 691, + 444, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 691, + 444, + 701 + ], + "score": 1.0, + "content": "MEASURING DISCRIMINATOR MEMORIZATION OF TASK-IRRELEVANT FEATURES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "In this section, we experimentally confirm that memorization is a limiting factor of GAIL methods.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "We equipped the discriminator with two extra heads whose inputs are the final spatial layer of the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "To understand this effect, we conducted an additional experiment (lift distracted seeded). Here the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "initial block positions are randomly drawn from the expert demonstrations. Therefore, it is impossible", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to discriminate between expert and actor episodes using the first few frames of an episode. Note", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "this initialization procedure is not applied to the evaluation actor, keeping the evaluation scores", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 352, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 352, + 139 + ], + "score": 1.0, + "content": "comparable between lift distracted and lift distracted seeded.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 505, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "This experiment exposes one major culprit behind the performance degradation of GAIL: memoriza-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "tion. The discriminator can achieve perfect accuracy by memorizing all 100 initial positions from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "the demonstration set, making the reward function uninformative. By constraining the discriminator,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "TRAIL squeezes out this irrelevant information and succeeds in solving the task in the presence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "distractions. TRAIL is the only method that is able to handle the variety of initial cube positions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 408, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 408, + 211 + ], + "score": 1.0, + "content": "during training, achieving better than expert performance on lift distracted.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 142, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "Interestingly, random performs reasonably on lift distracted. Given random’s strong performance,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "we conducted additional experiments to evaluate its effectiveness when trained with adaptive early", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 410, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 410, + 249 + ], + "score": 1.0, + "content": "stopping and present the results in Figure 12 of the supplementary material.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 214, + 506, + 249 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 262, + 272, + 272 + ], + "lines": [ + { + "bbox": [ + 107, + 262, + 273, + 273 + ], + "spans": [ + { + "bbox": [ + 107, + 262, + 264, + 273 + ], + "score": 1.0, + "content": "CONSTRUCTING THE INVARIANT SET", + "type": "text" + }, + { + "bbox": [ + 265, + 262, + 273, + 272 + ], + "score": 0.4, + "content": "\\mathcal { T }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 506, + 359 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "In the previous subsection, early frames were used to construct the invariant set (TRAIL-early). Here,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 392, + 305 + ], + "score": 1.0, + "content": "we evaluate another previously mentioned approach for constructing", + "type": "text" + }, + { + "bbox": [ + 392, + 293, + 400, + 302 + ], + "score": 0.69, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "; random policy (TRAIL-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "random). The lift distracted task caused all baselines to fail, but was solved by TRAIL. We introduce", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "a harder version of the task, where the expert appearance is different, to tease out the differences", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "between TRAIL-early and TRAIL-random. The difference in appearance between the expert and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "imitator allows the GAIL discriminator to trivially distinguish them. The results and the differences", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 310, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 310, + 360 + ], + "score": 1.0, + "content": "in the expert appearance are presented in Figure 4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 281, + 506, + 360 + ] + }, + { + "type": "image", + "bbox": [ + 125, + 371, + 479, + 458 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 371, + 479, + 458 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 371, + 479, + 458 + ], + "spans": [ + { + "bbox": [ + 125, + 371, + 479, + 458 + ], + "score": 0.964, + "type": "image", + "image_path": "6b8cf409f166f495f4aaf1c532472a37e5e94c105788ca3abc316459da5a2a6d.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 125, + 371, + 479, + 400.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 125, + 400.0, + 479, + 429.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 125, + 429.0, + 479, + 458.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 464, + 506, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 476 + ], + "score": 1.0, + "content": "Figure 4: Lift red block, where expert has a different body appearance, and with distractor blocks.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 475, + 411, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 411, + 487 + ], + "score": 1.0, + "content": "TRAIL-random outperforms GAIL, and performs on par with TRAIL-early.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + } + ], + "index": 24.25 + }, + { + "type": "text", + "bbox": [ + 108, + 499, + 503, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 512 + ], + "score": 1.0, + "content": "The new task is indeed harder and it takes longer for TRAIL methods to achieve performance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "better than BC baseline, which is not affected by the different body appearance. GAIL is clearly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 521, + 251, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 251, + 532 + ], + "score": 1.0, + "content": "outperformed and does not take off.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 497, + 505, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 537, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "The difference between TRAIL methods is negligible. We also tried to mix them but the differences", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "remain imperceptible. Hence, in the following experiments we simply use early frames. This choice", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "is pragmatic as it does not require that we collect any extra data, and hence the comparison with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "GAIL and other baselines is fair. It is also very simple to apply in practice, even if one does not have", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "access to the expert setup anymore. Finally, it is general and powerful enough to be successfully used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 352, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 352, + 605 + ], + "score": 1.0, + "content": "across all robotic manipulation tasks considered in this work.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 537, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 363, + 622 + ], + "score": 1.0, + "content": "To decide how many initial frames should be used to construct", + "type": "text" + }, + { + "bbox": [ + 363, + 610, + 371, + 620 + ], + "score": 0.71, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 610, + 505, + 622 + ], + "score": 1.0, + "content": ", we conducted an ablation study", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "and found out that the method is not very sensitive to this choice (see supplementary material A.2).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Hence, we chose 10 initial frames, and intentionally used the same number for all tasks to further", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 252, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 252, + 656 + ], + "score": 1.0, + "content": "emphasize generality of this choice.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 610, + 506, + 656 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 669, + 216, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 217, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 217, + 681 + ], + "score": 1.0, + "content": "5.2 ABLATION STUDIES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 443, + 701 + ], + "lines": [ + { + "bbox": [ + 107, + 691, + 444, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 691, + 444, + 701 + ], + "score": 1.0, + "content": "MEASURING DISCRIMINATOR MEMORIZATION OF TASK-IRRELEVANT FEATURES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "In this section, we experimentally confirm that memorization is a limiting factor of GAIL methods.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "We equipped the discriminator with two extra heads whose inputs are the final spatial layer of the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "ResNet for the lift distracted task. The first head is trained on the first frames only, and has the same", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "target as the main head (i.e. discriminating between agent and expert). To train the second head, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "randomly divide the expert demonstrations into two equi-numerous subsets, and the task is to predict", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "to which of these randomly chosen sets the demonstration was assigned to. Both heads are trained", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "via backpropagation but their gradient is not propagated to the ResNet so they do not influence the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 217, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 217, + 150 + ], + "score": 1.0, + "content": "training procedure directly.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "If our claim is correct, we expect the extra heads to have higher accuracy for TRAIL-0 as compared", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "to TRAIL, since TRAIL representations are penalized for having features triggering memorization,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "i.e. the features should not aid discriminating based on the first frames or predicting a random label", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 432, + 200 + ], + "score": 1.0, + "content": "for each expert demonstration. No reasonably performing method is able to force", + "type": "text" + }, + { + "bbox": [ + 433, + 187, + 452, + 198 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "accuracy for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 494, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 494, + 211 + ], + "score": 1.0, + "content": "extra heads since some features are important to solve the task (e.g. the position of the red cube).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 504, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "We also collected 25 extra holdout demonstrations and visualize the average discriminator prediction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 390, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 390, + 238 + ], + "score": 1.0, + "content": "on them, and compare with predictions on the training demonstrations.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "image", + "bbox": [ + 114, + 251, + 488, + 407 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 251, + 488, + 407 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 251, + 488, + 407 + ], + "spans": [ + { + "bbox": [ + 114, + 251, + 488, + 407 + ], + "score": 0.969, + "type": "image", + "image_path": "f3d0ef0720ecd274b57537e078d92668d28a832dccb1c8bad24328a24d337bc8.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 114, + 251, + 488, + 303.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 114, + 303.0, + 488, + 355.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 114, + 355.0, + 488, + 407.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 414, + 505, + 470 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "Figure 5: Demonstrating the memorization problem on the lift distracted task (here higher accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "score": 1.0, + "content": "is worse). Accuracy of different discriminator heads is presented (A-D). In A, the overall accuracy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 225, + 449 + ], + "score": 1.0, + "content": "for all timesteps. Then main", + "type": "text" + }, + { + "bbox": [ + 225, + 437, + 239, + 447 + ], + "score": 0.29, + "content": "\\mathrm { ( m ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "and extra (e) heads accuracy for the first steps are presented in B", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "and C, respectively. Accuracy of the head predicting randomly assigned demonstration class is shown", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "in D. Average discriminator predictions for training and holdout demonstration are shown in E and F.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "In Fig. 5, we see the overall accuracy of the main head for TRAIL-0 is significantly higher, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 388, + 511 + ], + "score": 1.0, + "content": "the difference is larger for early steps (when TRAIL achieves only", + "type": "text" + }, + { + "bbox": [ + 388, + 498, + 408, + 509 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ", as expected). TRAIL", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "representations are less helpful for extra heads (see Figure 5 C-D). Finally, TRAIL-0 clearly overfits", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 412, + 532 + ], + "score": 1.0, + "content": "on training demonstrations, predicting almost the maximum score, while only", + "type": "text" + }, + { + "bbox": [ + 413, + 520, + 442, + 531 + ], + "score": 0.85, + "content": "\\sim 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "is predicted for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 531, + 486, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 486, + 543 + ], + "score": 1.0, + "content": "holdout demonstrations. The TRAIL average predictions for both datasets are almost identical.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 561, + 264, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 265, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 265, + 574 + ], + "score": 1.0, + "content": "ACTOR EARLY STOPPING (TRAIL-0)", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "In this section, we analyze the importance of adaptive early stopping on 3 tasks in the Jaco work-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 188, + 606 + ], + "score": 1.0, + "content": "space: lift red cube", + "type": "text" + }, + { + "bbox": [ + 189, + 595, + 207, + 605 + ], + "score": 0.32, + "content": "( l i f t )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 593, + 294, + 606 + ], + "score": 1.0, + "content": ", put red cube in box", + "type": "text" + }, + { + "bbox": [ + 294, + 595, + 316, + 605 + ], + "score": 0.63, + "content": "( b o x )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 593, + 506, + 606 + ], + "score": 1.0, + "content": ", and stack red cube on blue cube (stack). We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "consider D4PGfD and three GAIL-based models with varying termination policies: a) fixed step", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "(50), b) based on ground truth task rewards, and c) based on adaptive early stopping (TRAIL-0).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "Using ground truth task rewards, an episode is terminated if the reward at the current step exceeds the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 639, + 488, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 488, + 650 + ], + "score": 1.0, + "content": "median reward of the episode so far for 10 consecutive steps. Results are presented in Figure 6.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Termination based on task reward is clearly superior; although unrealistic in practice, it defines the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "performance upper-bound and clearly shows that early stopping is beneficial. TRAIL-0 is robust", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "and reaches human performance on all tasks. A fixed termination policy, when tuned, can be very", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "effective. The same fixed termination step, however, does not work for all tasks. See Figure 13 in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "supplementary material for the effects of varying termination steps. Finally, as can be inferred from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "stack results, the dense rewards provided by TRAIL-0 are helpful in solving this challenging problem", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 453, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 453, + 732 + ], + "score": 1.0, + "content": "which is unsolved with D4PGfD even though D4PGfD uses ground truth task rewards.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "ResNet for the lift distracted task. The first head is trained on the first frames only, and has the same", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "target as the main head (i.e. discriminating between agent and expert). To train the second head, we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "randomly divide the expert demonstrations into two equi-numerous subsets, and the task is to predict", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "to which of these randomly chosen sets the demonstration was assigned to. Both heads are trained", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "via backpropagation but their gradient is not propagated to the ResNet so they do not influence the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 217, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 217, + 150 + ], + "score": 1.0, + "content": "training procedure directly.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 505, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "If our claim is correct, we expect the extra heads to have higher accuracy for TRAIL-0 as compared", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "to TRAIL, since TRAIL representations are penalized for having features triggering memorization,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "i.e. the features should not aid discriminating based on the first frames or predicting a random label", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 432, + 200 + ], + "score": 1.0, + "content": "for each expert demonstration. No reasonably performing method is able to force", + "type": "text" + }, + { + "bbox": [ + 433, + 187, + 452, + 198 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "accuracy for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 494, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 494, + 211 + ], + "score": 1.0, + "content": "extra heads since some features are important to solve the task (e.g. the position of the red cube).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 153, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 215, + 504, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "We also collected 25 extra holdout demonstrations and visualize the average discriminator prediction", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 390, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 390, + 238 + ], + "score": 1.0, + "content": "on them, and compare with predictions on the training demonstrations.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 214, + 505, + 238 + ] + }, + { + "type": "image", + "bbox": [ + 114, + 251, + 488, + 407 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 251, + 488, + 407 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 251, + 488, + 407 + ], + "spans": [ + { + "bbox": [ + 114, + 251, + 488, + 407 + ], + "score": 0.969, + "type": "image", + "image_path": "f3d0ef0720ecd274b57537e078d92668d28a832dccb1c8bad24328a24d337bc8.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 114, + 251, + 488, + 303.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 114, + 303.0, + 488, + 355.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 114, + 355.0, + 488, + 407.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 414, + 505, + 470 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "Figure 5: Demonstrating the memorization problem on the lift distracted task (here higher accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 439 + ], + "score": 1.0, + "content": "is worse). Accuracy of different discriminator heads is presented (A-D). In A, the overall accuracy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 225, + 449 + ], + "score": 1.0, + "content": "for all timesteps. Then main", + "type": "text" + }, + { + "bbox": [ + 225, + 437, + 239, + 447 + ], + "score": 0.29, + "content": "\\mathrm { ( m ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "and extra (e) heads accuracy for the first steps are presented in B", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "and C, respectively. Accuracy of the head predicting randomly assigned demonstration class is shown", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "in D. Average discriminator predictions for training and holdout demonstration are shown in E and F.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "In Fig. 5, we see the overall accuracy of the main head for TRAIL-0 is significantly higher, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 388, + 511 + ], + "score": 1.0, + "content": "the difference is larger for early steps (when TRAIL achieves only", + "type": "text" + }, + { + "bbox": [ + 388, + 498, + 408, + 509 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 497, + 506, + 511 + ], + "score": 1.0, + "content": ", as expected). TRAIL", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "representations are less helpful for extra heads (see Figure 5 C-D). Finally, TRAIL-0 clearly overfits", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 412, + 532 + ], + "score": 1.0, + "content": "on training demonstrations, predicting almost the maximum score, while only", + "type": "text" + }, + { + "bbox": [ + 413, + 520, + 442, + 531 + ], + "score": 0.85, + "content": "\\sim 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "is predicted for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 531, + 486, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 486, + 543 + ], + "score": 1.0, + "content": "holdout demonstrations. The TRAIL average predictions for both datasets are almost identical.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 487, + 506, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 561, + 264, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 265, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 265, + 574 + ], + "score": 1.0, + "content": "ACTOR EARLY STOPPING (TRAIL-0)", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 561, + 265, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "In this section, we analyze the importance of adaptive early stopping on 3 tasks in the Jaco work-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 188, + 606 + ], + "score": 1.0, + "content": "space: lift red cube", + "type": "text" + }, + { + "bbox": [ + 189, + 595, + 207, + 605 + ], + "score": 0.32, + "content": "( l i f t )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 593, + 294, + 606 + ], + "score": 1.0, + "content": ", put red cube in box", + "type": "text" + }, + { + "bbox": [ + 294, + 595, + 316, + 605 + ], + "score": 0.63, + "content": "( b o x )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 593, + 506, + 606 + ], + "score": 1.0, + "content": ", and stack red cube on blue cube (stack). We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "consider D4PGfD and three GAIL-based models with varying termination policies: a) fixed step", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "(50), b) based on ground truth task rewards, and c) based on adaptive early stopping (TRAIL-0).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "Using ground truth task rewards, an episode is terminated if the reward at the current step exceeds the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 639, + 488, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 488, + 650 + ], + "score": 1.0, + "content": "median reward of the episode so far for 10 consecutive steps. Results are presented in Figure 6.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 582, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Termination based on task reward is clearly superior; although unrealistic in practice, it defines the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "performance upper-bound and clearly shows that early stopping is beneficial. TRAIL-0 is robust", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "and reaches human performance on all tasks. A fixed termination policy, when tuned, can be very", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "effective. The same fixed termination step, however, does not work for all tasks. See Figure 13 in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "supplementary material for the effects of varying termination steps. Finally, as can be inferred from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "stack results, the dense rewards provided by TRAIL-0 are helpful in solving this challenging problem", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 453, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 453, + 732 + ], + "score": 1.0, + "content": "which is unsolved with D4PGfD even though D4PGfD uses ground truth task rewards.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 655, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 75, + 493, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 75, + 493, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 75, + 493, + 168 + ], + "spans": [ + { + "bbox": [ + 111, + 75, + 493, + 168 + ], + "score": 0.967, + "type": "image", + "image_path": "236f966ae10dec709f176a7097a72d5e3722052c12c4e9e616a6e0ea34d38c6a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 75, + 493, + 106.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 106.0, + 493, + 137.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 137.0, + 493, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 181, + 174, + 429, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 181, + 173, + 430, + 187 + ], + "spans": [ + { + "bbox": [ + 181, + 173, + 430, + 187 + ], + "score": 1.0, + "content": "Figure 6: Results for lift, box, and stack on Jaco environment.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 108, + 211, + 202, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 203, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 203, + 222 + ], + "score": 1.0, + "content": "DATA AUGMENTATION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 506, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "In Table 5.2, we report the best reward obtained in the first 12 hours of training (averaged for all seeds;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 243, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 507, + 255 + ], + "score": 1.0, + "content": "see Figure 14 for full curves). The results show that data augmentation is needed in lift distracted.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 254, + 453, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 453, + 267 + ], + "score": 1.0, + "content": "For the lift alone task, TRAIL-0 with data augmentation performs on par with TRAIL.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "table", + "bbox": [ + 155, + 281, + 454, + 349 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 155, + 281, + 454, + 349 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 281, + 454, + 349 + ], + "spans": [ + { + "bbox": [ + 155, + 281, + 454, + 349 + ], + "score": 0.98, + "html": "
TaskRegularizationData augmentationNo data augmentation
lift aloneTRAIL-0 TRAIL~165~115
lift distractedTRAIL-0~155~165
TRAIL~30 ~180~5 ~10
", + "type": "table", + "image_path": "44e435f0cddc322fb45ee6a3331dd739fc71ea12dd406b3c24ab878b32347dd0.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 155, + 281, + 454, + 303.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 155, + 303.6666666666667, + 454, + 326.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 155, + 326.33333333333337, + 454, + 349.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 111, + 353, + 494, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 351, + 496, + 366 + ], + "spans": [ + { + "bbox": [ + 113, + 351, + 496, + 366 + ], + "score": 1.0, + "content": "Table 1: Influence of data augmentation (evaluated on rewards) for lift alone and lift distracted.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + } + ], + "index": 10.0 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "The performance of TRAIL is not affected by the lack of data augmentation on the lift alone task,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 392, + 270, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 270, + 404 + ], + "score": 1.0, + "content": "whereas the performance of TRAIL-0 is.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 422, + 330, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 330, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 330, + 433 + ], + "score": 1.0, + "content": "LEARNING WITH A FIXED, PERFECT DISCRIMINATOR", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "To assess whether learned discriminators are necessary, we compare TRAIL against agents using", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 278, + 468 + ], + "score": 1.0, + "content": "a fixed reward function corresponding to", + "type": "text" + }, + { + "bbox": [ + 278, + 454, + 331, + 466 + ], + "score": 0.91, + "content": "R _ { e x p e r t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 453, + 351, + 468 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 454, + 401, + 466 + ], + "score": 0.91, + "content": "R _ { a g e n t } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "for the lift alone and lift", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "distracted tasks. This baseline simulates an oracle discriminator with perfect generalization, but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 498, + 405, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 405, + 510 + ], + "score": 1.0, + "content": "rewards are less than 5). See supplementary Figure 16 for learning curves.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 529, + 350, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 352, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 352, + 541 + ], + "score": 1.0, + "content": "5.3 LEARNING FROM OTHER EMBODIMENTS AND PROPS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 506, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "Since the TRAIL discriminator is trained to ignore task-irrelevant features, it can learn from demon-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "strations with different embodiments and props. Figure 7 shows that GAIL even with augmentation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "fails to learn block lifting from a different embodiment, and performs worse when the expert uses a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 584, + 469, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 469, + 596 + ], + "score": 1.0, + "content": "different prop color. 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The results show that data augmentation is needed in lift distracted.", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 254, + 453, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 453, + 267 + ], + "score": 1.0, + "content": "For the lift alone task, TRAIL-0 with data augmentation performs on par with TRAIL.", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 6, + "bbox_fs": [ + 105, + 231, + 507, + 267 + ] + }, + { + "type": "table", + "bbox": [ + 155, + 281, + 454, + 349 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 155, + 281, + 454, + 349 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 281, + 454, + 349 + ], + "spans": [ + { + "bbox": [ + 155, + 281, + 454, + 349 + ], + "score": 0.98, + "html": "
TaskRegularizationData augmentationNo data augmentation
lift aloneTRAIL-0 TRAIL~165~115
lift distractedTRAIL-0~155~165
TRAIL~30 ~180~5 ~10
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This baseline simulates an oracle discriminator with perfect generalization, but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "which is agnostic to behavior. On the lift alone task, agents using this fixed reward achieve roughly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "half the reward of TRAIL asymptotically, and on lift distracted they do not solve the task (average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 498, + 405, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 405, + 510 + ], + "score": 1.0, + "content": "rewards are less than 5). See supplementary Figure 16 for learning curves.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 442, + 506, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 529, + 350, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 352, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 352, + 541 + ], + "score": 1.0, + "content": "5.3 LEARNING FROM OTHER EMBODIMENTS AND PROPS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 506, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "Since the TRAIL discriminator is trained to ignore task-irrelevant features, it can learn from demon-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "strations with different embodiments and props. Figure 7 shows that GAIL even with augmentation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "fails to learn block lifting from a different embodiment, and performs worse when the expert uses a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 584, + 469, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 469, + 596 + ], + "score": 1.0, + "content": "different prop color. 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We test our proposed", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "algorithms in a variety of different envrionments using simulated Kinova Jaco2, and Sawyer robot", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 146, + 470, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 470, + 161 + ], + "score": 1.0, + "content": "arms3; see Figure 9. 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The gripper actions are implemented via the buttons on the controller.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "All demonstrations in our experiments are provided via human teleoperation and we collected 100", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 334, + 254, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 254, + 347 + ], + "score": 1.0, + "content": "demonstrations for each experiment.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 301, + 507, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "score": 1.0, + "content": "Jaco: When using the Jaco arm, we use joint velocity control (9DOF) where we control all 6 joints of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "arm and all 3 joints of the hand. 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The simulation is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "run with a numerical time step of 10 milliseconds, integrating 10 steps, to get a control frequency", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 322, + 605 + ], + "score": 1.0, + "content": "of 10HZ. 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Feature NameDimensions
frontal camera base force and torque sensors64×64×3 6
arm joints position6
arm joints velocity6
wrist force and torque sensors6
hand finger joints position3
hand finger joints velocity
hand fingertip sensors3
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grip site position3
pinch site position3
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b) Sawyer
Feature Name frontleft cameraDimensions 64×64×3
front right camera64×64×3
arm joint position7
arm joint velocity7
wrist force sensor3
wrist torque sensor3
hand grasp sensor
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hand joint position1
tool center point cartesian orientation9
tool center point cartesian position
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Feature NameDimensions
frontal camera base force and torque sensors64×64×3 6
arm joints position6
arm joints velocity6
wrist force and torque sensors6
hand finger joints position3
hand finger joints velocity
hand fingertip sensors3
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grip site position3
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b) Sawyer
Feature Name frontleft cameraDimensions 64×64×3
front right camera64×64×3
arm joint position7
arm joint velocity7
wrist force sensor3
wrist torque sensor3
hand grasp sensor
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hand joint position1
tool center point cartesian orientation9
tool center point cartesian position
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The learners could then draw samples from the replay memory for learning.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "The learner also serves as a parameter server to the actors which periodically update their policy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 221, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 221, + 365 + ], + "score": 1.0, + "content": "parameters from the learner.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "In our experiments, we always have access to expert demonstrations. 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ParametersValues
Actor/Critic Input Width64
Actor/Critic Input Height64
D4PG Parameters
Vmin-50
Vmax150
Vbins21
N step1
Actor learning rate10-4
Critic learning rate10-4
OptimizerAdam (Kingma & Ba (2014))
Batch size256
Target update period100
Discount factor (y)0.99
Replay capacity106
Number of actors32 or 128
Imitation Parameters
Discriminator learning rate10-4
Discriminator Input Width48
Discriminator Input Height48
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ParametersValues
Actor/Critic Input Width64
Actor/Critic Input Height64
D4PG Parameters
Vmin-50
Vmax150
Vbins21
N step1
Actor learning rate10-4
Critic learning rate10-4
OptimizerAdam (Kingma & Ba (2014))
Batch size256
Target update period100
Discount factor (y)0.99
Replay capacity106
Number of actors32 or 128
Imitation Parameters
Discriminator learning rate10-4
Discriminator Input Width48
Discriminator Input Height48
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When distractor blocks are added, the fixed reward agent fails to learn completely.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 8.75 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "image", + "bbox": [ + 180, + 116, + 424, + 234 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 180, + 116, + 424, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 180, + 116, + 424, + 234 + ], + "spans": [ + { + "bbox": [ + 180, + 116, + 424, + 234 + ], + "score": 0.969, + "type": "image", + "image_path": "73412ac4784215fb9164009bf367fa91028931791c266c531f38108dcd112e50.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 180, + 116, + 424, + 129.11111111111111 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 180, + 129.11111111111111, + 424, + 142.22222222222223 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 180, + 142.22222222222223, + 424, + 155.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 180, + 155.33333333333334, + 424, + 168.44444444444446 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 180, + 168.44444444444446, + 424, + 181.55555555555557 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 180, + 181.55555555555557, + 424, + 194.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 180, + 194.66666666666669, + 424, + 207.7777777777778 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 180, + 207.7777777777778, + 424, + 220.8888888888889 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 180, + 220.8888888888889, + 424, + 234.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 247, + 505, + 270 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "Figure 16: With fixed rewards, the agent is able to learn lift alone somewhat, but performs worse than", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 470, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 470, + 271 + ], + "score": 1.0, + "content": "TRAIL. 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lift aloneTRAIL-0 TRAIL~165~115
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hand grasp sensor
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ParametersValues
Actor/Critic Input Width64
Actor/Critic Input Height64
D4PG Parameters
Vmin-50
Vmax150
Vbins21
N step1
Actor learning rate10-4
Critic learning rate10-4
OptimizerAdam (Kingma & Ba (2014))
Batch size256
Target update period100
Discount factor (y)0.99
Replay capacity106
Number of actors32 or 128
Imitation Parameters
Discriminator learning rate10-4
Discriminator Input Width48
Discriminator Input Height48
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+ "page_info": { + "page_no": 16, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ.md b/parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ.md new file mode 100644 index 0000000000000000000000000000000000000000..f151b9b1065bd567f26ded1b987be97a45e0d906 --- /dev/null +++ b/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 + +
Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)
Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do
3 ←GMM(D,B)
4 ← GMM(D,A) 5 for k={A,Bj do
Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6
S={(Ii,Ti)|ui<0.5,∀(Ii,Ti) ∈(D,W)} 7
8 for j=num_steps do
9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);
Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10
Train the network k on (B, Bη） by optimizing Lsoft. 11
+ +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. + +
Flickr30KMS-COCO
Image→TextText→ImageImage→TextText→Image
NoiseMethodsR@1R@5R@10R@1 R@5R@10R@1R@5R@10R@1R@5R@10
0%SCAN67.490.395.848.677.785.269.2 93.697.656.086.593.5
VSRN71.390.696.054.781.8 88.276.294.898.262.889.795.1
IMRAM74.193.096.653.979.487.276.7 95.698.561.789.195.0
SAF73.793.396.356.181.588.076.1 95.498.361.889.495.3
SGR75.293.396.656.281.086.578.0 95.898.261.489.395.4
SGRAF NCR77.894.197.458.583.088.879.6 96.298.563.290.796.1
77.394.097.559.684.489.978.7 95.898.563.390.495.8
20%SCAN59.183.490.436.667.077.5 78.266.2 25.191.0 96.445.080.289.3
VSRN IMRAM58.1 63.082.689.340.768.759.074.817.649.064.1
SAF51.086.091.341.471.280.5 68.692.897.655.785.091.0
SGR*62.879.388.038.366.576.267.3 92.5 91.796.6 96.253.4 52.984.592.4
SGR-C72.886.2 90.892.2 95.444.4 56.472.380.467.883.590.1
NCR75.093.997.582.188.675.4 95.297.960.188.5 89.394.8
58.383.089.077.795.598.262.595.3 21.0
50%SCAN27.757.668.816.239.349.840.873.5 84.95.415.1
VSRN14.337.650.012.130.039.423.5 54.769.316.047.865.9
IMRAM9.126.638.22.78.412.721.3 60.275.922.352.864.3
SAF30.363.675.427.953.765.130.4 67.882.333.569.082.8
SGR*36.968.180.229.356.267.060.6 87.493.646.074.279.0
SGR-C69.890.394.850.177.585.271.7 94.197.757.086.693.7
NCR72.993.096.354.379.886.574.694.697.8 59.187.894.5
+ +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. + +
Image→ TextText→Image
MethodsR@1R@5R@10R@1R@5R@10
SCAN (ECCV'18)30.555.365.326.953.064.7
VSRN (ICCV'19)32.661.370.532.559.470.4
IMRAM (CVPR'20)33.157.668.129.056.867.4
SAF (AAAI'21)31.759.368.231.959.067.9
SGR (AAAI'21)11.329.739.613.130.141.6
SGR* (AAAI'21)35.063.473.334.963.072.8
NCR39.564.573.540.364.673.2
+ +# 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. + +
Image→ TextText→Image
Noise RatioMethodsR@1R@5R@10R@1R@5R@10
0%, Zero-ShotCLIP (ViT-L/14+)58.481.588.137.862.472.2
CLIP (ViT-B/32)50.274.683.630.456.066.8
20%,Fine-tuneNCR58.284.291.541.771.081.3
CLIP (ViT-B/32) NCR21.4 56.949.6 83.663.3 91.014.8 40.637.6 69.849.6 80.1
50%, Fine-tuneCLIP (ViT-B/32)10.927.838.37.819.5
NCR53.180.788.537.966.626.8 77.8
+ +# 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. + +
MethodImage →TextText→Image
Co-divideCo-rectifyWarmupR@1R@5R@10R@1R@5R@10
?√<<>72.993.096.354.379.886.5
71.490.895.754.180.386.5
?16.038.451.712.631.442.8
√0.30.61.00.20.51.1
+ +# 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 + +[1] Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. Bottom-up and top-down attention for image captioning and visual question answering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 6077–6086, 2018. +[2] Eric Arazo, Diego Ortego, Paul Albert, Noel O’Connor, and Kevin McGuinness. Unsupervised label noise modeling and loss correction. In International Conference on Machine Learning, pages 312–321. 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In IJCAI, pages 3518–3524, 2017. \ No newline at end of file diff --git a/parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ_content_list.json b/parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..217073437369aabd74bd0e22365c3e432b9a98c7 --- /dev/null +++ b/parse/train/S9ZyhWC17wJ/S9ZyhWC17wJ_content_list.json @@ -0,0 +1,1227 @@ +[ + { + "type": "text", + "text": "Learning with Noisy Correspondence for Cross-modal Matching ", + "text_level": 1, + "bbox": [ + 272, + 122, + 725, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Guocheng Niu Baidu Inc., China niuguocheng@baidu.com ", + "bbox": [ + 429, + 226, + 612, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zhenyu Huang∗ College of Computer Science Sichuan University, China zyhuang.gm@gmail.com ", + "bbox": [ + 200, + 226, + 397, + 282 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xiao Liu TAL Education Group liuxiao15@tal.com ", + "bbox": [ + 647, + 226, + 797, + 267 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wenbiao Ding TAL Education Group dingwenbiao@tal.com ", + "bbox": [ + 212, + 303, + 379, + 344 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xinyan Xiao Baidu Inc., China xiaoxinyan@baidu.com ", + "bbox": [ + 424, + 303, + 598, + 344 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hua Wu Baidu Inc., China wu_hua@baidu.com ", + "bbox": [ + 643, + 303, + 784, + 344 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xi Peng† College of Computer Science Sichuan University, China pengx.gm@gmail.com ", + "bbox": [ + 400, + 366, + 596, + 422 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 457, + 535, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "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. ", + "bbox": [ + 233, + 488, + 766, + 765 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 790, + 310, + 808 + ], + "page_idx": 0 + }, + { + "type": "text", + "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]. ", + "bbox": [ + 176, + 821, + 823, + 863 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 823, + 119 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 126, + 825, + 263 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/8fe147a50fe0a4e139ee18a498d586b8c26880aa0b567f8e71692812be7e2a62.jpg", + "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. " + ], + "image_footnote": [], + "bbox": [ + 205, + 280, + 789, + 402 + ], + "page_idx": 1 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 484, + 825, + 621 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 173, + 628, + 825, + 753 + ], + "page_idx": 1 + }, + { + "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. ", + "bbox": [ + 173, + 758, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related works ", + "text_level": 1, + "bbox": [ + 174, + 89, + 326, + 106 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we briefly introduce some recent developments in cross-modal matching and learning with noisy labels. ", + "bbox": [ + 176, + 121, + 823, + 148 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Cross-modal Matching ", + "text_level": 1, + "bbox": [ + 176, + 166, + 372, + 181 + ], + "page_idx": 2 + }, + { + "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": [ + 174, + 191, + 825, + 412 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 420, + 825, + 544 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Learning with Noisy Labels ", + "text_level": 1, + "bbox": [ + 176, + 560, + 405, + 575 + ], + "page_idx": 2 + }, + { + "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": [ + 174, + 587, + 825, + 794 + ], + "page_idx": 2 + }, + { + "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. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "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 $\\hat { \\alpha } _ { 2 }$ . " + ], + "image_footnote": [], + "bbox": [ + 171, + 89, + 825, + 248 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 463, + 825, + 506 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 The Proposed Method ", + "text_level": 1, + "bbox": [ + 174, + 525, + 390, + 542 + ], + "page_idx": 3 + }, + { + "type": "text", + "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. ", + "bbox": [ + 174, + 558, + 825, + 641 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Co-divide ", + "text_level": 1, + "bbox": [ + 174, + 659, + 281, + 674 + ], + "page_idx": 3 + }, + { + "type": "text", + "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. ", + "bbox": [ + 173, + 684, + 825, + 768 + ], + "page_idx": 3 + }, + { + "type": "text", + "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: ", + "bbox": [ + 173, + 773, + 825, + 886 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6fc5284745d0ad1346714b29253da56ab6a8e8fa66ab3747454a6042d2257c71.jpg", + "text": "$$\n\\ell _ { ( f , g , S ) } = \\{ \\ell _ { i } \\} _ { i = 1 } ^ { N } = \\{ L _ { w } ( I _ { i } , T _ { i } ) \\} _ { i = 1 } ^ { N }\n$$", + "text_format": "latex", + "bbox": [ + 370, + 893, + 625, + 914 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $L _ { w }$ is defined as: ", + "bbox": [ + 174, + 92, + 331, + 106 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3ee3df26f574057fe0d6067ba99e575d516cb6ef402949d4cdaccedc8036cc87.jpg", + "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", + "bbox": [ + 246, + 108, + 750, + 145 + ], + "page_idx": 4 + }, + { + "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]: ", + "bbox": [ + 173, + 148, + 825, + 210 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3b664f0344f0c21be2caace0bac7efb85763fbbc8bd31b5bf4a18615525f996b.jpg", + "text": "$$\np ( \\ell | \\theta ) = \\sum _ { k = 1 } ^ { K } \\beta _ { k } \\phi ( \\ell | k ) ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 213, + 576, + 257 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 261, + 825, + 359 + ], + "page_idx": 4 + }, + { + "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. ", + "bbox": [ + 173, + 364, + 825, + 463 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Co-Rectify ", + "text_level": 1, + "bbox": [ + 174, + 477, + 287, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "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: ", + "bbox": [ + 173, + 502, + 826, + 579 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 279, + 583, + 717, + 626 + ], + "page_idx": 4 + }, + { + "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": [ + 173, + 628, + 826, + 700 + ], + "page_idx": 4 + }, + { + "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, ", + "bbox": [ + 173, + 705, + 826, + 818 + ], + "page_idx": 4 + }, + { + "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", + "bbox": [ + 303, + 819, + 691, + 880 + ], + "page_idx": 4 + }, + { + "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 ", + "bbox": [ + 173, + 882, + 821, + 912 + ], + "page_idx": 4 + }, + { + "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 $[ 0 , 1 )$ . ", + "bbox": [ + 174, + 90, + 825, + 133 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1: Noisy Correspondence Rectifier ", + "text_level": 1, + "bbox": [ + 176, + 154, + 478, + 167 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/5594c7893fc3111e104dd235ec8d9a648eb8693d4b754bb39fa6af022f171953.jpg", + "table_caption": [], + "table_footnote": [ + "Result: Matching models $( A , B )$ " + ], + "table_body": "
Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)
Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do
3 ←GMM(D,B)
4 ← GMM(D,A) 5 for k={A,Bj do
Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6
S={(Ii,Ti)|ui<0.5,∀(Ii,Ti) ∈(D,W)} 7
8 for j=num_steps do
9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);
Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10
Train the network k on (B, Bη） by optimizing Lsoft. 11
", + "bbox": [ + 158, + 170, + 799, + 357 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 Robust Cross-modal Matching ", + "text_level": 1, + "bbox": [ + 174, + 402, + 426, + 417 + ], + "page_idx": 5 + }, + { + "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, ", + "bbox": [ + 174, + 428, + 825, + 470 + ], + "page_idx": 5 + }, + { + "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", + "bbox": [ + 236, + 477, + 761, + 497 + ], + "page_idx": 5 + }, + { + "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": [ + 174, + 505, + 825, + 551 + ], + "page_idx": 5 + }, + { + "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": [ + 441, + 550, + 555, + 583 + ], + "page_idx": 5 + }, + { + "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 $\\hat { \\alpha } _ { i }$ . ", + "bbox": [ + 174, + 585, + 825, + 641 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 173, + 647, + 825, + 746 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 Discussions on Matching Loss ", + "text_level": 1, + "bbox": [ + 176, + 761, + 419, + 776 + ], + "page_idx": 5 + }, + { + "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. ", + "bbox": [ + 174, + 786, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 823, + 133 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 Experiment ", + "text_level": 1, + "bbox": [ + 174, + 154, + 303, + 171 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 185, + 826, + 241 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Datasets and Performance Measurements ", + "text_level": 1, + "bbox": [ + 173, + 258, + 501, + 273 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 285, + 825, + 452 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 457, + 825, + 526 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 Implementation Details ", + "text_level": 1, + "bbox": [ + 174, + 545, + 377, + 559 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 570, + 825, + 681 + ], + "page_idx": 6 + }, + { + "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. ", + "bbox": [ + 174, + 688, + 825, + 770 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Comparisons with State of The Arts ", + "text_level": 1, + "bbox": [ + 176, + 787, + 464, + 803 + ], + "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": [ + 174, + 814, + 825, + 911 + ], + "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": [ + 173, + 90, + 825, + 174 + ], + "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": "
Flickr30KMS-COCO
Image→TextText→ImageImage→TextText→Image
NoiseMethodsR@1R@5R@10R@1 R@5R@10R@1R@5R@10R@1R@5R@10
0%SCAN67.490.395.848.677.785.269.2 93.697.656.086.593.5
VSRN71.390.696.054.781.8 88.276.294.898.262.889.795.1
IMRAM74.193.096.653.979.487.276.7 95.698.561.789.195.0
SAF73.793.396.356.181.588.076.1 95.498.361.889.495.3
SGR75.293.396.656.281.086.578.0 95.898.261.489.395.4
SGRAF NCR77.894.197.458.583.088.879.6 96.298.563.290.796.1
77.394.097.559.684.489.978.7 95.898.563.390.495.8
20%SCAN59.183.490.436.667.077.5 78.266.2 25.191.0 96.445.080.289.3
VSRN IMRAM58.1 63.082.689.340.768.759.074.817.649.064.1
SAF51.086.091.341.471.280.5 68.692.897.655.785.091.0
SGR*62.879.388.038.366.576.267.3 92.5 91.796.6 96.253.4 52.984.592.4
SGR-C72.886.2 90.892.2 95.444.4 56.472.380.467.883.590.1
NCR75.093.997.582.188.675.4 95.297.960.188.5 89.394.8
58.383.089.077.795.598.262.595.3 21.0
50%SCAN27.757.668.816.239.349.840.873.5 84.95.415.1
VSRN14.337.650.012.130.039.423.5 54.769.316.047.865.9
IMRAM9.126.638.22.78.412.721.3 60.275.922.352.864.3
SAF30.363.675.427.953.765.130.4 67.882.333.569.082.8
SGR*36.968.180.229.356.267.060.6 87.493.646.074.279.0
SGR-C69.890.394.850.177.585.271.7 94.197.757.086.693.7
NCR72.993.096.354.379.886.574.694.697.8 59.187.894.5
", + "bbox": [ + 176, + 200, + 823, + 477 + ], + "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": [ + 173, + 484, + 825, + 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, + 825, + 671 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ee4da61e843468259b8e4aa2049d759bd9f314376e9352e94ed997753b97cea1.jpg", + "table_caption": [ + "Table 2: Image-Text Retrieval on CC152K. " + ], + "table_footnote": [], + "table_body": "
Image→ TextText→Image
MethodsR@1R@5R@10R@1R@5R@10
SCAN (ECCV'18)30.555.365.326.953.064.7
VSRN (ICCV'19)32.661.370.532.559.470.4
IMRAM (CVPR'20)33.157.668.129.056.867.4
SAF (AAAI'21)31.759.368.231.959.067.9
SGR (AAAI'21)11.329.739.613.130.141.6
SGR* (AAAI'21)35.063.473.334.963.072.8
NCR39.564.573.540.364.673.2
", + "bbox": [ + 250, + 698, + 748, + 839 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Comparison to pre-trained model ", + "text_level": 1, + "bbox": [ + 174, + 857, + 446, + 872 + ], + "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": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 188 + ], + "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": "
Image→ TextText→Image
Noise RatioMethodsR@1R@5R@10R@1R@5R@10
0%, Zero-ShotCLIP (ViT-L/14+)58.481.588.137.862.472.2
CLIP (ViT-B/32)50.274.683.630.456.066.8
20%,Fine-tuneNCR58.284.291.541.771.081.3
CLIP (ViT-B/32) NCR21.4 56.949.6 83.663.3 91.014.8 40.637.6 69.849.6 80.1
50%, Fine-tuneCLIP (ViT-B/32)10.927.838.37.819.5
NCR53.180.788.537.966.626.8 77.8
", + "bbox": [ + 199, + 361, + 799, + 515 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 Experimental Analysis ", + "text_level": 1, + "bbox": [ + 174, + 535, + 370, + 551 + ], + "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": [ + 174, + 560, + 825, + 616 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5.1 Study on Robustness and Generalizability ", + "text_level": 1, + "bbox": [ + 174, + 631, + 516, + 646 + ], + "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": [ + 174, + 655, + 825, + 738 + ], + "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, + 825, + 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": "
MethodImage →TextText→Image
Co-divideCo-rectifyWarmupR@1R@5R@10R@1R@5R@10
?√<<>72.993.096.354.379.886.5
71.490.895.754.180.386.5
?16.038.451.712.631.442.8
√0.30.61.00.20.51.1
", + "bbox": [ + 205, + 425, + 794, + 536 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.5.4 Noisy Samples ", + "text_level": 1, + "bbox": [ + 173, + 558, + 326, + 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": [ + 173, + 582, + 825, + 638 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/c2ea469f89d768984f1c578d0e0317f6d920992f64f93e122eb3703e75d9c2da.jpg", + "image_caption": [ + "Figure 4: Some noisy examples correctly divided by NCR. " + ], + "image_footnote": [], + "bbox": [ + 174, + 647, + 823, + 750 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 173, + 795, + 299, + 811 + ], + "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": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 9 + }, + { + "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. ", + "bbox": [ + 173, + 119, + 826, + 272 + ], + "page_idx": 10 + }, + { + "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. ", + "bbox": [ + 174, + 321, + 825, + 406 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 425, + 266, + 441 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. Bottom-up and top-down attention for image captioning and visual question answering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 6077–6086, 2018. \n[2] Eric Arazo, Diego Ortego, Paul Albert, Noel O’Connor, and Kevin McGuinness. 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Although a huge number of cross-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 421, + 469, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 421, + 469, + 433 + ], + "score": 1.0, + "content": "modal matching methods have been proposed and achieved remarkable progress", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 432, + 470, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 470, + 443 + ], + "score": 1.0, + "content": "in recent years, almost all of these methods implicitly assume that the multimodal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 442, + 469, + 454 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 469, + 454 + ], + "score": 1.0, + "content": "training data are correctly aligned. In practice, however, such an assumption", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 454, + 470, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 470, + 466 + ], + "score": 1.0, + "content": "is extremely expensive even impossible to satisfy. Based on this observation,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 464, + 471, + 477 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 471, + 477 + ], + "score": 1.0, + "content": "we reveal and study a latent and challenging direction in cross-modal matching,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 475, + 470, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 470, + 487 + ], + "score": 1.0, + "content": "named noisy correspondence, which could be regarded as a new paradigm of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "score": 1.0, + "content": "noisy labels. Different from the traditional noisy labels which mainly refer to the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 497, + 469, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 469, + 509 + ], + "score": 1.0, + "content": "errors in category labels, our noisy correspondence refers to the mismatch paired", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 508, + 470, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 508, + 470, + 520 + ], + "score": 1.0, + "content": "samples. To solve this new problem, we propose a novel method for learning with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 519, + 470, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 519, + 470, + 531 + ], + "score": 1.0, + "content": "noisy correspondence, named Noisy Correspondence Rectifier (NCR). In brief,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 529, + 469, + 542 + ], + "spans": [ + { + "bbox": [ + 141, + 529, + 469, + 542 + ], + "score": 1.0, + "content": "NCR divides the data into clean and noisy partitions based on the memorization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 540, + 469, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 469, + 553 + ], + "score": 1.0, + "content": "effect of neural networks and then rectifies the correspondence via an adaptive", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 552, + 470, + 563 + ], + "spans": [ + { + "bbox": [ + 141, + 552, + 470, + 563 + ], + "score": 1.0, + "content": "prediction model in a co-teaching manner. To verify the effectiveness of our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 562, + 470, + 575 + ], + "spans": [ + { + "bbox": [ + 141, + 562, + 470, + 575 + ], + "score": 1.0, + "content": "method, we conduct experiments by using the image-text matching as a showcase.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 572, + 470, + 586 + ], + "spans": [ + { + "bbox": [ + 141, + 572, + 470, + 586 + ], + "score": 1.0, + "content": "Extensive experiments on Flickr30K, MS-COCO, and Conceptual Captions verify", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 583, + 471, + 597 + ], + "spans": [ + { + "bbox": [ + 141, + 583, + 471, + 597 + ], + "score": 1.0, + "content": "the effectiveness of our method. The code could be accessed from www.pengxi.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 597, + 158, + 607 + ], + "spans": [ + { + "bbox": [ + 141, + 597, + 158, + 607 + ], + "score": 1.0, + "content": "me.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 35.5, + "bbox_fs": [ + 141, + 388, + 471, + 607 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 626, + 190, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 192, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 192, + 642 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 108, + 651, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "As one of the most fundamental techniques in multimodal learning, cross-modal matching aims to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 675 + ], + "score": 1.0, + "content": "bridge different modalities. In recent years, some cross-modal matching methods [19, 11, 7, 26] have", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 673, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 506, + 687 + ], + "score": 1.0, + "content": "been proposed based on Deep Neural Networks (DNNs), which achieved remarkable progress in a", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "variety of applications, such as clustering [29, 24], image/video captioning [1, 44, 22], cross-modal", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 333, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 333, + 96 + ], + "score": 1.0, + "content": "retrieval [40, 19, 13], and visual question answering [9].", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 650, + 506, + 687 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "variety of applications, such as clustering [29, 24], image/video captioning [1, 44, 22], cross-modal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 333, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 333, + 96 + ], + "score": 1.0, + "content": "retrieval [40, 19, 13], and visual question answering [9].", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "In general, most existing cross-modal matching methods embed different modalities into a common", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "space wherein the similarity of positive cross-modal pairs is maximized and that of the negative ones", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 104, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "is minimized. Although these methods have achieved promising results, their success depends on an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "implicit data assumption, i.e., the training data are correctly aligned across modalities. For example,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "in the vision-and-language tasks, the text needs to accurately describe the image content, and vice", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "versa. In practice, however, it is extremely expensive and time-consuming to annotate or collect such", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "data pairs. Especially, considering the data collected from the Internet [35, 14], it is inevitable to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "collect some mismatched pairs which are wrongly treated as the matched ones. To the best of our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "knowledge, such a special noisy label (correspondence) problem has been ignored so far, which will", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "score": 1.0, + "content": "remarkably degrade the performance of matching methods as shown in our experiments.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5 + }, + { + "type": "image", + "bbox": [ + 126, + 222, + 483, + 319 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 222, + 483, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 222, + 483, + 319 + ], + "spans": [ + { + "bbox": [ + 126, + 222, + 483, + 319 + ], + "score": 0.972, + "type": "image", + "image_path": "8fe147a50fe0a4e139ee18a498d586b8c26880aa0b567f8e71692812be7e2a62.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 126, + 222, + 483, + 254.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 126, + 254.33333333333334, + 483, + 286.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 126, + 286.6666666666667, + 483, + 319.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 327, + 505, + 382 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Figure 1: Noisy labels vs. Noisy Correspondence. We denote the noisy samples with red lines and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 352 + ], + "score": 1.0, + "content": "clean samples with green lines. The traditional noisy labels mainly refer to the errors in category", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 363 + ], + "score": 1.0, + "content": "labels, while the noisy correspondence refers to the alignment errors in paired data. For the noisy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "correspondence in cross-modal matching, the true positive pair correctly guides the cross-modal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 425, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 425, + 384 + ], + "score": 1.0, + "content": "matching, while the false positive pair causes incorrect supervision for training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + } + ], + "index": 15.0 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 397 + ], + "score": 1.0, + "content": "Based on the above observation, we reveal a new paradigm for the noisy labels, named noisy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "correspondence. Different from the traditional noisy labels, the noisy correspondence refers to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "the alignment errors in paired data rather than the errors in category annotations (see Fig. 1). To", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "the best of our knowledge, there is no effort has been devoted to study this new problem and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "closest paradigm might be the partially view-aligned problem (PVP) [12, 41]. However, PVP is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "remarkably different from noisy correspondence, and the latter is more practical than the former. To", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "be specific, PVP focuses on that the cross-modal alignment is totally unavailable, whereas the noisy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "correspondence focuses on that some correspondences are incorrect. In addition, PVP assumes that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "some correctly aligned data are available for training, whereas our noisy correspondence assumes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 266, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 266, + 493 + ], + "score": 1.0, + "content": "that the clean and noisy data are mixed.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "To solve the noisy correspondence problem in cross-modal matching, we propose a novel method,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "named Noisy Correspondence Rectifier (NCR). Our method is based on the memorization effect", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "of DNNs observed in [3, 39], i.e., DNNs tend to learn the simple patterns before fitting noisy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "samples. Motivated by this empirical observation, NCR divides the data into two relative accurate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "data partitions, i.e., “noisy” and “clean” subsets, based on their loss difference. After that, NCR", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "employs an adaptive prediction function for label rectifying so that the false positives and the true", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "positives could be identified from the “clean” and the “noisy” subsets, respectively. Furthermore, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "propose a novel triplet loss for robust cross-modal matching by recasting the rectified labels as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 157, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 157, + 599 + ], + "score": 1.0, + "content": "soft margin.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "The main contributions and novelties of this paper could be summarized as below. i) We reveal a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "new problem in cross-modal analysis, which is also a new paradigm for noisy labels, termed noisy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "alignment errors in paired data instead of the errors in category annotations. To the best of our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "knowledge, this work could be the first study on this problem. ii) To solve the noisy correspondence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "problem, we propose a novel method for learning with noisy correspondence, named Noisy Corre-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "spondence Rectifier (NCR). One major novelty of NCR is that the rectified label is elegantly recasted", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "as the soft margin of a triplet loss so that the robust cross-modal matching could be achieved. iii)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "To verify the effectiveness of our method, we conduct experiments on the image-text matching task.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "Extensive experiments on three challenging datasets verify the effectiveness of our method in the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 220, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 220, + 723 + ], + "score": 1.0, + "content": "synthesized and real noises.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "In general, most existing cross-modal matching methods embed different modalities into a common", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "space wherein the similarity of positive cross-modal pairs is maximized and that of the negative ones", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 104, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "is minimized. Although these methods have achieved promising results, their success depends on an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "implicit data assumption, i.e., the training data are correctly aligned across modalities. For example,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "in the vision-and-language tasks, the text needs to accurately describe the image content, and vice", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "versa. In practice, however, it is extremely expensive and time-consuming to annotate or collect such", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "data pairs. Especially, considering the data collected from the Internet [35, 14], it is inevitable to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "collect some mismatched pairs which are wrongly treated as the matched ones. To the best of our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "knowledge, such a special noisy label (correspondence) problem has been ignored so far, which will", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 461, + 211 + ], + "score": 1.0, + "content": "remarkably degrade the performance of matching methods as shown in our experiments.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 100, + 506, + 211 + ] + }, + { + "type": "image", + "bbox": [ + 126, + 222, + 483, + 319 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 222, + 483, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 222, + 483, + 319 + ], + "spans": [ + { + "bbox": [ + 126, + 222, + 483, + 319 + ], + "score": 0.972, + "type": "image", + "image_path": "8fe147a50fe0a4e139ee18a498d586b8c26880aa0b567f8e71692812be7e2a62.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 126, + 222, + 483, + 254.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 126, + 254.33333333333334, + 483, + 286.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 126, + 286.6666666666667, + 483, + 319.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 327, + 505, + 382 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Figure 1: Noisy labels vs. Noisy Correspondence. We denote the noisy samples with red lines and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 352 + ], + "score": 1.0, + "content": "clean samples with green lines. The traditional noisy labels mainly refer to the errors in category", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 363 + ], + "score": 1.0, + "content": "labels, while the noisy correspondence refers to the alignment errors in paired data. For the noisy", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "correspondence in cross-modal matching, the true positive pair correctly guides the cross-modal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 425, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 425, + 384 + ], + "score": 1.0, + "content": "matching, while the false positive pair causes incorrect supervision for training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + } + ], + "index": 15.0 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 397 + ], + "score": 1.0, + "content": "Based on the above observation, we reveal a new paradigm for the noisy labels, named noisy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "correspondence. Different from the traditional noisy labels, the noisy correspondence refers to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "the alignment errors in paired data rather than the errors in category annotations (see Fig. 1). To", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "the best of our knowledge, there is no effort has been devoted to study this new problem and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "closest paradigm might be the partially view-aligned problem (PVP) [12, 41]. However, PVP is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "remarkably different from noisy correspondence, and the latter is more practical than the former. To", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "be specific, PVP focuses on that the cross-modal alignment is totally unavailable, whereas the noisy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "correspondence focuses on that some correspondences are incorrect. In addition, PVP assumes that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "some correctly aligned data are available for training, whereas our noisy correspondence assumes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 266, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 266, + 493 + ], + "score": 1.0, + "content": "that the clean and noisy data are mixed.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 381, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "To solve the noisy correspondence problem in cross-modal matching, we propose a novel method,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "named Noisy Correspondence Rectifier (NCR). Our method is based on the memorization effect", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "of DNNs observed in [3, 39], i.e., DNNs tend to learn the simple patterns before fitting noisy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "samples. Motivated by this empirical observation, NCR divides the data into two relative accurate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "data partitions, i.e., “noisy” and “clean” subsets, based on their loss difference. After that, NCR", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "employs an adaptive prediction function for label rectifying so that the false positives and the true", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "positives could be identified from the “clean” and the “noisy” subsets, respectively. Furthermore, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "propose a novel triplet loss for robust cross-modal matching by recasting the rectified labels as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 157, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 157, + 599 + ], + "score": 1.0, + "content": "soft margin.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 497, + 506, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "The main contributions and novelties of this paper could be summarized as below. i) We reveal a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "new problem in cross-modal analysis, which is also a new paradigm for noisy labels, termed noisy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "correspondence. Different from the traditional noisy labels, the noisy correspondence refers to the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "alignment errors in paired data instead of the errors in category annotations. To the best of our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "knowledge, this work could be the first study on this problem. ii) To solve the noisy correspondence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "problem, we propose a novel method for learning with noisy correspondence, named Noisy Corre-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "spondence Rectifier (NCR). One major novelty of NCR is that the rectified label is elegantly recasted", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "as the soft margin of a triplet loss so that the robust cross-modal matching could be achieved. iii)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "To verify the effectiveness of our method, we conduct experiments on the image-text matching task.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "Extensive experiments on three challenging datasets verify the effectiveness of our method in the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 220, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 220, + 723 + ], + "score": 1.0, + "content": "synthesized and real noises.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 602, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 200, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 200, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 200, + 86 + ], + "score": 1.0, + "content": "2 Related works", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 96, + 504, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 94, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 110 + ], + "score": 1.0, + "content": "In this section, we briefly introduce some recent developments in cross-modal matching and learning", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 178, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 178, + 119 + ], + "score": 1.0, + "content": "with noisy labels.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 132, + 228, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 230, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 230, + 146 + ], + "score": 1.0, + "content": "2.1 Cross-modal Matching", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 152, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "score": 1.0, + "content": "Most existing cross-modal matching works seek to learn a common space wherein different modalities", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "are comparable. In general, existing works could be roughly divided into two categories: 1) Coarse-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "grained Matching. It often utilizes multiple neural networks to compute a global feature and each", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "network is used for a specific modality [17, 37, 8]. For example, Kiros et al. [17] use a Convolutional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "score": 1.0, + "content": "Neural Network (CNN) and a Gated Recurrent Unit (GRU) to obtain the image and text features,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "while enforcing the similarity of positive pairs larger than that of the negative ones. To further", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 246, + 231 + ], + "score": 1.0, + "content": "boost the matching performance,", + "type": "text" + }, + { + "bbox": [ + 246, + 218, + 279, + 229 + ], + "score": 0.82, + "content": "{ \\mathrm { V S E } } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "[8] uses some representative negatives to improve the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "discrimination of the model. 2) Fine-grained Matching. It seeks to measure the fine-grained similarity", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 253 + ], + "score": 1.0, + "content": "for cross-modal matching [19, 21, 7]. For example, SCAN [19] proposes learning the latent semantic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "correspondence between the image regions and words that are extracted by bottom-up attention [1]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 507, + 276 + ], + "score": 1.0, + "content": "and GRU, respectively. VSRN [21] adopts a graph convolutional network for semantic reasoning.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "SGRAF [7] proposes constructing a similarity graph to reason the similarity and adopting an attention", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "filtration technique to eliminate the less-meaningful alignments. Recently Chun et al. [6] introduce a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "new paradigm for cross-modal matching, i.e. possible many-to-many correspondence that existed in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "score": 1.0, + "content": "the image and captions. To achieve this, they propose to use probabilistic representations to model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 277, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 277, + 329 + ], + "score": 1.0, + "content": "the possible one-to-many correspondence.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "Although promising results have been achieved in recent years, the existing methods heavily rely", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "on the correctly aligned data. In practice, however, such well-matched data is expensive and time-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "consuming to collect. Moreover, some recent works [35, 14] show that a large-scale dataset collected", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "from the wild could remarkably improve the performance of the model. However, such a data will", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "inevitably contain some mismatched pairs. Hence, it is highly expected to develop some methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "which are robust against the noisy correspondence, which has not been studied as far as we know.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "score": 1.0, + "content": "Different from the many-to-many correspondence [6] between image and captions, NCR reveals the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "noisy correspondence problem which refers to the alignment errors of image-text pairs and proposes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 421, + 393, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 393, + 432 + ], + "score": 1.0, + "content": "to eliminate the negative impact from noisy pairs for downstream tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 248, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 249, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 249, + 458 + ], + "score": 1.0, + "content": "2.2 Learning with Noisy Labels", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "To handle the possible noisy annotations in the training data, a large number of methods have been", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "proposed and almost all of them focus on the classification task [36, 27]. To reduce the negative", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "impact of the noisy labels, the existing works often resort to robust architecture design, regularization,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "loss adjustment, or sample selection methods. Here, we mainly introduce the last two approaches", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "which are most related to this work. To be specific, the loss adjustment achieves robustness by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "adjusting the contribution of clean and noisy samples w.r.t. the loss. For example, Reed et al. [32]", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "proposed a bootstrapping loss based on the model predictions for loss correction. Zhang et al. [45]", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "provided some theoretical explanations for the label correction along with a new label correction", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "algorithm. Different from the loss adjustment methods, sample selection methods aim to select clean", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "samples from a noisy dataset. For example, Arpit et al. [3] showed that DNNs tend to learn simple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "patterns before fitting noisy samples, namely the memorization effect. Motivated from this, Arazo", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 597 + ], + "score": 1.0, + "content": "et al. [2] proposed treating the samples with small loss as the clean samples. To avoid the selection", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "bias of clean samples, Co-teaching methods [10, 43] use the samples with small loss to iteratively", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "train two networks. In recent, DivideMix [20] adopts the MixMatch method [4] for semi-supervised", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 618, + 278, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 278, + 631 + ], + "score": 1.0, + "content": "learning with the clean and noisy samples.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "Unlike the above noisy label studies, this paper focuses on the noisy correspondence problem which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "considers mismatched multimodal data pairs instead of incorrectly annotated data points. Besides", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "the difference in the problem, this work is also different from the aforementioned studies in the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "methodology. To be specific, in cross-modal matching, it is impossible to directly adopt these noise", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 507, + 691 + ], + "score": 1.0, + "content": "label learning methods to solve the noisy correspondence problem due to the following two reasons.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "First, most of the noisy label learning methods propose to use the model’s prediction for label", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 701, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 504, + 712 + ], + "score": 1.0, + "content": "rectifying in the scenario of classification, while it is intractable to directly predict the correspondence", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "of given pairs in matching models. Second, even if we can rectify the noisy correspondence somehow,", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 200, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 200, + 86 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 200, + 86 + ], + "score": 1.0, + "content": "2 Related works", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 96, + 504, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 94, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 110 + ], + "score": 1.0, + "content": "In this section, we briefly introduce some recent developments in cross-modal matching and learning", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 178, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 178, + 119 + ], + "score": 1.0, + "content": "with noisy labels.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 94, + 506, + 119 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 132, + 228, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 230, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 230, + 146 + ], + "score": 1.0, + "content": "2.1 Cross-modal Matching", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 152, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 165 + ], + "score": 1.0, + "content": "Most existing cross-modal matching works seek to learn a common space wherein different modalities", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "are comparable. In general, existing works could be roughly divided into two categories: 1) Coarse-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "grained Matching. It often utilizes multiple neural networks to compute a global feature and each", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "network is used for a specific modality [17, 37, 8]. For example, Kiros et al. [17] use a Convolutional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 507, + 209 + ], + "score": 1.0, + "content": "Neural Network (CNN) and a Gated Recurrent Unit (GRU) to obtain the image and text features,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "while enforcing the similarity of positive pairs larger than that of the negative ones. To further", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 246, + 231 + ], + "score": 1.0, + "content": "boost the matching performance,", + "type": "text" + }, + { + "bbox": [ + 246, + 218, + 279, + 229 + ], + "score": 0.82, + "content": "{ \\mathrm { V S E } } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "[8] uses some representative negatives to improve the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "discrimination of the model. 2) Fine-grained Matching. It seeks to measure the fine-grained similarity", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 253 + ], + "score": 1.0, + "content": "for cross-modal matching [19, 21, 7]. For example, SCAN [19] proposes learning the latent semantic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "correspondence between the image regions and words that are extracted by bottom-up attention [1]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 507, + 276 + ], + "score": 1.0, + "content": "and GRU, respectively. VSRN [21] adopts a graph convolutional network for semantic reasoning.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "SGRAF [7] proposes constructing a similarity graph to reason the similarity and adopting an attention", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "filtration technique to eliminate the less-meaningful alignments. Recently Chun et al. [6] introduce a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "new paradigm for cross-modal matching, i.e. possible many-to-many correspondence that existed in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "score": 1.0, + "content": "the image and captions. To achieve this, they propose to use probabilistic representations to model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 277, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 277, + 329 + ], + "score": 1.0, + "content": "the possible one-to-many correspondence.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 152, + 507, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "Although promising results have been achieved in recent years, the existing methods heavily rely", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "on the correctly aligned data. In practice, however, such well-matched data is expensive and time-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "consuming to collect. Moreover, some recent works [35, 14] show that a large-scale dataset collected", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "from the wild could remarkably improve the performance of the model. However, such a data will", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "inevitably contain some mismatched pairs. Hence, it is highly expected to develop some methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "which are robust against the noisy correspondence, which has not been studied as far as we know.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "score": 1.0, + "content": "Different from the many-to-many correspondence [6] between image and captions, NCR reveals the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "noisy correspondence problem which refers to the alignment errors of image-text pairs and proposes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 421, + 393, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 393, + 432 + ], + "score": 1.0, + "content": "to eliminate the negative impact from noisy pairs for downstream tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 333, + 506, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 248, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 249, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 249, + 458 + ], + "score": 1.0, + "content": "2.2 Learning with Noisy Labels", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "To handle the possible noisy annotations in the training data, a large number of methods have been", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "proposed and almost all of them focus on the classification task [36, 27]. To reduce the negative", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "impact of the noisy labels, the existing works often resort to robust architecture design, regularization,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "loss adjustment, or sample selection methods. Here, we mainly introduce the last two approaches", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "which are most related to this work. To be specific, the loss adjustment achieves robustness by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "adjusting the contribution of clean and noisy samples w.r.t. the loss. For example, Reed et al. [32]", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "proposed a bootstrapping loss based on the model predictions for loss correction. Zhang et al. [45]", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "provided some theoretical explanations for the label correction along with a new label correction", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "algorithm. Different from the loss adjustment methods, sample selection methods aim to select clean", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "samples from a noisy dataset. For example, Arpit et al. [3] showed that DNNs tend to learn simple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "patterns before fitting noisy samples, namely the memorization effect. Motivated from this, Arazo", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 597 + ], + "score": 1.0, + "content": "et al. [2] proposed treating the samples with small loss as the clean samples. To avoid the selection", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "bias of clean samples, Co-teaching methods [10, 43] use the samples with small loss to iteratively", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "train two networks. In recent, DivideMix [20] adopts the MixMatch method [4] for semi-supervised", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 618, + 278, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 278, + 631 + ], + "score": 1.0, + "content": "learning with the clean and noisy samples.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 464, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "Unlike the above noisy label studies, this paper focuses on the noisy correspondence problem which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "considers mismatched multimodal data pairs instead of incorrectly annotated data points. Besides", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "the difference in the problem, this work is also different from the aforementioned studies in the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "methodology. To be specific, in cross-modal matching, it is impossible to directly adopt these noise", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 507, + 691 + ], + "score": 1.0, + "content": "label learning methods to solve the noisy correspondence problem due to the following two reasons.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "First, most of the noisy label learning methods propose to use the model’s prediction for label", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 701, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 504, + 712 + ], + "score": 1.0, + "content": "rectifying in the scenario of classification, while it is intractable to directly predict the correspondence", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "of given pairs in matching models. Second, even if we can rectify the noisy correspondence somehow,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "the rectified real-valued labels are incompatible with the existing matching methods since most of", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "them assume the given labels are binary. 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(a) Training pipeline of NCR. NCR consists of two", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "individual networks (A, B) which work in the manner of co-teaching. 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Finally,", + "type": "text" + }, + { + "bbox": [ + 262, + 274, + 276, + 287 + ], + "score": 0.89, + "content": "\\hat { \\cal S } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 275, + 295, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 295, + 274, + 310, + 286 + ], + "score": 0.9, + "content": "\\hat { S ^ { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 275, + 448, + 289 + ], + "score": 1.0, + "content": "will be used to train the network", + "type": "text" + }, + { + "bbox": [ + 448, + 277, + 458, + 286 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 275, + 477, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 277, + 486, + 286 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 275, + 507, + 289 + ], + "score": 1.0, + "content": "in a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "swapping way. (b) Robust image-text matching network. For example, the network A projects the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 300, + 312 + ], + "score": 1.0, + "content": "image and text by the modal-specific networks", + "type": "text" + }, + { + "bbox": [ + 300, + 299, + 307, + 310 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 298, + 326, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 300, + 333, + 310 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 298, + 473, + 312 + ], + "score": 1.0, + "content": ", respectively. Then the similarity", + "type": "text" + }, + { + "bbox": [ + 473, + 298, + 505, + 311 + ], + "score": 0.92, + "content": "S ( I , T )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 308, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 261, + 323 + ], + "score": 1.0, + "content": "is computed on the extracted features", + "type": "text" + }, + { + "bbox": [ + 262, + 309, + 281, + 321 + ], + "score": 0.92, + "content": "f ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 308, + 300, + 323 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 300, + 309, + 321, + 321 + ], + "score": 0.92, + "content": "g ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 308, + 506, + 323 + ], + "score": 1.0, + "content": ". To achieve robust image-text matching, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 318, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 335, + 335 + ], + "score": 1.0, + "content": "rectified soft labels are recast as the soft margin of our loss", + "type": "text" + }, + { + "bbox": [ + 335, + 321, + 358, + 333 + ], + "score": 0.92, + "content": "\\scriptstyle L _ { s o f t }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 318, + 480, + 335 + ], + "score": 1.0, + "content": ". As shown, for a given anchor,", + "type": "text" + }, + { + "bbox": [ + 480, + 321, + 504, + 333 + ], + "score": 0.9, + "content": "\\boldsymbol { L _ { s o f t } }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 413, + 345 + ], + "score": 1.0, + "content": "will enforce the true positive to closer to it than the negative by a large margin", + "type": "text" + }, + { + "bbox": [ + 413, + 332, + 425, + 342 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 331, + 506, + 345 + ], + "score": 1.0, + "content": ", and meanwhile the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 342, + 272, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 256, + 355 + ], + "score": 1.0, + "content": "false positive will has a small margin", + "type": "text" + }, + { + "bbox": [ + 257, + 343, + 268, + 353 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 342, + 272, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "the rectified real-valued labels are incompatible with the existing matching methods since most of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "them assume the given labels are binary. To address these problems, NCR proposes an adaptive", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 389, + 452, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 452, + 403 + ], + "score": 1.0, + "content": "prediction function and a novel triplet loss by recasting the soft labels as soft margins.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 416, + 239, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 241, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 241, + 433 + ], + "score": 1.0, + "content": "3 The Proposed Method", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "In this section, we elaborate on the proposed method, i.e., Noisy Correspondence Rectifier (NCR)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "which could be the first work to solve the noisy correspondence problem in cross-modal matching. In", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 462, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 478 + ], + "score": 1.0, + "content": "Section 3.1, we introduce the co-divide module which splits the training data into the clean and noisy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "subsets. After that, we introduce how to rectify the labels with an adaptive prediction function in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "Section 3.2. Finally, we detail how to combine the co-divide and co-rectify modules to achieve robust", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 258, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 258, + 509 + ], + "score": 1.0, + "content": "cross-modal matching in Section 3.3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 172, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 173, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 173, + 536 + ], + "score": 1.0, + "content": "3.1 Co-divide", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "Without loss of generality, we first introduce the cross-modal matching task by taking the image-text", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 551, + 508, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 308, + 569 + ], + "score": 1.0, + "content": "matching as a showcase. Given the training data", + "type": "text" + }, + { + "bbox": [ + 308, + 553, + 397, + 567 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ ( I _ { i } , T _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { \\overline { { N } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 551, + 429, + 569 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 429, + 555, + 439, + 564 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 551, + 508, + 569 + ], + "score": 1.0, + "content": "is the data size,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 135, + 577 + ], + "score": 0.91, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 565, + 244, + 578 + ], + "score": 1.0, + "content": "is an image-text pair and", + "type": "text" + }, + { + "bbox": [ + 244, + 565, + 293, + 577 + ], + "score": 0.93, + "content": "y _ { i } \\in \\{ 1 , \\bar { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "indicates that the pair belong to the same instance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "(positive) or not (negative). For the noisy correspondence case, it defines that an unknown portion of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 116, + 596 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 586, + 195, + 600 + ], + "score": 1.0, + "content": "is mismatched, i.e.,", + "type": "text" + }, + { + "bbox": [ + 196, + 587, + 225, + 599 + ], + "score": 0.92, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 586, + 386, + 600 + ], + "score": 1.0, + "content": "is a negative pair but wrongly labeled as", + "type": "text" + }, + { + "bbox": [ + 387, + 587, + 414, + 598 + ], + "score": 0.92, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 586, + 505, + 600 + ], + "score": 1.0, + "content": ". To solve such a noisy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 444, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 444, + 612 + ], + "score": 1.0, + "content": "correspondence problem, we propose NCR to achieve robust cross-modal matching.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "To begin, we project the visual and textual modalities into a shared space via two modal-specific", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 145, + 637 + ], + "score": 1.0, + "content": "networks", + "type": "text" + }, + { + "bbox": [ + 145, + 626, + 152, + 636 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 625, + 169, + 637 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 170, + 627, + 176, + 636 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 625, + 505, + 637 + ], + "score": 1.0, + "content": ", respectively. Then the similarity of the given image-text pairs is computed through", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 107, + 636, + 165, + 648 + ], + "score": 0.92, + "content": "S ( f ( I ) , g ( T ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 636, + 279, + 649 + ], + "score": 1.0, + "content": ". For simplicity, we denote", + "type": "text" + }, + { + "bbox": [ + 279, + 636, + 337, + 648 + ], + "score": 0.93, + "content": "S ( f ( \\dot { I } ) , g ( T ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 636, + 350, + 649 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 351, + 636, + 383, + 648 + ], + "score": 0.93, + "content": "S ( I , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "in the following. Some early", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "empirical studies [3] show that DNNs tend to first learn simple samples and then gradually fit the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "noisy samples. This so-called memorization effect of DNNs will lead to a relatively low loss for the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "clean samples. 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(a) Training pipeline of NCR. NCR consists of two", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "individual networks (A, B) which work in the manner of co-teaching. 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Then, based on the memorization effect of DNclean and noisy subsets at each epoch using either A or B, i.e.,", + "type": "text" + }, + { + "bbox": [ + 352, + 248, + 417, + 261 + ], + "score": 0.93, + "content": "S ^ { A } = ( S _ { c } ^ { A } , S _ { n } ^ { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 238, + 435, + 267 + ], + "score": 1.0, + "content": "the and", + "type": "text" + }, + { + "bbox": [ + 435, + 248, + 503, + 261 + ], + "score": 0.92, + "content": "S ^ { B } = \\mathsf { \\bar { ( } } S _ { c } ^ { B } , S _ { n } ^ { B } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 260, + 507, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 332, + 277 + ], + "score": 1.0, + "content": "After that, NCR will co-rectify the correspondence of", + "type": "text" + }, + { + "bbox": [ + 333, + 262, + 375, + 275 + ], + "score": 0.94, + "content": "\\{ { \\cal S } ^ { A } , { \\cal S } ^ { B } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 260, + 423, + 277 + ], + "score": 1.0, + "content": "and obtain", + "type": "text" + }, + { + "bbox": [ + 424, + 261, + 466, + 275 + ], + "score": 0.93, + "content": "\\{ \\hat { S ^ { A } } , \\hat { S ^ { B } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 260, + 507, + 277 + ], + "score": 1.0, + "content": "using an", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 274, + 507, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 261, + 289 + ], + "score": 1.0, + "content": "adaptive prediction function. Finally,", + "type": "text" + }, + { + "bbox": [ + 262, + 274, + 276, + 287 + ], + "score": 0.89, + "content": "\\hat { \\cal S } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 275, + 295, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 295, + 274, + 310, + 286 + ], + "score": 0.9, + "content": "\\hat { S ^ { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 275, + 448, + 289 + ], + "score": 1.0, + "content": "will be used to train the network", + "type": "text" + }, + { + "bbox": [ + 448, + 277, + 458, + 286 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 275, + 477, + 289 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 277, + 486, + 286 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 275, + 507, + 289 + ], + "score": 1.0, + "content": "in a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "swapping way. (b) Robust image-text matching network. For example, the network A projects the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 300, + 312 + ], + "score": 1.0, + "content": "image and text by the modal-specific networks", + "type": "text" + }, + { + "bbox": [ + 300, + 299, + 307, + 310 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 298, + 326, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 300, + 333, + 310 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 298, + 473, + 312 + ], + "score": 1.0, + "content": ", respectively. Then the similarity", + "type": "text" + }, + { + "bbox": [ + 473, + 298, + 505, + 311 + ], + "score": 0.92, + "content": "S ( I , T )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 308, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 261, + 323 + ], + "score": 1.0, + "content": "is computed on the extracted features", + "type": "text" + }, + { + "bbox": [ + 262, + 309, + 281, + 321 + ], + "score": 0.92, + "content": "f ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 308, + 300, + 323 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 300, + 309, + 321, + 321 + ], + "score": 0.92, + "content": "g ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 308, + 506, + 323 + ], + "score": 1.0, + "content": ". To achieve robust image-text matching, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 318, + 504, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 335, + 335 + ], + "score": 1.0, + "content": "rectified soft labels are recast as the soft margin of our loss", + "type": "text" + }, + { + "bbox": [ + 335, + 321, + 358, + 333 + ], + "score": 0.92, + "content": "\\scriptstyle L _ { s o f t }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 318, + 480, + 335 + ], + "score": 1.0, + "content": ". As shown, for a given anchor,", + "type": "text" + }, + { + "bbox": [ + 480, + 321, + 504, + 333 + ], + "score": 0.9, + "content": "\\boldsymbol { L _ { s o f t } }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 413, + 345 + ], + "score": 1.0, + "content": "will enforce the true positive to closer to it than the negative by a large margin", + "type": "text" + }, + { + "bbox": [ + 413, + 332, + 425, + 342 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 331, + 506, + 345 + ], + "score": 1.0, + "content": ", and meanwhile the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 342, + 272, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 256, + 355 + ], + "score": 1.0, + "content": "false positive will has a small margin", + "type": "text" + }, + { + "bbox": [ + 257, + 343, + 268, + 353 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 342, + 272, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + } + ], + "index": 4.75 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 401 + ], + "lines": [], + "index": 16, + "bbox_fs": [ + 105, + 366, + 506, + 403 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 416, + 239, + 430 + ], + "lines": [ + { + "bbox": [ + 104, + 415, + 241, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 241, + 433 + ], + "score": 1.0, + "content": "3 The Proposed Method", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "In this section, we elaborate on the proposed method, i.e., Noisy Correspondence Rectifier (NCR)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "which could be the first work to solve the noisy correspondence problem in cross-modal matching. In", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 462, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 478 + ], + "score": 1.0, + "content": "Section 3.1, we introduce the co-divide module which splits the training data into the clean and noisy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "subsets. After that, we introduce how to rectify the labels with an adaptive prediction function in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "Section 3.2. Finally, we detail how to combine the co-divide and co-rectify modules to achieve robust", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 497, + 258, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 258, + 509 + ], + "score": 1.0, + "content": "cross-modal matching in Section 3.3.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 442, + 506, + 509 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 172, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 173, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 173, + 536 + ], + "score": 1.0, + "content": "3.1 Co-divide", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "Without loss of generality, we first introduce the cross-modal matching task by taking the image-text", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 551, + 508, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 308, + 569 + ], + "score": 1.0, + "content": "matching as a showcase. Given the training data", + "type": "text" + }, + { + "bbox": [ + 308, + 553, + 397, + 567 + ], + "score": 0.93, + "content": "\\mathcal { D } = \\{ ( I _ { i } , T _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { \\overline { { N } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 551, + 429, + 569 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 429, + 555, + 439, + 564 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 551, + 508, + 569 + ], + "score": 1.0, + "content": "is the data size,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 135, + 577 + ], + "score": 0.91, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 565, + 244, + 578 + ], + "score": 1.0, + "content": "is an image-text pair and", + "type": "text" + }, + { + "bbox": [ + 244, + 565, + 293, + 577 + ], + "score": 0.93, + "content": "y _ { i } \\in \\{ 1 , \\bar { 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "indicates that the pair belong to the same instance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 588 + ], + "score": 1.0, + "content": "(positive) or not (negative). For the noisy correspondence case, it defines that an unknown portion of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 107, + 588, + 116, + 596 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 586, + 195, + 600 + ], + "score": 1.0, + "content": "is mismatched, i.e.,", + "type": "text" + }, + { + "bbox": [ + 196, + 587, + 225, + 599 + ], + "score": 0.92, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 586, + 386, + 600 + ], + "score": 1.0, + "content": "is a negative pair but wrongly labeled as", + "type": "text" + }, + { + "bbox": [ + 387, + 587, + 414, + 598 + ], + "score": 0.92, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 586, + 505, + 600 + ], + "score": 1.0, + "content": ". To solve such a noisy", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 444, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 444, + 612 + ], + "score": 1.0, + "content": "correspondence problem, we propose NCR to achieve robust cross-modal matching.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 542, + 508, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "To begin, we project the visual and textual modalities into a shared space via two modal-specific", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 145, + 637 + ], + "score": 1.0, + "content": "networks", + "type": "text" + }, + { + "bbox": [ + 145, + 626, + 152, + 636 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 625, + 169, + 637 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 170, + 627, + 176, + 636 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 625, + 505, + 637 + ], + "score": 1.0, + "content": ", respectively. Then the similarity of the given image-text pairs is computed through", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 107, + 636, + 165, + 648 + ], + "score": 0.92, + "content": "S ( f ( I ) , g ( T ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 636, + 279, + 649 + ], + "score": 1.0, + "content": ". For simplicity, we denote", + "type": "text" + }, + { + "bbox": [ + 279, + 636, + 337, + 648 + ], + "score": 0.93, + "content": "S ( f ( \\dot { I } ) , g ( T ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 636, + 350, + 649 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 351, + 636, + 383, + 648 + ], + "score": 0.93, + "content": "S ( I , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "in the following. Some early", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "empirical studies [3] show that DNNs tend to first learn simple samples and then gradually fit the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "noisy samples. This so-called memorization effect of DNNs will lead to a relatively low loss for the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "clean samples. Motivated by this, we utilize the difference of loss distribution between the clean", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 679, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 507, + 693 + ], + "score": 1.0, + "content": "samples and noisy samples to divide the training data like [10, 43, 2, 20]. Specifically, given a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 690, + 379, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 174, + 703 + ], + "score": 1.0, + "content": "matching model", + "type": "text" + }, + { + "bbox": [ + 174, + 690, + 208, + 702 + ], + "score": 0.93, + "content": "( \\bar { f } , g , \\bar { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 691, + 379, + 703 + ], + "score": 1.0, + "content": ", we compute the per-sample loss through:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 614, + 507, + 703 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 708, + 383, + 724 + ], + "lines": [ + { + "bbox": [ + 227, + 708, + 383, + 724 + ], + "spans": [ + { + "bbox": [ + 227, + 708, + 383, + 724 + ], + "score": 0.93, + "content": "\\ell _ { ( f , g , S ) } = \\{ \\ell _ { i } \\} _ { i = 1 } ^ { N } = \\{ L _ { w } ( I _ { i } , T _ { i } ) \\} _ { i = 1 } ^ { N }", + "type": "interline_equation", + "image_path": "6fc5284745d0ad1346714b29253da56ab6a8e8fa66ab3747454a6042d2257c71.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 227, + 708, + 383, + 724 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 203, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 204, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 133, + 86 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 74, + 147, + 84 + ], + "score": 0.88, + "content": "L _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 71, + 204, + 86 + ], + "score": 1.0, + "content": "is defined as:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 86, + 459, + 115 + ], + "lines": [ + { + "bbox": [ + 151, + 86, + 459, + 115 + ], + "spans": [ + { + "bbox": [ + 151, + 86, + 459, + 115 + ], + "score": 0.94, + "content": "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 ) ] _ { + } ,", + "type": "interline_equation", + "image_path": "3ee3df26f574057fe0d6067ba99e575d516cb6ef402949d4cdaccedc8036cc87.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 151, + 86, + 459, + 95.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 151, + 95.66666666666667, + 459, + 105.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 151, + 105.33333333333334, + 459, + 115.00000000000001 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 505, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 133, + 131 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 119, + 158, + 131 + ], + "score": 0.92, + "content": "( I , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 118, + 227, + 131 + ], + "score": 1.0, + "content": "is a positive pair,", + "type": "text" + }, + { + "bbox": [ + 228, + 119, + 254, + 129 + ], + "score": 0.9, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 118, + 367, + 131 + ], + "score": 1.0, + "content": "denotes a given margin, and", + "type": "text" + }, + { + "bbox": [ + 367, + 118, + 441, + 131 + ], + "score": 0.92, + "content": "[ x ] _ { + } = m a x ( x , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 118, + 505, + 131 + ], + "score": 1.0, + "content": ". 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Then, we fit the per-sample loss of all training data by using a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 310, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 310, + 167 + ], + "score": 1.0, + "content": "two-component Gaussian Mixture Model [20, 30]:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 169, + 353, + 204 + ], + "lines": [ + { + "bbox": [ + 257, + 169, + 353, + 204 + ], + "spans": [ + { + "bbox": [ + 257, + 169, + 353, + 204 + ], + "score": 0.95, + "content": "p ( \\ell | \\theta ) = \\sum _ { k = 1 } ^ { K } \\beta _ { k } \\phi ( \\ell | k ) ,", + "type": "interline_equation", + "image_path": "3b664f0344f0c21be2caace0bac7efb85763fbbc8bd31b5bf4a18615525f996b.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 257, + 169, + 353, + 186.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 257, + 186.5, + 353, + 204.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 133, + 221 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 208, + 145, + 219 + ], + "score": 0.88, + "content": "\\beta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 207, + 163, + 221 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 207, + 191, + 219 + ], + "score": 0.93, + "content": "\\phi ( \\ell | k )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 207, + 437, + 221 + ], + "score": 1.0, + "content": "are the mixture coefficient and the probability density of the", + "type": "text" + }, + { + "bbox": [ + 437, + 208, + 444, + 217 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "-th component,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "respectively. Based on the memorization effect of DNNs, we treat the component with a lower mean", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "value (i.e., lower loss) as the clean set, and the other as the noisy set. To optimize the GMM, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "adopt the Expectation-Maximization algorithm. Moreover, we compute the posterior probability", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 107, + 251, + 245, + 263 + ], + "score": 0.93, + "content": "w _ { i } \\stackrel { - } { = } p ( k | \\ell _ { i } ) ^ { - } = p ( k ) p ( \\ell _ { i } | k ) / p ( \\ell _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 251, + 355, + 264 + ], + "score": 1.0, + "content": "as the clean probability of", + "type": "text" + }, + { + "bbox": [ + 355, + 252, + 359, + 261 + ], + "score": 0.79, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 251, + 433, + 264 + ], + "score": 1.0, + "content": "-th sample, where", + "type": "text" + }, + { + "bbox": [ + 433, + 252, + 440, + 261 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "is the Gaussian", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 258, + 508, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 336, + 278 + ], + "score": 1.0, + "content": "component with the lower mean. By setting a threshold to", + "type": "text" + }, + { + "bbox": [ + 336, + 262, + 370, + 274 + ], + "score": 0.92, + "content": "\\{ w _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 258, + 508, + 278 + ], + "score": 1.0, + "content": ", we divide the data into clean and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 273, + 432, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 432, + 286 + ], + "score": 1.0, + "content": "noisy subsets. For simplicity, we set the threshold to 0.5 through all experiments.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "As observed in [10], it probably introduces error accumulation if the neural network is trained in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "a self-divide manner. To avoid such a situation, we adopt the co-teaching paradigm. Specifically,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 306, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 102, + 306, + 256, + 326 + ], + "score": 1.0, + "content": "we individually train two networks", + "type": "text" + }, + { + "bbox": [ + 257, + 310, + 339, + 323 + ], + "score": 0.93, + "content": "A = \\{ f ^ { A } , g ^ { A } , S ^ { A } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 306, + 359, + 326 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 360, + 310, + 444, + 323 + ], + "score": 0.92, + "content": "B = \\{ f ^ { B } , \\bar { g } ^ { B } , S ^ { \\bar { B } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 306, + 507, + 326 + ], + "score": 1.0, + "content": "with different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 367, + 334 + ], + "score": 1.0, + "content": "initializations and batch sequences. At each epoch, the network", + "type": "text" + }, + { + "bbox": [ + 367, + 323, + 376, + 332 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 321, + 388, + 334 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 389, + 323, + 398, + 332 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "will model its per-sample", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "loss distribution with a GMM and divide the dataset into clean and noisy subsets which are then", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "used for training the other network, i.e., co-divide. Note that, before co-divide, a warmup process is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 355, + 463, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 368, + 367 + ], + "score": 1.0, + "content": "conducted on all training data to achieve initial convergence with", + "type": "text" + }, + { + "bbox": [ + 368, + 355, + 383, + 366 + ], + "score": 0.89, + "content": "L _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 355, + 463, + 367 + ], + "score": 1.0, + "content": "as defined in Eq. 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 378, + 176, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 375, + 178, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 178, + 394 + ], + "score": 1.0, + "content": "3.2 Co-Rectify", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 102, + 393, + 503, + 419 + ], + "spans": [ + { + "bbox": [ + 102, + 393, + 234, + 419 + ], + "score": 1.0, + "content": "For either of A and B, the data", + "type": "text" + }, + { + "bbox": [ + 234, + 401, + 243, + 410 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 393, + 394, + 419 + ], + "score": 1.0, + "content": "will be divided into the clean subset", + "type": "text" + }, + { + "bbox": [ + 394, + 399, + 503, + 412 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mathcal { S } } _ { c } = \\{ ( \\boldsymbol { I } _ { i } ^ { c } , \\boldsymbol { T } _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ) \\} _ { i = 1 } ^ { N _ { c } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 102, + 406, + 510, + 432 + ], + "spans": [ + { + "bbox": [ + 102, + 406, + 174, + 432 + ], + "score": 1.0, + "content": "and noisy subset", + "type": "text" + }, + { + "bbox": [ + 174, + 412, + 252, + 425 + ], + "score": 0.92, + "content": "\\boldsymbol { \\mathcal { S } } _ { n } = \\{ I _ { i } ^ { n } , T _ { i } ^ { n } \\} _ { i = 1 } ^ { N _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 406, + 510, + 432 + ], + "score": 1.0, + "content": ". Then, the co-rectify module will correct the labels to recall the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 217, + 437 + ], + "score": 1.0, + "content": "possible true positives from", + "type": "text" + }, + { + "bbox": [ + 217, + 425, + 229, + 435 + ], + "score": 0.86, + "content": "S _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 424, + 492, + 437 + ], + "score": 1.0, + "content": "and eliminate the negative impact of the possible false positives in", + "type": "text" + }, + { + "bbox": [ + 503, + 424, + 507, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 198, + 449 + ], + "score": 1.0, + "content": "Formally, the network", + "type": "text" + }, + { + "bbox": [ + 198, + 437, + 205, + 446 + ], + "score": 0.65, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 436, + 261, + 448 + ], + "score": 0.87, + "content": "( k \\in \\{ A , B \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 436, + 362, + 449 + ], + "score": 1.0, + "content": "will rectify the labels of", + "type": "text" + }, + { + "bbox": [ + 362, + 436, + 399, + 448 + ], + "score": 0.93, + "content": "\\{ S _ { c } , S _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 436, + 419, + 449 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 419, + 435, + 455, + 448 + ], + "score": 0.94, + "content": "\\{ \\hat { \\cal S } _ { c } , \\hat { \\cal S } _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "for training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 446, + 288, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 288, + 461 + ], + "score": 1.0, + "content": "itself. The rectified labels are determined by:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 462, + 439, + 496 + ], + "lines": [ + { + "bbox": [ + 171, + 462, + 439, + 496 + ], + "spans": [ + { + "bbox": [ + 171, + 462, + 439, + 496 + ], + "score": 0.92, + "content": "\\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.", + "type": "interline_equation", + "image_path": "4e3b5a31990709244ff49efc174e8e2f4deeb4362d006d1e7b51bcade2efc57a.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 171, + 462, + 439, + 473.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 171, + 473.3333333333333, + 439, + 484.66666666666663 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 171, + 484.66666666666663, + 439, + 495.99999999999994 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 506, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 133, + 512 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 498, + 217, + 511 + ], + "score": 0.93, + "content": "P ^ { A } ( I , T ) / P ^ { B } ( I , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 497, + 403, + 512 + ], + "score": 1.0, + "content": "denotes the predictions given by the network", + "type": "text" + }, + { + "bbox": [ + 403, + 499, + 425, + 511 + ], + "score": 0.91, + "content": "A / B", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 497, + 506, + 512 + ], + "score": 1.0, + "content": ". The roles of Eq. 4", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 295, + 522 + ], + "score": 1.0, + "content": "are as below. On the one hand, as most pairs of", + "type": "text" + }, + { + "bbox": [ + 295, + 511, + 306, + 521 + ], + "score": 0.88, + "content": " { \\boldsymbol { S } } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "are true positive, Eq. 4 will use the original labels", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 520, + 507, + 534 + ], + "spans": [ + { + "bbox": [ + 107, + 522, + 117, + 533 + ], + "score": 0.88, + "content": "y _ { i } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 520, + 265, + 534 + ], + "score": 1.0, + "content": "together with the model’s prediction", + "type": "text" + }, + { + "bbox": [ + 266, + 522, + 307, + 533 + ], + "score": 0.94, + "content": "P ( I _ { i } ^ { c } , T _ { i } ^ { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 520, + 507, + 534 + ], + "score": 1.0, + "content": "to rectify the correspondence. 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By setting a threshold to", + "type": "text" + }, + { + "bbox": [ + 336, + 262, + 370, + 274 + ], + "score": 0.92, + "content": "\\{ w _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 258, + 508, + 278 + ], + "score": 1.0, + "content": ", we divide the data into clean and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 273, + 432, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 432, + 286 + ], + "score": 1.0, + "content": "noisy subsets. For simplicity, we set the threshold to 0.5 through all experiments.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 207, + 508, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "As observed in [10], it probably introduces error accumulation if the neural network is trained in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "a self-divide manner. To avoid such a situation, we adopt the co-teaching paradigm. 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Note that, before co-divide, a warmup process is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 355, + 463, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 368, + 367 + ], + "score": 1.0, + "content": "conducted on all training data to achieve initial convergence with", + "type": "text" + }, + { + "bbox": [ + 368, + 355, + 383, + 366 + ], + "score": 0.89, + "content": "L _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 355, + 463, + 367 + ], + "score": 1.0, + "content": "as defined in Eq. 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 102, + 289, + 507, + 367 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 378, + 176, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 375, + 178, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 178, + 394 + ], + "score": 1.0, + "content": "3.2 Co-Rectify", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 102, + 393, + 503, + 419 + ], + "spans": [ + { + "bbox": [ + 102, + 393, + 234, + 419 + ], + "score": 1.0, + "content": "For either of A and B, the data", + "type": "text" + }, + { + "bbox": [ + 234, + 401, + 243, + 410 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 393, + 394, + 419 + ], + "score": 1.0, + "content": "will be divided into the clean subset", + "type": "text" + }, + { + "bbox": [ + 394, + 399, + 503, + 412 + ], + "score": 0.93, + "content": "\\boldsymbol { \\mathcal { S } } _ { c } = \\{ ( \\boldsymbol { I } _ { i } ^ { c } , \\boldsymbol { T } _ { i } ^ { c } , y _ { i } ^ { c } , w _ { i } ) \\} _ { i = 1 } ^ { N _ { c } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 102, + 406, + 510, + 432 + ], + "spans": [ + { + "bbox": [ + 102, + 406, + 174, + 432 + ], + "score": 1.0, + "content": "and noisy subset", + "type": "text" + }, + { + "bbox": [ + 174, + 412, + 252, + 425 + ], + "score": 0.92, + "content": "\\boldsymbol { \\mathcal { S } } _ { n } = \\{ I _ { i } ^ { n } , T _ { i } ^ { n } \\} _ { i = 1 } ^ { N _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 406, + 510, + 432 + ], + "score": 1.0, + "content": ". 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Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)
Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do
3 ←GMM(D,B)
4 ← GMM(D,A) 5 for k={A,Bj do
Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6
S={(Ii,Ti)|ui<0.5,∀(Ii,Ti) ∈(D,W)} 7
8 for j=num_steps do
9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);
Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10
Train the network k on (B, Bη） by optimizing Lsoft. 11
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To achieve robust image-text matching, we propose a novel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 361, + 446, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 151, + 375 + ], + "score": 1.0, + "content": "triplet loss", + "type": "text" + }, + { + "bbox": [ + 151, + 362, + 174, + 374 + ], + "score": 0.92, + "content": "\\scriptstyle { L _ { s o f t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 361, + 446, + 375 + ], + "score": 1.0, + "content": "by recasting the rectified labels as the soft margin. Mathematically,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 378, + 466, + 394 + ], + "lines": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "spans": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "score": 0.91, + "content": "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 } ) ] _ { + } ,", + "type": "interline_equation", + "image_path": "b89b0624abfbe280229b638d5dd7358f01553e7b11c483fbe264ed86470ec138.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 399, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 132, + 417 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 400, + 243, + 416 + ], + "score": 0.93, + "content": "\\hat { I } _ { h } = \\mathrm { a r g m a x } _ { I _ { j } \\ne I _ { i } } S ( I _ { j } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 399, + 261, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 400, + 375, + 416 + ], + "score": 0.92, + "content": "\\hat { T } _ { h } = \\mathrm { a r g m a x } _ { T _ { j } \\neq T _ { i } } S ( I _ { i } , T _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 399, + 506, + 417 + ], + "score": 1.0, + "content": "are the most similar negatives in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 259, + 429 + ], + "score": 1.0, + "content": "the mini-batch for a given positive pair", + "type": "text" + }, + { + "bbox": [ + 259, + 415, + 289, + 427 + ], + "score": 0.91, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 414, + 329, + 429 + ], + "score": 1.0, + "content": "similar to", + "type": "text" + }, + { + "bbox": [ + 330, + 416, + 361, + 426 + ], + "score": 0.75, + "content": "{ \\mathrm { V S E } } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 414, + 442, + 429 + ], + "score": 1.0, + "content": "[8]. The soft margin", + "type": "text" + }, + { + "bbox": [ + 442, + 415, + 453, + 426 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "is adaptively", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 169, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 169, + 439 + ], + "score": 1.0, + "content": "determined by:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 436, + 340, + 462 + ], + "lines": [ + { + "bbox": [ + 270, + 436, + 340, + 462 + ], + "spans": [ + { + "bbox": [ + 270, + 436, + 340, + 462 + ], + "score": 0.94, + "content": "\\hat { \\alpha } _ { i } = \\frac { m ^ { \\hat { y } _ { i } } - 1 } { m - 1 } \\alpha ,", + "type": "interline_equation", + "image_path": "a876de5ed3ec8e6b239cb30ae499449cd6fc1e4190ca088b63be19780b097d3d.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 270, + 436, + 340, + 462 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 133, + 476 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 466, + 144, + 474 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 464, + 255, + 476 + ], + "score": 1.0, + "content": "is the curve parameter, and", + "type": "text" + }, + { + "bbox": [ + 256, + 465, + 265, + 476 + ], + "score": 0.88, + "content": "\\hat { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "is the rectified label. The above formulation is designed to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 232, + 487 + ], + "score": 1.0, + "content": "achieve the following goal, i.e.,", + "type": "text" + }, + { + "bbox": [ + 232, + 476, + 243, + 486 + ], + "score": 0.88, + "content": "\\hat { \\alpha } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 475, + 373, + 487 + ], + "score": 1.0, + "content": "will be assigned a small value if", + "type": "text" + }, + { + "bbox": [ + 373, + 476, + 383, + 487 + ], + "score": 0.88, + "content": "\\hat { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "is close to 0, and a large value", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 324, + 499 + ], + "score": 1.0, + "content": "otherwise. 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[15] proposes a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "log-ratio matching loss with a regularization defined by the label distance ratio, which is computed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "by the continuous labels. Liu et al. [25] introduces the hubness problem in image-text matching and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "proposes to consider all samples in a mini-batch and weights them according to both local and global", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "statistics. Wray et al. [38] recasts the soft similarity into binary labels by directly setting a threshold", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "on the predicted similarity. 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This implies that the data have at least", + "type": "text" + }, + { + "bbox": [ + 440, + 73, + 459, + 83 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "clean pairs", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "which could be regarded as a similarity anchor for prediction. 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Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)
Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do
3 ←GMM(D,B)
4 ← GMM(D,A) 5 for k={A,Bj do
Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6
S={(Ii,Ti)|ui<0.5,∀(Ii,Ti) ∈(D,W)} 7
8 for j=num_steps do
9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);
Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10
Train the network k on (B, Bη） by optimizing Lsoft. 11
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To achieve robust image-text matching, we propose a novel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 361, + 446, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 151, + 375 + ], + "score": 1.0, + "content": "triplet loss", + "type": "text" + }, + { + "bbox": [ + 151, + 362, + 174, + 374 + ], + "score": 0.92, + "content": "\\scriptstyle { L _ { s o f t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 361, + 446, + 375 + ], + "score": 1.0, + "content": "by recasting the rectified labels as the soft margin. Mathematically,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 339, + 505, + 375 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 378, + 466, + 394 + ], + "lines": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "spans": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "score": 0.91, + "content": "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 } ) ] _ { + } ,", + "type": "interline_equation", + "image_path": "b89b0624abfbe280229b638d5dd7358f01553e7b11c483fbe264ed86470ec138.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 145, + 378, + 466, + 394 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 399, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 132, + 417 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 400, + 243, + 416 + ], + "score": 0.93, + "content": "\\hat { I } _ { h } = \\mathrm { a r g m a x } _ { I _ { j } \\ne I _ { i } } S ( I _ { j } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 399, + 261, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 400, + 375, + 416 + ], + "score": 0.92, + "content": "\\hat { T } _ { h } = \\mathrm { a r g m a x } _ { T _ { j } \\neq T _ { i } } S ( I _ { i } , T _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 399, + 506, + 417 + ], + "score": 1.0, + "content": "are the most similar negatives in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 259, + 429 + ], + "score": 1.0, + "content": "the mini-batch for a given positive pair", + "type": "text" + }, + { + "bbox": [ + 259, + 415, + 289, + 427 + ], + "score": 0.91, + "content": "( I _ { i } , T _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 414, + 329, + 429 + ], + "score": 1.0, + "content": "similar to", + "type": "text" + }, + { + "bbox": [ + 330, + 416, + 361, + 426 + ], + "score": 0.75, + "content": "{ \\mathrm { V S E } } { + } { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 414, + 442, + 429 + ], + "score": 1.0, + "content": "[8]. 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[15] proposes a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "log-ratio matching loss with a regularization defined by the label distance ratio, which is computed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "by the continuous labels. Liu et al. [25] introduces the hubness problem in image-text matching and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "proposes to consider all samples in a mini-batch and weights them according to both local and global", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "statistics. Wray et al. [38] recasts the soft similarity into binary labels by directly setting a threshold", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "on the predicted similarity. 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As a result and more importantly, our loss is specifically designed to solve the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 354, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 354, + 107 + ], + "score": 1.0, + "content": "noisy correspondence problem whereas existing ones are not.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 622, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "soft margins in the triplet loss by assigning large margins to the true positive pairs and small ones to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "the false positive pairs. As a result and more importantly, our loss is specifically designed to solve the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 354, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 354, + 107 + ], + "score": 1.0, + "content": "noisy correspondence problem whereas existing ones are not.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 122, + 186, + 136 + ], + "lines": [ + { + "bbox": [ + 104, + 119, + 189, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 119, + 189, + 140 + ], + "score": 1.0, + "content": "4 Experiment", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 147, + 506, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 507, + 160 + ], + "score": 1.0, + "content": "In this section, we carry out experiments to verify the effectiveness of NCR in robust image-text match-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "ing. In the experiments, we use three benchmark datasets including Flickr30K [42], MS-COCO [23],", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "and Conceptual Captions [35]. Among them, Conceptual Captions is with real noisy correspondence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 458, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 458, + 193 + ], + "score": 1.0, + "content": "from the wild, and Flickr30K and MS-COCO are with simulated noisy correspondence.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 106, + 205, + 307, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 308, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 308, + 218 + ], + "score": 1.0, + "content": "4.1 Datasets and Performance Measurements", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "Three datasets are used in our experiments. To be specific, Flickr30K contains 31,000 images", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "collected from the Flickr website with five captions each. Following [19], we use 1,000 images for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "score": 1.0, + "content": "validation, 1,000 images for testing, and the rest for training. MS-COCO contains 123,287 images", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "score": 1.0, + "content": "with five captions each. We follow the data partition in [19] which consists of 113,287 training images,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "5,000 validation images, and 5,000 test images. As Flickr30K and MS-COCO are well annotated,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 282, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 293 + ], + "score": 1.0, + "content": "we simulate the noisy correspondence by randomly shuffling the captions of training images for a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "specific percentage, denoted by noise ratio. Conceptual Captions is a large-scale data consisting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "of 3.3M images with a single caption each. As this data set is harvested from the Internet, about", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 153, + 324 + ], + "score": 0.94, + "content": "3 \\% \\sim 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "correspondences are incorrect [35]. In our experiments, we use a subset of Conceptual", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "Captions for evaluation, named CC152K. Specifically, we randomly select 150,000 samples from the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "training split for training, 1,000 samples from the validation split for validation, and 1,000 samples", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 250, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 250, + 360 + ], + "score": 1.0, + "content": "from the validation split for testing.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "Following [19], for all images, we take the Faster-RCNN [33] detector provided by [1] to extract the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "top 36 region proposals of which each is encoded as a 2048-dimensional feature. For evaluation, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 182, + 397 + ], + "score": 1.0, + "content": "take the recall at K", + "type": "text" + }, + { + "bbox": [ + 183, + 385, + 216, + 396 + ], + "score": 0.71, + "content": ". ( \\mathbb { R } ^ { \\ @ \\mathbb { K } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 385, + 337, + 397 + ], + "score": 1.0, + "content": "as the measurement. In short,", + "type": "text" + }, + { + "bbox": [ + 338, + 385, + 363, + 395 + ], + "score": 0.83, + "content": "{ \\mathrm { R @ K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "is the fraction of queries for which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 480, + 407 + ], + "score": 1.0, + "content": "the correct item is retrieved in the closest K points to the query. In the experiments, we report", + "type": "text" + }, + { + "bbox": [ + 480, + 396, + 502, + 406 + ], + "score": 0.81, + "content": "\\mathbf { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 396, + 506, + 407 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 307, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 128, + 417 + ], + "score": 0.84, + "content": "\\mathbf { R } @ 5", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 405, + 149, + 419 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 149, + 406, + 177, + 416 + ], + "score": 0.87, + "content": "\\mathbb { R } \\ @ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 405, + 307, + 419 + ], + "score": 1.0, + "content": "for a comprehensive evaluation.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 107, + 432, + 231, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 231, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 231, + 445 + ], + "score": 1.0, + "content": "4.2 Implementation Details", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "NCR is a general framework which could enable almost all existing cross-modal matching methods", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 475 + ], + "score": 1.0, + "content": "robust against noisy correspondence. To verify the effectiveness of our framework, SGR [7] is chosen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "to guarantee the robustness because it is the state of the art in image-text matching. In brief, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 497 + ], + "score": 1.0, + "content": "image regions and words are projected into a shared embedding space through a full-connected layer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 137, + 507 + ], + "score": 0.25, + "content": "( i . e . , f )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 496, + 218, + 508 + ], + "score": 1.0, + "content": "and a Bi-GRU [34]", + "type": "text" + }, + { + "bbox": [ + 219, + 496, + 249, + 507 + ], + "score": 0.4, + "content": "( i . e . , g )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 496, + 415, + 508 + ], + "score": 1.0, + "content": ", respectively. For the similarity function", + "type": "text" + }, + { + "bbox": [ + 415, + 496, + 423, + 506 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 496, + 505, + 508 + ], + "score": 1.0, + "content": ", it will compute the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "similarity between the given image and text by combining the local and global features with the help", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "of a graph reasoning technique proposed in [18]. Due to the space limitation, we leave more details", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 529, + 270, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 270, + 541 + ], + "score": 1.0, + "content": "and results in the supplemental material.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "We train our network using the Adam optimizer [16] with the default parameters and a batch size of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 270, + 568 + ], + "score": 1.0, + "content": "128. For fair comparisons, the networks", + "type": "text" + }, + { + "bbox": [ + 271, + 556, + 278, + 568 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 556, + 295, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 558, + 303, + 568 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "are the same with SGR, i.e., the word embedding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 567, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 468, + 578 + ], + "score": 1.0, + "content": "size is 300 and the joint embedding space size is 2048. In addition, we fix the margin", + "type": "text" + }, + { + "bbox": [ + 468, + 567, + 504, + 577 + ], + "score": 0.87, + "content": "\\alpha = 0 . 2", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 124, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 578, + 159, + 588 + ], + "score": 0.89, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "for the soft margin through the experiments. At the inference stage, we average the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 237, + 600 + ], + "score": 1.0, + "content": "similarities predicted by network", + "type": "text" + }, + { + "bbox": [ + 237, + 589, + 246, + 598 + ], + "score": 0.66, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 587, + 263, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 589, + 272, + 598 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "for the retrieval evaluation. To avoid overfitting, we choose", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 405, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 405, + 611 + ], + "score": 1.0, + "content": "the best checkpoint in terms of the sum of the recalls on the validation set.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 108, + 624, + 284, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 285, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 285, + 638 + ], + "score": 1.0, + "content": "4.3 Comparisons with State of The Arts", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "In this section, we conduct comparisons on the three datasets. The baselines include SCAN [19],", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "VSRN [21], IMRAM [5], SGRAF, SGR and SAF [7]. For Flickr30K and MS-COCO, we report the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 277, + 680 + ], + "score": 1.0, + "content": "results with three different noise ratios, i.e.,", + "type": "text" + }, + { + "bbox": [ + 277, + 667, + 291, + 678 + ], + "score": 0.81, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 666, + 295, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 295, + 667, + 315, + 678 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 666, + 334, + 680 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 335, + 667, + 353, + 678 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 666, + 505, + 680 + ], + "score": 1.0, + "content": ". In addition, we also report the results", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "of SGR on the clean Flickr30K and MS-COCO by discarding the noisy pairs, denoted by SGR-C.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "Clearly, SGR-C is a quite strong baseline since the used data does not contain noisy correspondence.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "We do not report the results of SGRAF and SAF on the clean datasets since our framework only", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 215, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 215, + 725 + ], + "score": 1.0, + "content": "extends SGR in this paper.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 753 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 72, + 506, + 107 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 122, + 186, + 136 + ], + "lines": [ + { + "bbox": [ + 104, + 119, + 189, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 119, + 189, + 140 + ], + "score": 1.0, + "content": "4 Experiment", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 147, + 506, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 507, + 160 + ], + "score": 1.0, + "content": "In this section, we carry out experiments to verify the effectiveness of NCR in robust image-text match-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "ing. In the experiments, we use three benchmark datasets including Flickr30K [42], MS-COCO [23],", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "and Conceptual Captions [35]. Among them, Conceptual Captions is with real noisy correspondence", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 458, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 458, + 193 + ], + "score": 1.0, + "content": "from the wild, and Flickr30K and MS-COCO are with simulated noisy correspondence.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 147, + 507, + 193 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 205, + 307, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 308, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 308, + 218 + ], + "score": 1.0, + "content": "4.1 Datasets and Performance Measurements", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "Three datasets are used in our experiments. To be specific, Flickr30K contains 31,000 images", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "collected from the Flickr website with five captions each. Following [19], we use 1,000 images for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 262 + ], + "score": 1.0, + "content": "validation, 1,000 images for testing, and the rest for training. MS-COCO contains 123,287 images", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "score": 1.0, + "content": "with five captions each. We follow the data partition in [19] which consists of 113,287 training images,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "5,000 validation images, and 5,000 test images. As Flickr30K and MS-COCO are well annotated,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 282, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 293 + ], + "score": 1.0, + "content": "we simulate the noisy correspondence by randomly shuffling the captions of training images for a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "specific percentage, denoted by noise ratio. Conceptual Captions is a large-scale data consisting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "of 3.3M images with a single caption each. As this data set is harvested from the Internet, about", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 153, + 324 + ], + "score": 0.94, + "content": "3 \\% \\sim 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "correspondences are incorrect [35]. In our experiments, we use a subset of Conceptual", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "Captions for evaluation, named CC152K. Specifically, we randomly select 150,000 samples from the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "training split for training, 1,000 samples from the validation split for validation, and 1,000 samples", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 250, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 250, + 360 + ], + "score": 1.0, + "content": "from the validation split for testing.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 225, + 507, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "Following [19], for all images, we take the Faster-RCNN [33] detector provided by [1] to extract the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "top 36 region proposals of which each is encoded as a 2048-dimensional feature. For evaluation, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 182, + 397 + ], + "score": 1.0, + "content": "take the recall at K", + "type": "text" + }, + { + "bbox": [ + 183, + 385, + 216, + 396 + ], + "score": 0.71, + "content": ". ( \\mathbb { R } ^ { \\ @ \\mathbb { K } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 385, + 337, + 397 + ], + "score": 1.0, + "content": "as the measurement. In short,", + "type": "text" + }, + { + "bbox": [ + 338, + 385, + 363, + 395 + ], + "score": 0.83, + "content": "{ \\mathrm { R @ K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "is the fraction of queries for which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 480, + 407 + ], + "score": 1.0, + "content": "the correct item is retrieved in the closest K points to the query. In the experiments, we report", + "type": "text" + }, + { + "bbox": [ + 480, + 396, + 502, + 406 + ], + "score": 0.81, + "content": "\\mathbf { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 396, + 506, + 407 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 307, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 128, + 417 + ], + "score": 0.84, + "content": "\\mathbf { R } @ 5", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 405, + 149, + 419 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 149, + 406, + 177, + 416 + ], + "score": 0.87, + "content": "\\mathbb { R } \\ @ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 405, + 307, + 419 + ], + "score": 1.0, + "content": "for a comprehensive evaluation.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 361, + 506, + 419 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 432, + 231, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 231, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 231, + 445 + ], + "score": 1.0, + "content": "4.2 Implementation Details", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "NCR is a general framework which could enable almost all existing cross-modal matching methods", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 475 + ], + "score": 1.0, + "content": "robust against noisy correspondence. To verify the effectiveness of our framework, SGR [7] is chosen", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "to guarantee the robustness because it is the state of the art in image-text matching. In brief, the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 497 + ], + "score": 1.0, + "content": "image regions and words are projected into a shared embedding space through a full-connected layer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 137, + 507 + ], + "score": 0.25, + "content": "( i . e . , f )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 496, + 218, + 508 + ], + "score": 1.0, + "content": "and a Bi-GRU [34]", + "type": "text" + }, + { + "bbox": [ + 219, + 496, + 249, + 507 + ], + "score": 0.4, + "content": "( i . e . , g )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 496, + 415, + 508 + ], + "score": 1.0, + "content": ", respectively. For the similarity function", + "type": "text" + }, + { + "bbox": [ + 415, + 496, + 423, + 506 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 496, + 505, + 508 + ], + "score": 1.0, + "content": ", it will compute the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "similarity between the given image and text by combining the local and global features with the help", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "of a graph reasoning technique proposed in [18]. Due to the space limitation, we leave more details", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 529, + 270, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 270, + 541 + ], + "score": 1.0, + "content": "and results in the supplemental material.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 452, + 506, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "We train our network using the Adam optimizer [16] with the default parameters and a batch size of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 270, + 568 + ], + "score": 1.0, + "content": "128. For fair comparisons, the networks", + "type": "text" + }, + { + "bbox": [ + 271, + 556, + 278, + 568 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 556, + 295, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 558, + 303, + 568 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "are the same with SGR, i.e., the word embedding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 567, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 468, + 578 + ], + "score": 1.0, + "content": "size is 300 and the joint embedding space size is 2048. In addition, we fix the margin", + "type": "text" + }, + { + "bbox": [ + 468, + 567, + 504, + 577 + ], + "score": 0.87, + "content": "\\alpha = 0 . 2", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 124, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 578, + 159, + 588 + ], + "score": 0.89, + "content": "m = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "for the soft margin through the experiments. At the inference stage, we average the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 237, + 600 + ], + "score": 1.0, + "content": "similarities predicted by network", + "type": "text" + }, + { + "bbox": [ + 237, + 589, + 246, + 598 + ], + "score": 0.66, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 587, + 263, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 589, + 272, + 598 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "for the retrieval evaluation. To avoid overfitting, we choose", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 405, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 405, + 611 + ], + "score": 1.0, + "content": "the best checkpoint in terms of the sum of the recalls on the validation set.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 544, + 506, + 611 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 624, + 284, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 285, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 285, + 638 + ], + "score": 1.0, + "content": "4.3 Comparisons with State of The Arts", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "In this section, we conduct comparisons on the three datasets. The baselines include SCAN [19],", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "VSRN [21], IMRAM [5], SGRAF, SGR and SAF [7]. For Flickr30K and MS-COCO, we report the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 277, + 680 + ], + "score": 1.0, + "content": "results with three different noise ratios, i.e.,", + "type": "text" + }, + { + "bbox": [ + 277, + 667, + 291, + 678 + ], + "score": 0.81, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 666, + 295, + 680 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 295, + 667, + 315, + 678 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 666, + 334, + 680 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 335, + 667, + 353, + 678 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 666, + 505, + 680 + ], + "score": 1.0, + "content": ". In addition, we also report the results", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "of SGR on the clean Flickr30K and MS-COCO by discarding the noisy pairs, denoted by SGR-C.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "Clearly, SGR-C is a quite strong baseline since the used data does not contain noisy correspondence.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "We do not report the results of SGRAF and SAF on the clean datasets since our framework only", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 215, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 215, + 725 + ], + "score": 1.0, + "content": "extends SGR in this paper.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 645, + 507, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 197, + 85 + ], + "score": 1.0, + "content": "When the noise rate is", + "type": "text" + }, + { + "bbox": [ + 197, + 73, + 212, + 83 + ], + "score": 0.87, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 72, + 505, + 85 + ], + "score": 1.0, + "content": ", we directly refer to the results reported in the corresponding papers. For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "the noisy cases, we train the baseline models with the recommended settings three times and report", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "the best result. Note that for SGR, we found it is very sensitive to the noisy correspondence, as shown", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "in Table. 2. To obtain a desirable result, we experimentally employ a pre-training process to SGR", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 156, + 128 + ], + "score": 1.0, + "content": "(denoted by", + "type": "text" + }, + { + "bbox": [ + 157, + 117, + 183, + 127 + ], + "score": 0.29, + "content": "{ \\mathrm { S G R } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 116, + 504, + 128 + ], + "score": 1.0, + "content": "), namely, training the model with the vanilla triplet loss without hard negatives", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 304, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 304, + 140 + ], + "score": 1.0, + "content": "and then following the standard pipeline of SGR.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 108, + 159, + 504, + 378 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 175, + 145, + 434, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 144, + 435, + 158 + ], + "spans": [ + { + "bbox": [ + 175, + 144, + 435, + 158 + ], + "score": 1.0, + "content": "Table 1: Image-Text Retrieval on Flickr30K and MS-COCO 1K.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 108, + 159, + 504, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 159, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 108, + 159, + 504, + 378 + ], + "score": 0.986, + "html": "
Flickr30KMS-COCO
Image→TextText→ImageImage→TextText→Image
NoiseMethodsR@1R@5R@10R@1 R@5R@10R@1R@5R@10R@1R@5R@10
0%SCAN67.490.395.848.677.785.269.2 93.697.656.086.593.5
VSRN71.390.696.054.781.8 88.276.294.898.262.889.795.1
IMRAM74.193.096.653.979.487.276.7 95.698.561.789.195.0
SAF73.793.396.356.181.588.076.1 95.498.361.889.495.3
SGR75.293.396.656.281.086.578.0 95.898.261.489.395.4
SGRAF NCR77.894.197.458.583.088.879.6 96.298.563.290.796.1
77.394.097.559.684.489.978.7 95.898.563.390.495.8
20%SCAN59.183.490.436.667.077.5 78.266.2 25.191.0 96.445.080.289.3
VSRN IMRAM58.1 63.082.689.340.768.759.074.817.649.064.1
SAF51.086.091.341.471.280.5 68.692.897.655.785.091.0
SGR*62.879.388.038.366.576.267.3 92.5 91.796.6 96.253.4 52.984.592.4
SGR-C72.886.2 90.892.2 95.444.4 56.472.380.467.883.590.1
NCR75.093.997.582.188.675.4 95.297.960.188.5 89.394.8
58.383.089.077.795.598.262.595.3 21.0
50%SCAN27.757.668.816.239.349.840.873.5 84.95.415.1
VSRN14.337.650.012.130.039.423.5 54.769.316.047.865.9
IMRAM9.126.638.22.78.412.721.3 60.275.922.352.864.3
SAF30.363.675.427.953.765.130.4 67.882.333.569.082.8
SGR*36.968.180.229.356.267.060.6 87.493.646.074.279.0
SGR-C69.890.394.850.177.585.271.7 94.197.757.086.693.7
NCR72.993.096.354.379.886.574.694.697.8 59.187.894.5
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Note that for MS-COCO, we only report the results by averaging over 5 folds of 1K test", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "images due to space limitation, and leave the results on the full 5K test images in the supplemental", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "material. From the results, one could observe that NCR is competitive to SGRAF in the noise-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "free case, namely, NCR could achieve state-of-the-art performance even though it is proposed to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "achieve robustness. When the data is contaminated by the noisy correspondence, NCR remarkably", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "outperforms all the baselines by a large margin. Even comparing with SGR-C which is trained on the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 461, + 465, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 213, + 473 + ], + "score": 1.0, + "content": "clean data, NCR improves", + "type": "text" + }, + { + "bbox": [ + 214, + 461, + 236, + 471 + ], + "score": 0.82, + "content": "\\mathbb { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 461, + 249, + 473 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 249, + 461, + 271, + 471 + ], + "score": 0.83, + "content": "2 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 461, + 275, + 473 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 461, + 297, + 472 + ], + "score": 0.81, + "content": "3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 461, + 300, + 473 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 301, + 461, + 323, + 471 + ], + "score": 0.81, + "content": "2 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 461, + 343, + 473 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 344, + 461, + 366, + 471 + ], + "score": 0.87, + "content": "2 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 461, + 465, + 473 + ], + "score": 1.0, + "content": "in these four valuations.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "score": 1.0, + "content": "Results on CC152K. Table 2 shows the quantitative results on the CC152K which is with real noisy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "correspondences. From the results, one could see that our NCR consistently outperforms the evaluated", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 414, + 512 + ], + "score": 1.0, + "content": "models by a considerable margin in terms of all metrics. Specifically, NCR is", + "type": "text" + }, + { + "bbox": [ + 414, + 499, + 436, + 510 + ], + "score": 0.88, + "content": "4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 499, + 453, + 512 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 454, + 499, + 477, + 510 + ], + "score": 0.87, + "content": "5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "higher", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 240, + 525 + ], + "score": 1.0, + "content": "than the best baseline in terms of", + "type": "text" + }, + { + "bbox": [ + 240, + 510, + 262, + 520 + ], + "score": 0.86, + "content": "\\mathbf { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 508, + 505, + 525 + ], + "score": 1.0, + "content": "in text and image retrieval, respectively. Moreover, the large", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 521, + 469, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 250, + 534 + ], + "score": 1.0, + "content": "performance gap between SGR and", + "type": "text" + }, + { + "bbox": [ + 251, + 521, + 276, + 532 + ], + "score": 0.65, + "content": "\\operatorname { S G R } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 521, + 469, + 534 + ], + "score": 1.0, + "content": "shows the noise sensitivity of the original SGR.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "table", + "bbox": [ + 153, + 553, + 458, + 665 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 220, + 538, + 392, + 549 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 218, + 537, + 392, + 550 + ], + "spans": [ + { + "bbox": [ + 218, + 537, + 392, + 550 + ], + "score": 1.0, + "content": "Table 2: Image-Text Retrieval on CC152K.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "table_body", + "bbox": [ + 153, + 553, + 458, + 665 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 153, + 553, + 458, + 665 + ], + "spans": [ + { + "bbox": [ + 153, + 553, + 458, + 665 + ], + "score": 0.985, + "html": "
Image→ TextText→Image
MethodsR@1R@5R@10R@1R@5R@10
SCAN (ECCV'18)30.555.365.326.953.064.7
VSRN (ICCV'19)32.661.370.532.559.470.4
IMRAM (CVPR'20)33.157.668.129.056.867.4
SAF (AAAI'21)31.759.368.231.959.067.9
SGR (AAAI'21)11.329.739.613.130.141.6
SGR* (AAAI'21)35.063.473.334.963.072.8
NCR39.564.573.540.364.673.2
", + "type": "table", + "image_path": "ee4da61e843468259b8e4aa2049d759bd9f314376e9352e94ed997753b97cea1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 153, + 553, + 458, + 590.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 153, + 590.3333333333334, + 458, + 627.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 153, + 627.6666666666667, + 458, + 665.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 273, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 275, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 275, + 693 + ], + "score": 1.0, + "content": "4.4 Comparison to pre-trained model", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "In this section, we perform comparison to the large pre-trained model CLIP [31]. In brief, CLIP", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "is trained on a massive dataset harvested from the Internet and thus presumably has a lot of noisy", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 300, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 197, + 85 + ], + "score": 1.0, + "content": "When the noise rate is", + "type": "text" + }, + { + "bbox": [ + 197, + 73, + 212, + 83 + ], + "score": 0.87, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 72, + 505, + 85 + ], + "score": 1.0, + "content": ", we directly refer to the results reported in the corresponding papers. For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "the noisy cases, we train the baseline models with the recommended settings three times and report", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "the best result. Note that for SGR, we found it is very sensitive to the noisy correspondence, as shown", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "in Table. 2. To obtain a desirable result, we experimentally employ a pre-training process to SGR", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 156, + 128 + ], + "score": 1.0, + "content": "(denoted by", + "type": "text" + }, + { + "bbox": [ + 157, + 117, + 183, + 127 + ], + "score": 0.29, + "content": "{ \\mathrm { S G R } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 116, + 504, + 128 + ], + "score": 1.0, + "content": "), namely, training the model with the vanilla triplet loss without hard negatives", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 304, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 304, + 140 + ], + "score": 1.0, + "content": "and then following the standard pipeline of SGR.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 72, + 505, + 140 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 159, + 504, + 378 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 175, + 145, + 434, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 144, + 435, + 158 + ], + "spans": [ + { + "bbox": [ + 175, + 144, + 435, + 158 + ], + "score": 1.0, + "content": "Table 1: Image-Text Retrieval on Flickr30K and MS-COCO 1K.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 108, + 159, + 504, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 159, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 108, + 159, + 504, + 378 + ], + "score": 0.986, + "html": "
Flickr30KMS-COCO
Image→TextText→ImageImage→TextText→Image
NoiseMethodsR@1R@5R@10R@1 R@5R@10R@1R@5R@10R@1R@5R@10
0%SCAN67.490.395.848.677.785.269.2 93.697.656.086.593.5
VSRN71.390.696.054.781.8 88.276.294.898.262.889.795.1
IMRAM74.193.096.653.979.487.276.7 95.698.561.789.195.0
SAF73.793.396.356.181.588.076.1 95.498.361.889.495.3
SGR75.293.396.656.281.086.578.0 95.898.261.489.395.4
SGRAF NCR77.894.197.458.583.088.879.6 96.298.563.290.796.1
77.394.097.559.684.489.978.7 95.898.563.390.495.8
20%SCAN59.183.490.436.667.077.5 78.266.2 25.191.0 96.445.080.289.3
VSRN IMRAM58.1 63.082.689.340.768.759.074.817.649.064.1
SAF51.086.091.341.471.280.5 68.692.897.655.785.091.0
SGR*62.879.388.038.366.576.267.3 92.5 91.796.6 96.253.4 52.984.592.4
SGR-C72.886.2 90.892.2 95.444.4 56.472.380.467.883.590.1
NCR75.093.997.582.188.675.4 95.297.960.188.5 89.394.8
58.383.089.077.795.598.262.595.3 21.0
50%SCAN27.757.668.816.239.349.840.873.5 84.95.415.1
VSRN14.337.650.012.130.039.423.5 54.769.316.047.865.9
IMRAM9.126.638.22.78.412.721.3 60.275.922.352.864.3
SAF30.363.675.427.953.765.130.4 67.882.333.569.082.8
SGR*36.968.180.229.356.267.060.6 87.493.646.074.279.0
SGR-C69.890.394.850.177.585.271.7 94.197.757.086.693.7
NCR72.993.096.354.379.886.574.694.697.8 59.187.894.5
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Note that for MS-COCO, we only report the results by averaging over 5 folds of 1K test", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "images due to space limitation, and leave the results on the full 5K test images in the supplemental", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "material. From the results, one could observe that NCR is competitive to SGRAF in the noise-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "free case, namely, NCR could achieve state-of-the-art performance even though it is proposed to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "achieve robustness. When the data is contaminated by the noisy correspondence, NCR remarkably", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "outperforms all the baselines by a large margin. Even comparing with SGR-C which is trained on the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 461, + 465, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 213, + 473 + ], + "score": 1.0, + "content": "clean data, NCR improves", + "type": "text" + }, + { + "bbox": [ + 214, + 461, + 236, + 471 + ], + "score": 0.82, + "content": "\\mathbb { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 461, + 249, + 473 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 249, + 461, + 271, + 471 + ], + "score": 0.83, + "content": "2 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 461, + 275, + 473 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 461, + 297, + 472 + ], + "score": 0.81, + "content": "3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 461, + 300, + 473 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 301, + 461, + 323, + 471 + ], + "score": 0.81, + "content": "2 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 461, + 343, + 473 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 344, + 461, + 366, + 471 + ], + "score": 0.87, + "content": "2 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 461, + 465, + 473 + ], + "score": 1.0, + "content": "in these four valuations.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 384, + 506, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 490 + ], + "score": 1.0, + "content": "Results on CC152K. Table 2 shows the quantitative results on the CC152K which is with real noisy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "correspondences. From the results, one could see that our NCR consistently outperforms the evaluated", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 414, + 512 + ], + "score": 1.0, + "content": "models by a considerable margin in terms of all metrics. Specifically, NCR is", + "type": "text" + }, + { + "bbox": [ + 414, + 499, + 436, + 510 + ], + "score": 0.88, + "content": "4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 499, + 453, + 512 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 454, + 499, + 477, + 510 + ], + "score": 0.87, + "content": "5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "higher", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 240, + 525 + ], + "score": 1.0, + "content": "than the best baseline in terms of", + "type": "text" + }, + { + "bbox": [ + 240, + 510, + 262, + 520 + ], + "score": 0.86, + "content": "\\mathbf { R } \\ @ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 508, + 505, + 525 + ], + "score": 1.0, + "content": "in text and image retrieval, respectively. Moreover, the large", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 521, + 469, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 250, + 534 + ], + "score": 1.0, + "content": "performance gap between SGR and", + "type": "text" + }, + { + "bbox": [ + 251, + 521, + 276, + 532 + ], + "score": 0.65, + "content": "\\operatorname { S G R } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 521, + 469, + 534 + ], + "score": 1.0, + "content": "shows the noise sensitivity of the original SGR.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 476, + 506, + 534 + ] + }, + { + "type": "table", + "bbox": [ + 153, + 553, + 458, + 665 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 220, + 538, + 392, + 549 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 218, + 537, + 392, + 550 + ], + "spans": [ + { + "bbox": [ + 218, + 537, + 392, + 550 + ], + "score": 1.0, + "content": "Table 2: Image-Text Retrieval on CC152K.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "table_body", + "bbox": [ + 153, + 553, + 458, + 665 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 153, + 553, + 458, + 665 + ], + "spans": [ + { + "bbox": [ + 153, + 553, + 458, + 665 + ], + "score": 0.985, + "html": "
Image→ TextText→Image
MethodsR@1R@5R@10R@1R@5R@10
SCAN (ECCV'18)30.555.365.326.953.064.7
VSRN (ICCV'19)32.661.370.532.559.470.4
IMRAM (CVPR'20)33.157.668.129.056.867.4
SAF (AAAI'21)31.759.368.231.959.067.9
SGR (AAAI'21)11.329.739.613.130.141.6
SGR* (AAAI'21)35.063.473.334.963.072.8
NCR39.564.573.540.364.673.2
", + "type": "table", + "image_path": "ee4da61e843468259b8e4aa2049d759bd9f314376e9352e94ed997753b97cea1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 153, + 553, + 458, + 590.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 153, + 590.3333333333334, + 458, + 627.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 153, + 627.6666666666667, + 458, + 665.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 273, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 275, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 275, + 693 + ], + "score": 1.0, + "content": "4.4 Comparison to pre-trained model", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "In this section, we perform comparison to the large pre-trained model CLIP [31]. In brief, CLIP", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "is trained on a massive dataset harvested from the Internet and thus presumably has a lot of noisy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "image-text pairs. Such a comparison is helpful in understanding, big data based model (CLIP)", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "or noisy correspondence modeling technique (NCR), which one is more favorable to handle the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "mismatching problem. More specifically, CLIP claims that using hundreds of million data could", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "ignore the existence of possible noise, while we believe that a well-designed algorithm is essential", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "to solve the noisy correspondence. Noticed, although some existing works including CLIP have", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "slightly/indirectly realized the existence of the noise, NONE of them explicitly give a solution to", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 426, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 426, + 151 + ], + "score": 1.0, + "content": "solve this problem and explores the characteristics of the noise correspondence.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 700, + 505, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "image-text pairs. Such a comparison is helpful in understanding, big data based model (CLIP)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "or noisy correspondence modeling technique (NCR), which one is more favorable to handle the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "mismatching problem. More specifically, CLIP claims that using hundreds of million data could", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "ignore the existence of possible noise, while we believe that a well-designed algorithm is essential", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "to solve the noisy correspondence. Noticed, although some existing works including CLIP have", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "slightly/indirectly realized the existence of the noise, NONE of them explicitly give a solution to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 426, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 426, + 151 + ], + "score": 1.0, + "content": "solve this problem and explores the characteristics of the noise correspondence.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "In the experiments, we conduct CLIP on the MS-COCO dataset under the following two settings,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "i.e., Zero-shot and Fine-tune. In brief, the first setting directly uses the released pre-trained CLIP to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "perform inference on MS-COCO, and the second fine-tunes the pre-trained model using the noisy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "training data of MS-COCO. As CLIP only released some pre-trained models and inference code 3,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "we use the non-official code 4 to fine-tune the model with 32 epochs for the fine-tune setting. Note", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "score": 1.0, + "content": "that CLIP (ViT-L/14†) is unreleased and we report the results from the original paper [31]. One could", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "observe that, although CLIP utilizes 400 million image-text pairs for pre-training, its performance", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "inevitably degenerates during fine-tuning. In contrast, NCR achieves the matching performance with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 439, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 439, + 255 + ], + "score": 1.0, + "content": "the presence of noisy correspondence, indicating the necessity of algorithm design.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 122, + 286, + 489, + 408 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 200, + 270, + 407, + 281 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 199, + 269, + 408, + 282 + ], + "spans": [ + { + "bbox": [ + 199, + 269, + 408, + 282 + ], + "score": 1.0, + "content": "Table 3: Comparison with CLIP on MS-COCO 5K.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 122, + 286, + 489, + 408 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 286, + 489, + 408 + ], + "spans": [ + { + "bbox": [ + 122, + 286, + 489, + 408 + ], + "score": 0.986, + "html": "
Image→ TextText→Image
Noise RatioMethodsR@1R@5R@10R@1R@5R@10
0%, Zero-ShotCLIP (ViT-L/14+)58.481.588.137.862.472.2
CLIP (ViT-B/32)50.274.683.630.456.066.8
20%,Fine-tuneNCR58.284.291.541.771.081.3
CLIP (ViT-B/32) NCR21.4 56.949.6 83.663.3 91.014.8 40.637.6 69.849.6 80.1
50%, Fine-tuneCLIP (ViT-B/32)10.927.838.37.819.5
NCR53.180.788.537.966.626.8 77.8
", + "type": "table", + "image_path": "4917d0b351b31f4228103f7f7cdef5414b86ca797b9f66407b7c2675314e7610.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 122, + 286, + 489, + 326.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 122, + 326.6666666666667, + 489, + 367.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 122, + 367.33333333333337, + 489, + 408.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 107, + 424, + 227, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 228, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 228, + 439 + ], + "score": 1.0, + "content": "4.5 Experimental Analysis", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In this section, we first conduct experiments to show the robustness and generalizability of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "proposed method. Then, we investigate the effect of co-divide and co-rectify with the visualization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "results. After that, we carry out the ablation study to verify different components of NCR. Finally, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 477, + 336, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 336, + 490 + ], + "score": 1.0, + "content": "visually demonstrate some noisy cases detected by NCR.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 316, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 316, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 316, + 514 + ], + "score": 1.0, + "content": "4.5.1 Study on Robustness and Generalizability", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "To show the robustness of NCR, we conduct experiments on Flickr30K by increasing the noise ratio", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 128, + 543 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 530, + 143, + 541 + ], + "score": 0.88, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 529, + 154, + 543 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 154, + 530, + 174, + 541 + ], + "score": 0.89, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 529, + 250, + 543 + ], + "score": 1.0, + "content": "with an interval of", + "type": "text" + }, + { + "bbox": [ + 251, + 530, + 270, + 541 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". In addition, to verify the generalizability of NCR to other", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "image-text matching methods, we extend SCAN [19] by NCR, denoted by NCR-SCAN. Fig. 4 shows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "that NCR and NCR-SCAN perform more stable than SGR and SCAN with increasing noise ratio.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "Moreover, NCR (NCR-SCAN) is remarkably superior to SGR (SCAN) in all tests, which shows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 207, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 207, + 586 + ], + "score": 1.0, + "content": "generalizability of NCR.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 330, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 331, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 331, + 611 + ], + "score": 1.0, + "content": "4.5.2 Visualization on the Co-divide and Co-rectify", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "To further investigate the influence of the co-divide and co-rectify modules in our method, we carry", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "out experiments on the Flickr30K dataset by visualizing the per-sample loss distribution and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 637, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 358, + 650 + ], + "score": 1.0, + "content": "model predictions on the noisy data, where the noisy ratio is", + "type": "text" + }, + { + "bbox": [ + 358, + 637, + 378, + 648 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 637, + 504, + 650 + ], + "score": 1.0, + "content": ". For better visualization, here", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "we show the result of NCR-SCAN and leave the result of NCR in the supplemental material. As", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "shown in Fig. 3(b), the loss of most noisy samples is larger than the clean loss, which verifies the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "memorization effect of DNNs. By fitting the per-sample loss with GMM, NCR could effectively", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "divide the data into clean and noisy splits. Regarding the analysis on the co-rectify, Fig. 3(c) shows", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 700, + 275, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 244, + 712 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 244, + 712 + ], + "score": 1.0, + "content": "3https://github.com/OpenAI/CLIP", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 710, + 277, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 277, + 723 + ], + "score": 1.0, + "content": "4https://github.com/Zasder3/train-CLIP-FT", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 149 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 73, + 506, + 151 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "In the experiments, we conduct CLIP on the MS-COCO dataset under the following two settings,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "i.e., Zero-shot and Fine-tune. In brief, the first setting directly uses the released pre-trained CLIP to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "perform inference on MS-COCO, and the second fine-tunes the pre-trained model using the noisy", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "training data of MS-COCO. As CLIP only released some pre-trained models and inference code 3,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "we use the non-official code 4 to fine-tune the model with 32 epochs for the fine-tune setting. Note", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "score": 1.0, + "content": "that CLIP (ViT-L/14†) is unreleased and we report the results from the original paper [31]. One could", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "observe that, although CLIP utilizes 400 million image-text pairs for pre-training, its performance", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "inevitably degenerates during fine-tuning. In contrast, NCR achieves the matching performance with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 439, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 439, + 255 + ], + "score": 1.0, + "content": "the presence of noisy correspondence, indicating the necessity of algorithm design.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 153, + 506, + 255 + ] + }, + { + "type": "table", + "bbox": [ + 122, + 286, + 489, + 408 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 200, + 270, + 407, + 281 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 199, + 269, + 408, + 282 + ], + "spans": [ + { + "bbox": [ + 199, + 269, + 408, + 282 + ], + "score": 1.0, + "content": "Table 3: Comparison with CLIP on MS-COCO 5K.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 122, + 286, + 489, + 408 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 286, + 489, + 408 + ], + "spans": [ + { + "bbox": [ + 122, + 286, + 489, + 408 + ], + "score": 0.986, + "html": "
Image→ TextText→Image
Noise RatioMethodsR@1R@5R@10R@1R@5R@10
0%, Zero-ShotCLIP (ViT-L/14+)58.481.588.137.862.472.2
CLIP (ViT-B/32)50.274.683.630.456.066.8
20%,Fine-tuneNCR58.284.291.541.771.081.3
CLIP (ViT-B/32) NCR21.4 56.949.6 83.663.3 91.014.8 40.637.6 69.849.6 80.1
50%, Fine-tuneCLIP (ViT-B/32)10.927.838.37.819.5
NCR53.180.788.537.966.626.8 77.8
", + "type": "table", + "image_path": "4917d0b351b31f4228103f7f7cdef5414b86ca797b9f66407b7c2675314e7610.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 122, + 286, + 489, + 326.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 122, + 326.6666666666667, + 489, + 367.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 122, + 367.33333333333337, + 489, + 408.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 107, + 424, + 227, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 228, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 228, + 439 + ], + "score": 1.0, + "content": "4.5 Experimental Analysis", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In this section, we first conduct experiments to show the robustness and generalizability of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "proposed method. Then, we investigate the effect of co-divide and co-rectify with the visualization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "results. After that, we carry out the ablation study to verify different components of NCR. Finally, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 477, + 336, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 336, + 490 + ], + "score": 1.0, + "content": "visually demonstrate some noisy cases detected by NCR.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 444, + 505, + 490 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 316, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 316, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 316, + 514 + ], + "score": 1.0, + "content": "4.5.1 Study on Robustness and Generalizability", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "To show the robustness of NCR, we conduct experiments on Flickr30K by increasing the noise ratio", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 128, + 543 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 530, + 143, + 541 + ], + "score": 0.88, + "content": "0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 529, + 154, + 543 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 154, + 530, + 174, + 541 + ], + "score": 0.89, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 529, + 250, + 543 + ], + "score": 1.0, + "content": "with an interval of", + "type": "text" + }, + { + "bbox": [ + 251, + 530, + 270, + 541 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ". In addition, to verify the generalizability of NCR to other", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "image-text matching methods, we extend SCAN [19] by NCR, denoted by NCR-SCAN. Fig. 4 shows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "that NCR and NCR-SCAN perform more stable than SGR and SCAN with increasing noise ratio.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "Moreover, NCR (NCR-SCAN) is remarkably superior to SGR (SCAN) in all tests, which shows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 207, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 207, + 586 + ], + "score": 1.0, + "content": "generalizability of NCR.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 518, + 506, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 330, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 331, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 331, + 611 + ], + "score": 1.0, + "content": "4.5.2 Visualization on the Co-divide and Co-rectify", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "To further investigate the influence of the co-divide and co-rectify modules in our method, we carry", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "out experiments on the Flickr30K dataset by visualizing the per-sample loss distribution and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 637, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 358, + 650 + ], + "score": 1.0, + "content": "model predictions on the noisy data, where the noisy ratio is", + "type": "text" + }, + { + "bbox": [ + 358, + 637, + 378, + 648 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 637, + 504, + 650 + ], + "score": 1.0, + "content": ". For better visualization, here", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "we show the result of NCR-SCAN and leave the result of NCR in the supplemental material. As", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "shown in Fig. 3(b), the loss of most noisy samples is larger than the clean loss, which verifies the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "memorization effect of DNNs. By fitting the per-sample loss with GMM, NCR could effectively", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "divide the data into clean and noisy splits. Regarding the analysis on the co-rectify, Fig. 3(c) shows", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 223, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 239 + ], + "score": 1.0, + "content": "that the rectified soft labels of most clean pairs range into [0.3, 1] and those of most noisy pairs range", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "into [0, 0.5]. In other words, one could enforce the similarity of true positives larger than that of the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 247, + 474, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 474, + 261 + ], + "score": 1.0, + "content": "negatives during training, thus eliminating the negative impact of the noisy correspondence.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 614, + 506, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 63, + 497, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 63, + 497, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 63, + 497, + 169 + ], + "spans": [ + { + "bbox": [ + 115, + 63, + 497, + 169 + ], + "score": 0.968, + "type": "image", + "image_path": "0503d947414e2b32158fb766cbf1a3b87cabce246d93f4abb1f73ad11d4aa0c3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 63, + 497, + 98.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 98.33333333333334, + 497, + 133.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 133.66666666666669, + 497, + 169.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 169, + 504, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "score": 1.0, + "content": "Figure 3: (a) Retrieval performance of NCR and NCR-SCAN on Flickr30K with varying noise ratio.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "score": 1.0, + "content": "(b) Per-sample loss distribution and GMM fitting visualization after warmup. (c) Model predictions", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 192, + 187, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 187, + 203 + ], + "score": 1.0, + "content": "on the noisy subset.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 239 + ], + "score": 1.0, + "content": "that the rectified soft labels of most clean pairs range into [0.3, 1] and those of most noisy pairs range", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "into [0, 0.5]. In other words, one could enforce the similarity of true positives larger than that of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 247, + 474, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 474, + 261 + ], + "score": 1.0, + "content": "negatives during training, thus eliminating the negative impact of the noisy correspondence.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 273, + 203, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 203, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 203, + 288 + ], + "score": 1.0, + "content": "4.5.3 Ablation Study", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 465, + 305 + ], + "score": 1.0, + "content": "In this section, we carry out the ablation study on the Flickr30K with the noise ratio of", + "type": "text" + }, + { + "bbox": [ + 466, + 293, + 486, + 303 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 292, + 506, + 305 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 303, + 477, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 477, + 316 + ], + "score": 1.0, + "content": "shown in Table 4, all these three components are important to achieving encouraging results.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "table", + "bbox": [ + 126, + 337, + 486, + 425 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 192, + 323, + 417, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 322, + 419, + 336 + ], + "spans": [ + { + "bbox": [ + 192, + 322, + 371, + 336 + ], + "score": 1.0, + "content": "Table 4: Ablation studies on Flickr30K with", + "type": "text" + }, + { + "bbox": [ + 372, + 324, + 391, + 334 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 322, + 419, + 336 + ], + "score": 1.0, + "content": "noise.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "table_body", + "bbox": [ + 126, + 337, + 486, + 425 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 337, + 486, + 425 + ], + "spans": [ + { + "bbox": [ + 126, + 337, + 486, + 425 + ], + "score": 0.98, + "html": "
MethodImage →TextText→Image
Co-divideCo-rectifyWarmupR@1R@5R@10R@1R@5R@10
?√<<>72.993.096.354.379.886.5
71.490.895.754.180.386.5
?16.038.451.712.631.442.8
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Input: A given training data D, matching models A = (fA,gA, SA) and B = (fB,gB,SB)
Warmup the model (A,B) using Lw· 2 forn=1:num_epoch do
3 ←GMM(D,B)
4 ← GMM(D,A) 5 for k={A,Bj do
Sk={(Ii,Ti,yi,Wi)lwi ≥0.5,∀(Ii,Ti,yi,Wi) ∈(D,Wk)} 6
S={(Ii,Ti)|ui<0.5,∀(Ii,Ti) ∈(D,W)} 7
8 for j=num_steps do
9 Sample a mini-batch (B,Bη) from(Sk, S𝑘);
Rectify the labels of (Bj, Bγ) into (Bg, B) using Eq. 4-5; 10
Train the network k on (B, Bη） by optimizing Lsoft. 11
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Flickr30KMS-COCO
Image→TextText→ImageImage→TextText→Image
NoiseMethodsR@1R@5R@10R@1 R@5R@10R@1R@5R@10R@1R@5R@10
0%SCAN67.490.395.848.677.785.269.2 93.697.656.086.593.5
VSRN71.390.696.054.781.8 88.276.294.898.262.889.795.1
IMRAM74.193.096.653.979.487.276.7 95.698.561.789.195.0
SAF73.793.396.356.181.588.076.1 95.498.361.889.495.3
SGR75.293.396.656.281.086.578.0 95.898.261.489.395.4
SGRAF NCR77.894.197.458.583.088.879.6 96.298.563.290.796.1
77.394.097.559.684.489.978.7 95.898.563.390.495.8
20%SCAN59.183.490.436.667.077.5 78.266.2 25.191.0 96.445.080.289.3
VSRN IMRAM58.1 63.082.689.340.768.759.074.817.649.064.1
SAF51.086.091.341.471.280.5 68.692.897.655.785.091.0
SGR*62.879.388.038.366.576.267.3 92.5 91.796.6 96.253.4 52.984.592.4
SGR-C72.886.2 90.892.2 95.444.4 56.472.380.467.883.590.1
NCR75.093.997.582.188.675.4 95.297.960.188.5 89.394.8
58.383.089.077.795.598.262.595.3 21.0
50%SCAN27.757.668.816.239.349.840.873.5 84.95.415.1
VSRN14.337.650.012.130.039.423.5 54.769.316.047.865.9
IMRAM9.126.638.22.78.412.721.3 60.275.922.352.864.3
SAF30.363.675.427.953.765.130.4 67.882.333.569.082.8
SGR*36.968.180.229.356.267.060.6 87.493.646.074.279.0
SGR-C69.890.394.850.177.585.271.7 94.197.757.086.693.7
NCR72.993.096.354.379.886.574.694.697.8 59.187.894.5
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Image→ TextText→Image
MethodsR@1R@5R@10R@1R@5R@10
SCAN (ECCV'18)30.555.365.326.953.064.7
VSRN (ICCV'19)32.661.370.532.559.470.4
IMRAM (CVPR'20)33.157.668.129.056.867.4
SAF (AAAI'21)31.759.368.231.959.067.9
SGR (AAAI'21)11.329.739.613.130.141.6
SGR* (AAAI'21)35.063.473.334.963.072.8
NCR39.564.573.540.364.673.2
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Image→ TextText→Image
Noise RatioMethodsR@1R@5R@10R@1R@5R@10
0%, Zero-ShotCLIP (ViT-L/14+)58.481.588.137.862.472.2
CLIP (ViT-B/32)50.274.683.630.456.066.8
20%,Fine-tuneNCR58.284.291.541.771.081.3
CLIP (ViT-B/32) NCR21.4 56.949.6 83.663.3 91.014.8 40.637.6 69.849.6 80.1
50%, Fine-tuneCLIP (ViT-B/32)10.927.838.37.819.5
NCR53.180.788.537.966.626.8 77.8
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297.0, + 355.0, + 297.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 13, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/SypU81Ole/SypU81Ole.md b/parse/train/SypU81Ole/SypU81Ole.md new file mode 100644 index 0000000000000000000000000000000000000000..6c2e5e6317b19cc5f003d325b82e0838fa282a17 --- /dev/null +++ b/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). + +
MaleNot MaleTotal
Smiling17%31%48%
Not Smiling25%27%52%
Total42%58%
+ +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). + +
Open MouthNot Open MouthTotal
Smiling36%12%48%
Not Smiling12%40%52%
Total48%52%
+ +![](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. + +
Approach /Attribute[1][2] W [2] L[3] ANet[4] LNet[4] FullVAE AtDotGAN AtDot
5 0.C. Shadow8582 888688918889
Arched Eyebrow7673 787574797575
Attractive7877 817977817169
Bags Under Eye7671 797773798079
Bald8992 969295989898
Bangs8889 929492959090
Big Lips6461 676366688576
Big Nose7470 757475787777
Black Hair7074 857784888381
Blond Hair8081 938691958790
Blurry8177 868380849595
Brown Hair6069 777478807681
BushyEyebrow8076 868085908786
Chubby8682 868686919494
Double Chin8885 889088929595
Eyeglasses9894 989696999594
Goatee9386 939292959394
Gray Hair9088 949393979596
Heavy Makeup8584 908785907675
High Cheekbone8480 868584878168
Male9193 979594988180
Mouth Open8782 938586927867
Mustache9183 938791959596
Narrow Eyes8279 848377819389
No Beard9087 939192958584
Oval Face6462 656563667271
Pale Skin8384 918987919696
Pointy Nose686571 6770727272
Recede Hair768285 8485899392
Rosy Cheeks848187 8587909393
Sideburns949093 9491969494
Smiling898992 9288928768
Straight Hair636769 7069738079
Wavy Hair737677 7975807575
Earring737278 7778828181
Hat899196 9396999697
Lipstick898893 9190937977
Necklace686767 7068718888
Necktie868891 9086939392
Young807784 8183877881
Average8179 858383878684
+ +![](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. +Gómez-Bombarelli, Rafael, Duvenaud, David, Miguel, Hernández-Lobato, José, Aguilera-Iparraguirre, Jorge, Hirzel, Timothy D., Adams, Ryan P., and Alán Aspuru-Guzik. Automatic chemical design using a data-driven continuous representation of molecules. https://arxiv.org/abs/1610.02415 2016 Kingma, Diederik P. and Welling, Max. Auto-encoding variational Bayes. In Proceedings of the International Conference on Learning Representations, 2014. +Kumar, N., Belhumeur, P., and Nayar, S. Facetracer: A search engine for large collections of images with faces. In ECCV, 2008. +Lamb, Alex, Dumoulin, Vincent, Courville, Aaron. Discriminative Regularization for Generative Models. http://arxiv.org/abs/1602.03220 2016 +Larsen, Anders Boesen Lindbo, Sønderby, Søren Kaae, Larochelle, Hugo, Winther, Ole. Autoencoding beyond pixels using a learned similarity metric. https://arxiv.org/abs/1512.09300 2016 +LeCun, Y. and Bengio., Y. Word-level training of a handwritten word recognizer based on convolutional neural networks. In ICPR, 1994. +Li, J., and Zhang, Y. Learning surf cascade for fast and accurate object detection. In CVPR, 2013. +Makhzani, Alireza, Shlens, Jonathon, Jaitly, Navdeep, Goodfellow, Ian, Brendan, Frey. Adversarial Autoencoders. http://arxiv.org/abs/1511.05644. 2016. +Mikolov, Tomas, Yih, Scott Wen-tau, Zweig, Geoffrey. Linguistic Regularities in Continuous Space Word Representations. NAACL-HLT, 2013. +Radford , Alec, Metz , Luke, Chintala, Soumith. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. Advances in Neural Information Processing Systems, 2015. +Reed, Scott E, Zhang, Yi, Zhang, Yuting, and Lee, Honglak. Deep visual analogy-making. In Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 28, pp. 1252–1260. Curran Associates, Inc., 2015 +Shoemake, Ken. Animating rotation with quaternion curves. In ACM Siggraph, 19(3):245–254, 1985. Zhang, N., Paluri, M., Ranzato, M., Darrell, T., and Bourdev, L. Panda: Pose aligned networks for deep attribute modeling. In CVPR, 2014. \ No newline at end of file diff --git a/parse/train/SypU81Ole/SypU81Ole_content_list.json b/parse/train/SypU81Ole/SypU81Ole_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..24c81487ba085977e64529bd7a0aab7af1d22e49 --- /dev/null +++ b/parse/train/SypU81Ole/SypU81Ole_content_list.json @@ -0,0 +1,935 @@ +[ + { + "type": "text", + "text": "SAMPLING GENERATIVE NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 616, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tom White \nSchool of Design \nVictoria University of Wellington \nWellington, New Zealand \ntom.white@vuw.ac.nz ", + "bbox": [ + 184, + 145, + 405, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 251, + 544, + 266 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 233, + 284, + 766, + 422 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 450, + 336, + 467 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 483, + 825, + 580 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/2555a5b238a3ab69e2c09451d964cdc3cd6249e1558a4a8be442906ffa2cf867.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 385, + 616, + 612, + 734 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 180, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 176, + 313, + 825, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 SAMPLING TECHNIQUES", + "text_level": 1, + "bbox": [ + 176, + 392, + 410, + 409 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 425, + 825, + 468 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 474, + 825, + 601 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 INTERPOLATION ", + "text_level": 1, + "bbox": [ + 176, + 621, + 328, + 635 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 648, + 825, + 704 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 710, + 825, + 823 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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: ", + "bbox": [ + 176, + 829, + 823, + 871 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/708bf31233a29b497978964975c5b0a0374f39ed8d7f70db8a566d24410a9165.jpg", + "text": "$$\nS l e r p ( q _ { 1 } , q _ { 2 } ; \\mu ) = \\frac { \\sin { ( 1 - \\mu ) \\theta } } { \\sin { \\theta } } q _ { 1 } + \\frac { \\sin { \\mu \\theta } } { \\sin { \\theta } } q _ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 338, + 886, + 656, + 917 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/8aae2d9c8a78ba60fe122cdffe2e1c90eb9cc17ebb47405d844f2f462451a069.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 312, + 157, + 684, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 ANALOGY ", + "text_level": 1, + "bbox": [ + 174, + 523, + 287, + 537 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 549, + 825, + 606 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Analogies are usually written in the form: ", + "bbox": [ + 176, + 613, + 447, + 627 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7515ff34d92558afaa0cf5987d8f1ba38f78a30fd8b261ad03c919fc04b64a21.jpg", + "text": "$$\nA : B : : C : ?\n$$", + "text_format": "latex", + "bbox": [ + 454, + 642, + 545, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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: ", + "bbox": [ + 173, + 671, + 826, + 700 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/bd560f14f7a2926ad033a750c2c01592e629c31ab24d696eedeb8083b91e3c07.jpg", + "text": "$$\n( B - A ) = ( ? - C )\n$$", + "text_format": "latex", + "bbox": [ + 429, + 714, + 568, + 732 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/777dfd3a142196be95405584ec75399d9096b7ee18c044ecc725a26974de998a.jpg", + "text": "$$\n? = C + B - A\n$$", + "text_format": "latex", + "bbox": [ + 442, + 752, + 555, + 768 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that an interesting property of this solution is an implied symmetry: ", + "bbox": [ + 173, + 781, + 651, + 797 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/93441d6800e52da0841aca98ed4cb183d97e215ac8eee6b23f88cbeac7786300.jpg", + "text": "$$\nA : C : : B : ?\n$$", + "text_format": "latex", + "bbox": [ + 452, + 811, + 545, + 828 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Because the same terms can be rearranged: ", + "bbox": [ + 174, + 840, + 455, + 856 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f326febc90f8b687bb59ca751d04afee6f1f0c7ec833343f67cec8348345e5d0.jpg", + "text": "$$\n( C - A ) = ( ? - B )\n$$", + "text_format": "latex", + "bbox": [ + 429, + 869, + 566, + 887 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a9461c48a768bb6554de598af283285d591990d0b8e984191ec6de1362c54495.jpg", + "text": "$$\n? = B + C - A\n$$", + "text_format": "latex", + "bbox": [ + 444, + 909, + 555, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 176, + 166, + 823, + 223 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/e1290feaf1c831efd10ce46a564ea1977174665ca522bfad469160b29a6b92b6.jpg", + "image_caption": [ + "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) " + ], + "image_footnote": [], + "bbox": [ + 312, + 234, + 683, + 440 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/e2c12f518e7a60e7756fcff5a47ee67e605eac0dece4b8a72e38308c67903fc1.jpg", + "image_caption": [ + "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) " + ], + "image_footnote": [], + "bbox": [ + 312, + 539, + 681, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 MANIFOLD TRAVERSAL ", + "text_level": 1, + "bbox": [ + 174, + 820, + 380, + 834 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 166, + 825, + 265 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/5c968d2e79efc68093073d2e767c0ac6d0414407c99248db2b212c59fc275721.jpg", + "image_caption": [ + "(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) " + ], + "image_footnote": [], + "bbox": [ + 254, + 276, + 743, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 ATTRIBUTE VECTORS ", + "text_level": 1, + "bbox": [ + 176, + 835, + 387, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/369499ddd963f3fcfa56e86da65b1e2f780c624c8dbdfb4bec9bf195d3b0fc9f.jpg", + "image_caption": [ + "Figure 6: Traversals along the smile vector. (model: GAN from Dumoulin 16 on CelebA) " + ], + "image_footnote": [], + "bbox": [ + 220, + 159, + 779, + 247 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 CORRELATED LABELS ", + "text_level": 1, + "bbox": [ + 176, + 297, + 370, + 311 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 173, + 323, + 826, + 393 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/13921031a6fc7f4a741d17f1a09d64c75ec3878b6bd667a0d83e202b74aa2ab9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MaleNot MaleTotal
Smiling17%31%48%
Not Smiling25%27%52%
Total42%58%
", + "bbox": [ + 352, + 406, + 643, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 476, + 828, + 518 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/12e22a423a2190d78b89c4905870ca763f27e7d34d28d2eb69364768552065dd.jpg", + "image_caption": [ + "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) " + ], + "image_footnote": [], + "bbox": [ + 312, + 541, + 684, + 761 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 844, + 825, + 900 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/1fc47c5f803af6629ad9fcf26dfc4ebc0c47909864f336c43d20ce2635e86732.jpg", + "table_caption": [ + "Table 2: CelebA smile versus open mouth attributes shows a strong symmetric correlation (greater than 3 to 1). " + ], + "table_footnote": [], + "table_body": "
Open MouthNot Open MouthTotal
Smiling36%12%48%
Not Smiling12%40%52%
Total48%52%
", + "bbox": [ + 303, + 205, + 694, + 266 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/69eab4601a0a204cd93f91d2d32563736b22537e7fffd0a22b831ef129643299.jpg", + "image_caption": [ + "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) " + ], + "image_footnote": [], + "bbox": [ + 220, + 337, + 777, + 553 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 SYNTHETIC ATTRIBUTES ", + "text_level": 1, + "bbox": [ + 174, + 645, + 388, + 659 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 671, + 825, + 756 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 173, + 762, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7bbf40f34aa3802b69e1e6f8f450dabf7543c69c4a76b250dbe87d112adffd28.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 220, + 101, + 779, + 246 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/4c1c224c51d82c101fea05e04fb4347d0849d65fb20d5665fa371f8f5ff6f53b.jpg", + "image_caption": [ + "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. " + ], + "image_footnote": [], + "bbox": [ + 218, + 299, + 779, + 585 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 QUANTITATIVE ANALYSIS WITH CLASSICICATION ", + "text_level": 1, + "bbox": [ + 174, + 694, + 560, + 708 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 719, + 825, + 832 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "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). ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2b719666fa1b998c6cab6803b6a5af1dea037009abfc7767ccaadde07b2a8585.jpg", + "table_caption": [ + "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_footnote": [], + "table_body": "
Approach /Attribute[1][2] W [2] L[3] ANet[4] LNet[4] FullVAE AtDotGAN AtDot
5 0.C. Shadow8582 888688918889
Arched Eyebrow7673 787574797575
Attractive7877 817977817169
Bags Under Eye7671 797773798079
Bald8992 969295989898
Bangs8889 929492959090
Big Lips6461 676366688576
Big Nose7470 757475787777
Black Hair7074 857784888381
Blond Hair8081 938691958790
Blurry8177 868380849595
Brown Hair6069 777478807681
BushyEyebrow8076 868085908786
Chubby8682 868686919494
Double Chin8885 889088929595
Eyeglasses9894 989696999594
Goatee9386 939292959394
Gray Hair9088 949393979596
Heavy Makeup8584 908785907675
High Cheekbone8480 868584878168
Male9193 979594988180
Mouth Open8782 938586927867
Mustache9183 938791959596
Narrow Eyes8279 848377819389
No Beard9087 939192958584
Oval Face6462 656563667271
Pale Skin8384 918987919696
Pointy Nose686571 6770727272
Recede Hair768285 8485899392
Rosy Cheeks848187 8587909393
Sideburns949093 9491969494
Smiling898992 9288928768
Straight Hair636769 7069738079
Wavy Hair737677 7975807575
Earring737278 7778828181
Hat899196 9396999697
Lipstick898893 9190937977
Necklace686767 7068718888
Necktie868891 9086939392
Young807784 8183877881
Average8179 858383878684
", + "bbox": [ + 174, + 94, + 903, + 720 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/23124b836cb1a63feda2684bd36b5d1d066add8214b0fa371e200f2960eb088d.jpg", + "image_caption": [ + "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). " + ], + "image_footnote": [], + "bbox": [ + 184, + 99, + 812, + 237 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4 FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 359, + 334, + 376 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 393, + 825, + 463 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 470, + 825, + 512 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 520, + 825, + 575 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 595, + 326, + 608 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. ", + "bbox": [ + 174, + 619, + 825, + 676 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 695, + 267, + 708 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "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. \nGoodfellow, 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. \nGómez-Bombarelli, Rafael, Duvenaud, David, Miguel, Hernández-Lobato, José, Aguilera-Iparraguirre, Jorge, Hirzel, Timothy D., Adams, Ryan P., and Alán Aspuru-Guzik. Automatic chemical design using a data-driven continuous representation of molecules. https://arxiv.org/abs/1610.02415 2016 Kingma, Diederik P. and Welling, Max. Auto-encoding variational Bayes. In Proceedings of the International Conference on Learning Representations, 2014. \nKumar, N., Belhumeur, P., and Nayar, S. Facetracer: A search engine for large collections of images with faces. In ECCV, 2008. \nLamb, Alex, Dumoulin, Vincent, Courville, Aaron. Discriminative Regularization for Generative Models. http://arxiv.org/abs/1602.03220 2016 \nLarsen, Anders Boesen Lindbo, Sønderby, Søren Kaae, Larochelle, Hugo, Winther, Ole. Autoencoding beyond pixels using a learned similarity metric. https://arxiv.org/abs/1512.09300 2016 \nLeCun, Y. and Bengio., Y. Word-level training of a handwritten word recognizer based on convolutional neural networks. In ICPR, 1994. \nLi, J., and Zhang, Y. Learning surf cascade for fast and accurate object detection. In CVPR, 2013. \nMakhzani, Alireza, Shlens, Jonathon, Jaitly, Navdeep, Goodfellow, Ian, Brendan, Frey. Adversarial Autoencoders. http://arxiv.org/abs/1511.05644. 2016. \nMikolov, Tomas, Yih, Scott Wen-tau, Zweig, Geoffrey. Linguistic Regularities in Continuous Space Word Representations. NAACL-HLT, 2013. \nRadford , Alec, Metz , Luke, Chintala, Soumith. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks. Advances in Neural Information Processing Systems, 2015. \nReed, Scott E, Zhang, Yi, Zhang, Yuting, and Lee, Honglak. Deep visual analogy-making. In Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 28, pp. 1252–1260. Curran Associates, Inc., 2015 \nShoemake, Ken. Animating rotation with quaternion curves. In ACM Siggraph, 19(3):245–254, 1985. Zhang, N., Paluri, M., Ranzato, M., Darrell, T., and Bourdev, L. Panda: Pose aligned networks for deep attribute modeling. In CVPR, 2014. 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On labeled", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "datasets with correlated labels, data replication can be used to create bias-corrected vectors. Synthetic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "attributes vectors also can be derived via data augmentation on unlabeled data. 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But often these can result in sampling", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 360, + 438, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 438, + 372 + ], + "score": 1.0, + "content": "the latent space from locations very far outside the manifold of probable locations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "Our work has followed two useful principles when sampling the latent space of a generative model.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "The first is to avoid sampling from locations that are highly unlikely given the prior of the model. 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Quantitative analysis", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 484, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 484, + 293 + ], + "score": 1.0, + "content": "of attribute vectors can be performed by using them as the basis for attribute binary classifiers.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 247, + 506, + 293 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 311, + 251, + 324 + ], + "lines": [ + { + "bbox": [ + 104, + 309, + 253, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 253, + 327 + ], + "score": 1.0, + "content": "2 SAMPLING TECHNIQUES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "Generative models are often evaluated by examining samples from the latent space. Techniques", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 363 + ], + "score": 1.0, + "content": "frequently used are random sampling and linear interpolation. But often these can result in sampling", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 360, + 438, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 438, + 372 + ], + "score": 1.0, + "content": "the latent space from locations very far outside the manifold of probable locations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 337, + 505, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "Our work has followed two useful principles when sampling the latent space of a generative model.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "The first is to avoid sampling from locations that are highly unlikely given the prior of the model. This", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "technique is very well established - including being used in the original VAE paper which adjusted", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "sampling through the inverse CDF of the Gaussian to accommodate the Gaussian prior (Kingma", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "& Welling, 2014). The second principle is to recognize that the dimensionality of the latent space", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 429, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 505, + 445 + ], + "score": 1.0, + "content": "is often artificially high and may contains dead zones that are not on the manifold learned during", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 457 + ], + "score": 1.0, + "content": "training. This has been demonstrated for VAE models (Makhzani et al., 2016) and implies that simply", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "matching the model’s prior will not always be sufficient to yield samples that appear to have been", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 220, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 220, + 478 + ], + "score": 1.0, + "content": "drawn from the training set.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 376, + 506, + 478 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 201, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 203, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 203, + 505 + ], + "score": 1.0, + "content": "2.1 INTERPOLATION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "Interpolation is used to traverse between two known locations in latent space. Research on gener-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "ative models often uses interpolation as a way of demonstrating that a generative model has not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 380, + 549 + ], + "score": 1.0, + "content": "simply memorized the training examples (eg: Radford et al., 2015,", + "type": "text" + }, + { + "bbox": [ + 381, + 536, + 401, + 547 + ], + "score": 0.5, + "content": "\\ S 6 . 1 \\AA ,", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "). In creative applications", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 453, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 453, + 560 + ], + "score": 1.0, + "content": "interpolations can be used to provide smooth transitions between two decoded images.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 513, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "Frequently linear interpolation is used, which is easily understood and implemented. But this is often", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 433, + 587 + ], + "score": 1.0, + "content": "inappropriate as the latent spaces of most generative models are high dimensional", + "type": "text" + }, + { + "bbox": [ + 433, + 575, + 454, + 585 + ], + "score": 0.79, + "content": "( > 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "dimensions)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "with a Gaussian or uniform prior. In such a space, linear interpolation traverses locations that are", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "extremely unlikely given the prior. As a concrete example, consider a 100 dimensional space with the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 167, + 620 + ], + "score": 1.0, + "content": "Gaussian prior", + "type": "text" + }, + { + "bbox": [ + 167, + 608, + 185, + 619 + ], + "score": 0.88, + "content": "\\scriptstyle \\mu = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 608, + 188, + 620 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 189, + 608, + 206, + 618 + ], + "score": 0.84, + "content": "\\sigma { = } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 608, + 505, + 620 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "each pair, the top series is linear interpolation and the bottom is spherical. Note the weak generations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 482, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 482, + 397 + ], + "score": 1.0, + "content": "produced by linear interpolation at the center, which are not present in spherical interpolation.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 16.0 + }, + { + "type": "title", + "bbox": [ + 107, + 415, + 176, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 177, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 177, + 428 + ], + "score": 1.0, + "content": "2.2 ANALOGY", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "Analogy has been shown to capture regularities in continuous space models. 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The resulting MINE grid represents a small contiguous manifold of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 628, + 363, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 363, + 640 + ], + "score": 1.0, + "content": "the larger latent space. (model: VAE from Lamb 16 on CelebA)", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 237, + 675 + ], + "lines": [ + { + "bbox": [ + 104, + 660, + 239, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 660, + 239, + 677 + ], + "score": 1.0, + "content": "3 ATTRIBUTE VECTORS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Many generative models result in a latent space that is highly structured, even on purely unsupervised", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "datasets (Radford et al., 2015). When combined with labeled data, attribute vectors can be computed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "using simple arithmetic. For example, a vector can be computed which represents the smile attribute,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "which by shorthand we call a smile vector. 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(model: GAN from Dumoulin 16 on CelebA)", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + } + ], + "index": 5.0 + }, + { + "type": "title", + "bbox": [ + 108, + 236, + 227, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 228, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 228, + 249 + ], + "score": 1.0, + "content": "3.1 CORRELATED LABELS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "The approach of building attribute vectors from means of labeled data has been noted to suffer from", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "correlated labels (Larsen et al., 2016). 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I thank Victoria University of Wellington School of Design for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "supporting research on Creative Intelligence. I also thank the vibrant machine learning and creative", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 524, + 381, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 381, + 537 + ], + "score": 1.0, + "content": "coding communities on twitter for their support and encouragement.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 551, + 164, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 551, + 165, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 165, + 562 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 506, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "Dumoulin, Vincent, Belghazi, Ishmael, Poole, Ben, Lamb, Alex, Arjovsky, Martin, Mastropietro, Olivier", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 581, + 489, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 489, + 591 + ], + "score": 1.0, + "content": "Courville, Aaron. 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We hope to continue to improve the library so", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "that the techniques are applicable across a broad range of generative models. 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I thank Victoria University of Wellington School of Design for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "supporting research on Creative Intelligence. 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b/parse/train/Wua2zjxJdYo/Wua2zjxJdYo.md new file mode 100644 index 0000000000000000000000000000000000000000..8c059849ceedde57b7953540df537944db3f4c95 --- /dev/null +++ b/parse/train/Wua2zjxJdYo/Wua2zjxJdYo.md @@ -0,0 +1,276 @@ +# Learning to Generate Realistic Noisy Images via Pixel-level Noise-aware Adversarial Training + +Yuanhao Cai 1,2, Xiaowan $\mathbf { H } \mathbf { u } ^ { \mathrm { ~ 1 , 2 ~ } }$ , Haoqian Wang $^ { 1 , 2 , * }$ , Yulun Zhang 3, Hanspeter Pfister 4, Donglai Wei 5 + +1 Shenzhen International Graduate School, Tsinghua University, 2 Shenzhen Institute of Future Media Technology, 3 ETH Zürich, 4 Harvard University, 5 Boston College + +# Abstract + +Existing deep learning real denoising methods require a large amount of noisyclean image pairs for supervision. Nonetheless, capturing a real noisy-clean dataset is an unacceptable expensive and cumbersome procedure. To alleviate this problem, this work investigates how to generate realistic noisy images. Firstly, we formulate a simple yet reasonable noise model that treats each real noisy pixel as a random variable. This model splits the noisy image generation problem into two subproblems: image domain alignment and noise domain alignment. Subsequently, we propose a novel framework, namely Pixel-level Noise-aware Generative Adversarial Network (PNGAN). PNGAN employs a pre-trained real denoiser to map the fake and real noisy images into a nearly noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a pixel-level adversarial training to conduct noise domain alignment. Additionally, for better noise fitting, we present an efficient architecture Simple Multi-scale Network (SMNet) as the generator. Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers trained with the generated noisy images achieve state-of-the-art (SOTA) results on four real denoising benchmarks. + +# 1 Introduction + +Image denoising is an important yet challenging problem in low-level vision. It aims to restore a clean image from its noisy counterpart. Traditional approaches concentrate on designing a rational maximum a posteriori (MAP) model, containing regularization and fidelity terms, from a Bayesian perspective [1]. Some image priors like low-rankness [2, 3, 4], sparsity [5], and non-local similarity [6, 7] are exploited to customize a better rational MAP model. However, these hand-crafted methods are inferior in representing capacity. With the development of deep learning, image denoising has witnessed significant progress. Deep convolutional neural network (CNN) applies a powerful learning model to eliminate noise and has achieved promising performance [8, 9, 10, 11, 12, 13, 14]. These deep CNN denoisers rely on a large-scale dataset of real-world noisy-clean image pairs. Nonetheless, collecting even small datasets is extremely tedious and labor-intensive. The process of acquiring real-world noisy-clean image pairs is to take hundreds of noisy images of the same scene and average them to get the clean image. To get more image pairs, researchers try to synthesize noisy images. + +In particular, there are two common settings for synthesizing noisy images. As shown in Fig. 1 (a1), setting1 directly adds the additive white Gaussian noise (AWGN) with the clean RGB image. For a long time, single image denoising [15, 16, 17, 18, 19, 10] is performed with setting1. Nevertheless, fundamentally different from AWGN, real camera noise is generally more sophisticated and signaldependent[20, 21]. The noise produced by photon sensing is further affected by the in-camera signal processing (ISP) pipeline (e.g., Gama correction, compression, and demosaicing). Models trained with setting1 are easily over-fitted to AWGN and fail in real noise removal. Setting2 is based on ISP-modeling CNN [22] and Poisson-Gaussian [21, 23] noise model that modeling photon sensing with Poisson and remaining stationary disturbances with Gaussian has been adopted in RAW denoising. As shown in Fig. 1 (a2), setting2 adds a Poisson-Gaussian noise with the clean RAW image and then passes the result through a pre-trained RAW2RGB CNN to obtain the RGB noisy counterpart. Notably, when the clean RAW image is unavailable, a pre-trained RGB2RAW CNN is utilized to transform the clean RGB image to its RAW counterpart [22]. However, setting2 has the following drawbacks: (i) The noise is assumed to obey a hand-crafted probability distribution. However, because of the randomness and complexity of real camera noise, it’s difficult to customize a hand-crafted probability distribution to model all the characteristics of real noise. (ii) The ISP pipeline is very sophisticated and hard to be completely modeled. The RAW2RGB branch only learns the mapping from the clean RAW domain to the clean RGB space. However, the mapping from the Poisson-Gaussian noisy RAW domain to the real noisy RGB space can not be ensured. (iii) The ISP pipelines of different devices vary significantly, which results in the poor generality and robustness of ISP modeling CNNs. Thus, whether noisy images are synthesized with setting1 or 2, there still remains a discrepancy between synthetic and real noisy datasets. We notice that GAN utilizes the internal information of the input image and external information from other images when modeling image priors. Hence, we propose to use GAN to adaptively learn the real noise distribution. + +GAN is firstly introduced in [24] and has been proven successful in image synthesis [25, 26, 27] and translation [26, 27]. Subsequently, GAN is applied to image restoration and enhancement, e.g., super resolution [28, 29, 30], style transfer [27, 31], enlighten [32, 33], deraining [34], dehazing [35], image inpainting [36, 37], image editing [38, 39], and mobile photo enhancement [40, 41]. Although GAN is widely applied in low-level vision tasks, few works are dedicated to investigating the realistic noise generation problem [42]. Chen et al. [43] propose a simple GAN that takes Gaussian noise as input to generate noisy patches. However, as in general, this GAN is image-level, i.e., it treats images as samples and attempts to approximate the probability distribution of real-world noisy images. This image-level GAN neglects that each pixel of a real noisy image is a random variable and the real noise is spatio-chromatically correlated, thus results in coarse learning of the real noise distribution. + +To alleviate the above problems, this work focuses on learning how to generate realistic noisy images so as to augment the training data for real denoisers. To begin with, we propose a simple yet reasonable noise model that treats each pixel of a real noisy image as a random variable. This noise model splits the noise generation problem into two sub-problems: image domain alignment and noise domain alignment. Subsequently, to tackle these two sub-problems, we propose a novel Pixel-level Noise-aware Generative Adversarial Network (PNGAN). During the training procedure of PNGAN, we employ a pre-trained real denoiser to map the generated and real noisy images into a nearly noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a pixel-level adversarial training that encourages the generator to adaptively simulate the real noise distribution so as to conduct the noise domain alignment. In addition, for better real noise fitting, we present a lightweight yet efficient CNN architecture, Simple Multi-scale Network (SMNet) as the generator. SMNet repeatedly aggregates multi-scale features to capture rich auto-correlation, which provides more sufficient spatial representations for noise simulating. Different from general image-level GAN, our discriminator is pixel-level. The discriminator outputs a score map. Each position on the score map indicates how realistic the corresponding noisy pixel is. With this pixellevel noise-aware adversarial training, the generator is encouraged to create solutions that are highly similar to real noisy images and thus difficult to be distinguished. + +In conclusion, our contributions can be summarized into four points: + +(1) We formulate a simple yet reasonable noise model. This model treats each noisy pixel as a random variable and then splits the noisy image generation into two parts: image and noise domain alignment. + +(2) We propose a novel framework, PNGAN. It establishes an effective pixel-level adversarial training to encourage the generator to favor solutions that reside on the manifold of real noisy images. + +(3) We customize an efficient CNN architecture, SMNet learning rich multi-scale auto-correlation for better noise fitting. SMNet serves as the generator in PNGAN costing only 0.8M parameters. + +(4) Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers finetuned with the generated noisy images achieve SOTA results on four real denoising benchmarks. + +![](images/09f874f387c331b9a197a98ffe2b51b7acdc68dca22e5b8d83d8a65e04c2d5c0.jpg) +Figure 1: The pipeline of using PNGAN to perform data augmentation. It is divided into: (a) synthesizing phase, (b) training phase, and (c) finetuning phase. Please refer to the text (Sec. 2) for more detailed descriptions. + +# 2 Proposed Method + +As shown in Fig. 1, the pipeline of using PNGAN to perform data augmentation consists of three phases. (a) is the synthesizing phase. (a1) and (a2) are two common synthetic settings. In this phase, we produce the synthetic noisy image from its clean RGB or RAW counterpart. (b) is the training phase of PNGAN. The generator $G$ adopts the synthetic image as input. Which synthetic setting is selected is controlled by the switch. By using a pre-trained real denoiser $D _ { d }$ , PNGAN establishes a pixel-level noise-aware adversarial training between the generator $G$ and discriminator $D$ so as to simultaneously conduct image and noise domain alignment. $D _ { d }$ is set as RIDNet [44] in this work. (c) is the finetuning phase. Firstly, in (c1), the generator creates extended fake noisy-clean image pairs. Secondly, in (c2), the fake and real data are jointly utilized to finetune a series of real denoisers. + +# 2.1 Pixel-level Noise Modelling + +Real camera noise is sophisticated and signal-dependent. Specifically, in the real camera system, the RAW noise produced by photon sensing comes from multiple sources (e.g., short noise, thermal noise, and dark current noise) and is further affected by the ISP pipeline. Besides, illumination changes and camera movement inevitably lead to spatial pixel misalignment and color or brightness deviation. Hence, hand-designed noise models based on mathematical assumptions are difficult to accurately and completely describe the properties of real noise. Different from previous methods, we don’t base our noise model on any mathematical assumptions. Instead, we use CNN to implicitly simulate the characteristics of real noise. We begin by noting that when taking multiple noisy images of the same scene, the noise intensity of the same pixel varies a lot. Simultaneously, affected by the ISP pipeline, the real noise is spatio-chromatically correlated. Thus, the correlation between different pixels of the same real noisy image should be considered. In light of these facts, we treat each pixel of a real noisy image as a random variable and formulate a simple yet reasonable noise model: + +$$ +\mathbf { I } _ { r n } [ i ] = \hat { \mathbf { I } } _ { c l e a n } [ i ] + \mathbf { N } [ i ] , \quad D _ { d } ( \mathbf { I } _ { r n } ) [ i ] = \hat { \mathbf { I } } _ { c l e a n } [ i ] , \quad 1 \leq i \leq H \times W , +$$ + +where $\hat { \mathbf { I } } _ { c l e a n } \in \mathbb { R } ^ { H \times W \times 3 }$ is the predicted clean counterpart of ${ \mathbf I } _ { r n }$ , it’s denoised by $D _ { d }$ . Each $\mathbf { N } [ i ]$ is a random noise variable with unknown probability distribution. Therefore, each $\mathbf { I } _ { \mathit { r n } } [ i ]$ can also be viewed as a distribution-unknown random variable. Now we aim to design a framework to generate a fake noisy image $\mathbf { I } _ { f n } \in \mathbb { R } ^ { H \times W \times 3 }$ such that the probability distribution of $\mathbf { I } _ { f n } [ i ]$ and $\mathbf { I } _ { \mathit { r n } } [ i ]$ is as close as possible. Please note that the mapping learned by $D _ { d }$ is precisely from ${ \mathbf I } _ { r n }$ to $\hat { \mathbf { I } } _ { c l e a n }$ . If the constant in Eq. (1) is set as the clean image $\mathbf { I } _ { c l e a n } \in \mathbb { R } ^ { \bar { H } \times W \times 3 }$ , the subsequent domain alignment will introduce unnecessary errors and eventually lead to inaccurate results. + +# 2.2 Pixel-level Noise-aware Adversarial Training + +Our goal is to generate realistic noisy images. According to the noise model in Eq. (1), we split this problem into two sub-problems: (i) Image domain alignment aims to align $\hat { \mathbf { I } } _ { c l e a n } [ i ]$ . (ii) Noise domain alignment targets at modeling the distribution of $\mathbf { N } [ i ]$ . To handle the sub-problems, PNGAN establishes a novel pixel-level noise-aware adversarial training between $G$ and $D$ in Fig. 1 (b). + +Image Domain Alignment. A very naive strategy to construct both image and noise domain alignment is to directly minimize the distance of $\mathbf { I } _ { f n }$ and ${ \mathbf I } _ { r n }$ . However, due to the intrinsic randomness, complexity, and irregularity of real noise, directly deploying $\mathcal { L } _ { 1 }$ loss between $\mathbf { I } _ { f n }$ and ${ \mathbf I } _ { r n }$ is unreasonable and drastically damages the quality of $\mathbf { I } _ { f n }$ . Besides, as analyzed in Sec. 2.1, each pixel of ${ \mathbf I } _ { r n }$ is a distribution-unknown random variable. This indicates that such a naive strategy challenges the training and may easily cause the non-convergence issue. Therefore, the noise interference should be eliminated while constructing the image domain alignment. To this end, we feed $\mathbf { I } _ { f n }$ and ${ \mathbf I } _ { r n }$ into $D _ { d }$ to obtain their denoised versions and then perform $\mathcal { L } _ { 1 }$ loss between $\mathbf { I } _ { f n }$ and ${ \mathbf I } _ { r n }$ : + +$$ +\mathcal { L } _ { 1 } = \sum _ { i = 1 } ^ { H \times W } \left| \left| D _ { d } ( \mathbf { I } _ { f n } ) [ i ] - D _ { d } ( \mathbf { I } _ { r n } ) [ i ] \right| \right| _ { 1 } = \sum _ { i = 1 } ^ { H \times W } \left| \left| D _ { d } ( \mathbf { I } _ { f n } ) [ i ] - \hat { \mathbf { I } } _ { c l e a n } [ i ] \right| \right| _ { 1 } . +$$ + +By using $D _ { d }$ , we can transfer ${ \mathbf I } _ { r n }$ and $\mathbf { I } _ { f n }$ into a nearly noise-free solution space. The value of $\hat { \mathbf { I } } _ { c l e a n }$ is relatively stable. Therefore, minimizing $\mathcal { L } _ { 1 }$ can encourage $G$ to favor solutions that after being denoised by $D _ { d }$ converge to $\hat { \mathbf { I } } _ { c l e a n }$ . In this way, the image domain alignment is constructed. + +Noise Domain Alignment. Becasue of the complexity and variability of real noise, it’s hard to completely seperate $\mathbf { N } [ i ]$ from $\mathbf { I } _ { \mathit { r n } } [ i ]$ in Eq (1). Fortunately, we note that on the basis of constructing the image domain alignment of $\hat { \mathbf { I } } _ { c l e a n } [ i ]$ , the noise domain alignment of $\mathbf { N } [ i ]$ is equivalent to the distribution estimation of $\mathbf { I } _ { \mathit { r n } } [ \bar { i } ]$ . Additionally, as the real noise is signaldependent, the alignment between $\mathbf { I } _ { f n } [ i ]$ and $\mathbf { I } _ { \mathit { r n } } [ i ]$ is more beneficial to capture the correlation between noise and scene. We denote the + +![](images/51ccb4cb837145780a480b4df93e5a343951a841cb8ba4022dc8e241605f5765.jpg) +Figure 2: Architecture of discriminator. + +distribution of $\mathbf { I } _ { \mathit { r n } } [ i ]$ as $P _ { d a t a } ( x _ { i } )$ , some real noisy pixel samples of $\mathbf { I } _ { \mathit { r n } } [ i ]$ as $\{ x _ { i } ^ { 1 } , x _ { i } ^ { 2 } , . . . , x _ { i } ^ { m } \}$ such that $x _ { i } ^ { k } \sim P _ { d a t a } ( x _ { i } )$ , and the distribution of $\mathbf { I } _ { f n } [ i ]$ as $P _ { G } ( x _ { i } ; \theta _ { G } )$ . Here $\theta _ { G }$ is the parameter of $G$ . Then we formulate the noise domain aligment into a maximum likelihood estimation problem: + +$$ +\theta _ { G } ^ { * } = a r g \operatorname * { m a x } _ { \theta _ { G } } \sum _ { i = 1 } ^ { H \times W } \sum _ { k = 1 } ^ { m } \log P _ { G } ( x _ { i } ^ { k } ; \theta _ { G } ) = a r g m a x \operatorname { \mathbb { E } } _ { i } \left[ \mathbb { E } _ { x _ { i } ^ { k } } \left[ \log P _ { G } ( x _ { i } ^ { k } ; \theta _ { G } ) \right] \right] , +$$ + +where $\mathbb { E }$ means taking the average value. To approach this upper bound as close as possible, we present $D$ and establish the pixel-level adversarial traininig between $G$ and $D$ . The architecture of $D$ is shown in Fig. 2. $D$ consists of 4 convolutional $( c o n v )$ layers and utilizes LeakyReLU activation $( \alpha = 0 . 2$ ). General discriminator treats a image as a sample and outputs a score indicating how realistic the image is. Instead, $D$ is a pixel-level classifier. $D$ adopts the fake and real noisy images as input in a mini-batch and outputs a score map $\mathbf { P } \in \mathbb { R } ^ { H \times W }$ for each image. Specifically, the information of $\mathbf { P } [ i ] \in [ 0 , 1 ]$ is the probability value indicating how realistic $P _ { G } ( x _ { i } ; \theta _ { G } )$ is. $G$ aims to generate more realistic $\mathbf { I } _ { f n } [ i ]$ to fool $D$ while $D$ targets at distinguishing $\mathbf { I } _ { f n } [ i ]$ from $\mathbf { I } _ { \mathit { r n } } [ i ]$ . According to Eq .(3), we formulate the adversarial training between $G$ and $D$ as a min-max problem: + +$$ +\operatorname* { m i n } _ { \theta _ { G } } \operatorname* { m } _ { \theta _ { D } } \mathbb { E } _ { i } \left[ \mathbb { E } _ { \mathbf { I } _ { r n } } \left[ \mathrm { l o g } ( D ( \mathbf { I } _ { r n } ; \theta _ { D } ) [ i ] ) \right] \right] + \mathbb { E } _ { i } \left[ \mathbb { E } _ { \mathbf { I } _ { f n } } \left[ \mathrm { l o g } ( 1 - D ( \mathbf { I } _ { f n } ; \theta _ { D } ) [ i ] ) \right] \right] , +$$ + +where $\mathbb { E } _ { \mathbf { I } _ { r n } }$ and $\mathbb { E } _ { \mathbf { I } _ { f n } }$ respectively represent the operation of taking the average for all fake and real data in the mini-batch. As analyzed in [45], to make GANs analogous to divergence minimization and produce sensible predictions based on the a priori knowledge that half of the samples in the mini-batch are fake, we utilize the recently proposed relativistic discriminator [45] as follow: + +$$ +\begin{array} { r l } & { D ( \mathbf { I } _ { r n } ; \theta _ { D } ) = \sigma ( C _ { D } ( \mathbf { I } _ { r n } ) ) , \quad D _ { R a } ( \mathbf { I } _ { r n } , \mathbf { I } _ { f n } ) = \sigma ( C _ { D } ( \mathbf { I } _ { r n } ) - \mathbb { E } _ { \mathbf { I } _ { f n } } ( C _ { D } ( \mathbf { I } _ { f n } ) ) ) , } \\ & { D ( \mathbf { I } _ { f n } ; \theta _ { D } ) = \sigma ( C _ { D } ( \mathbf { I } _ { f n } ) ) , \quad D _ { R a } ( \mathbf { I } _ { f n } , \mathbf { I } _ { r n } ) = \sigma ( C _ { D } ( \mathbf { I } _ { f n } ) - \mathbb { E } _ { \mathbf { I } _ { r n } } ( C _ { D } ( \mathbf { I } _ { r n } ) ) ) , } \end{array} +$$ + +where $D _ { R a }$ denotes the relativistic discriminator, $\sigma$ means the Sigmoid activation, and $C _ { D }$ represents the non-transformed discriminator output. $D _ { R a }$ estimates the probability that real data is more realistic than fake data and also directs the generator to create a fake image that is more realistic than real images. The loss functions of $D$ and $G$ are then defined in a symmetrical form: + +$$ +\begin{array} { r l } & { \mathcal { L } _ { D } = - \mathbb { E } _ { i } [ \mathbb { E } _ { { \mathbf { I } } _ { r n } } \big [ \mathrm { l o g } ( D _ { R a } ( { \mathbf { I } } _ { r n } , { \mathbf { I } } _ { f n } ) [ i ] ) \big ] + \mathbb { E } _ { { \mathbf { I } } _ { f n } } \big [ \mathrm { l o g } ( 1 - D _ { R a } ( { \mathbf { I } } _ { f n } , { \mathbf { I } } _ { r n } ) [ i ] ) \big ] \big ] , } \\ & { \mathcal { L } _ { G } = - \mathbb { E } _ { i } [ \mathbb { E } _ { { \mathbf { I } } _ { r n } } \big [ \mathrm { l o g } ( 1 - D _ { R a } ( { \mathbf { I } } _ { r n } , { \mathbf { I } } _ { f n } ) [ i ] ) \big ] + \mathbb { E } _ { { \mathbf { I } } _ { f n } } \big [ \mathrm { l o g } ( D _ { R a } ( { \mathbf { I } } _ { f n } , { \mathbf { I } } _ { r n } ) [ i ] ) \big ] ] . } \end{array} +$$ + +![](images/24b711af4a5f63491e29b6e034035d38283bbbffaf19904c2afc7a1fe897c1b6.jpg) +Figure 3: Details of the generator. (a) is the architecture of SMNet. (b) depicts the components of SRG. (c) shows the details of MAB. MAB is equipped with FCA, which is illustrated in (d). + +During the training procedure, we fix $D$ to train $G$ and fix $G$ to train $D$ iteratively. Minimizing $\mathcal { L } _ { G }$ and $\mathcal { L } _ { D }$ alternately allows us to train a generative model $G$ with the goal of fooling the pixel-level discriminator $D$ that is trained to distinguish fake noisy images from real noisy images. This pixellevel noise-aware adversarial training scheme encourages $G$ to favor perceptually natural solutions that reside on the manifold of real noisy images so as to construct the noise domain alignment. + +# 2.3 Noisy Image Generating + +In Sec. 2.1, we denote the probability distribution of $\mathbf { I } _ { f n } [ i ]$ as $P _ { G } ( x _ { i } ; \theta _ { G } )$ . Now we customize a light-weight yet efficient CNN architecture, SMNet as $G$ to generate $P _ { G } ( x _ { i } ; \theta _ { G } )$ . In this section, we firstly introduce the input setting of $G$ and subsequently detail the architecture of SMNet. + +Input Setting. We aim to generate a realistic noisy image from its clean counterpart. A naive setting is to directly adopt the clean image as the input to generate the noisy image. However, this naive setting is not in line with the fact. When we repeatedly feed the same clean image to a pre-trained $G$ , $G$ outputs completely the same noisy images. In contrast, when taking multiple pictures in the real world, the real noisy images vary a lot in the intensity of each pixel. This is caused by many factors (e.g., photon sensing noise, ISP pipelines, and illumination conditions). Hence, the naive input setting containing no distribution is unreasonable. We review that the general GANs sample from an initial random distribution (usually Gaussian) to generate a fake image. Hence, the input of $G$ should contain a random distribution so as to generate multiple noisy images of the same scene. We note that the two common synthetic settings meet this condition. Therefore, we utilize the two common settings to produce the synthetic image and then adopt the synthetic image as the input of $G$ Subsequently, we propose a light-weight yet efficient architecture, SMNet for better real noise fitting. + +SMNet Architecture. The architecture of SMNet is shown in Fig. 3 (a). SMNet involves $t$ Simple Residual Groups (SRG) and each SRG contains $n$ Multi-scale Attention Blocks (MAB). The synthetic input $\mathbf { I } _ { s y n } \in \dot { \mathbb { R } } ^ { H \times W \times 3 }$ continuously undergoes a conv layer $f _ { 1 }$ , $t$ SRGs, and a conv layer $f _ { 2 }$ , then adds with a long identity mapping for efficient residual learning to eventually generate the fake noisy counterpart $\mathbf { I } _ { f n } \in \mathbb { R } ^ { H \times W \times \dot { 3 } }$ . This process can be formulated as: + +$$ +\mathbf { I } _ { f n } = \mathbf { I } _ { s y n } + f _ { 2 } ( S _ { t } ( \mathbf { F } _ { S _ { t } } ) ) , ~ \mathbf { F } _ { S _ { j + 1 } } = S _ { j } ( \mathbf { F } _ { S _ { j } } ) , ~ \mathbf { F } _ { S _ { 1 } } = f _ { 1 } ( \mathbf { I } _ { f n } ) , +$$ + +where $S _ { j }$ denotes the $j _ { t h }$ SRG, $1 \leq j \leq t - 1$ . The components of SRG are depicted in Fig. 3 (b). We define the input feature of the $j _ { t h }$ SRG as $\mathbf { F } _ { S _ { j } } \in \mathbb { R } ^ { H \times W \times C }$ and its channel as $C$ . $\mathbf { F } _ { S _ { j } }$ continuously undergoes a conv layer, $n$ MABs, and a conv layer to add with an identity mapping: + +$$ +\mathbf { F } _ { S _ { j + 1 } } = \mathbf { F } _ { S _ { j } } + M _ { n } ^ { j } ( \mathbf { F } _ { M _ { n } ^ { j } } ) , ~ \mathbf { F } _ { M _ { k + 1 } ^ { j } } = M _ { k } ^ { j } ( \mathbf { F } _ { M _ { k } ^ { j } } ) , ~ \mathbf { F } _ { M _ { 1 } ^ { j } } = \mathbf { F } _ { S _ { j } } , +$$ + +where $M _ { k } ^ { j }$ denotes the $k _ { t h }$ MAB of the $j _ { t h }$ SRG, $1 \leq k \leq n - 1$ . MAB is the basic building block and the most significant component of SMNet. The details of MAB are depicted in Fig. 3 (c). We customize MAB with the following motivations: (i) Multi-scale feature fusion can increase the receptive field and multi-resolution contextual information can cover rich auto-correlation, which provides more sufficient spatial representations for noise fitting. (ii) The noise level decreases as the scale increases and nonlinear sampling operations can increase the richness of the mapping in the potential space of real noise. Therefore, we exploit parallel multi-resolution branch aggregation from top to bottom and bottom to top to facilitate the learning of complex real noise. (iii) Specifically, during the feature downsampling, general downsample operation damages the image information, resulting in pixel discontinuity and jagged artifact. To alleviate these issues, we exploit ShiftInvariant Downsample [46] that copes with the discontinuity by using continuous pooling and filtering operation, preserving rich cross-correlation information between original and downsampled images. (iv) To efficiently capture continuous channel correlation and avoid information loss, we use the 1D channel attention module, Fast Channel Attention (FCA) instead of the general 2D convolution attention module. The input feature, $\mathbf { F } _ { M _ { k } ^ { j } } \in \mathbb { R } ^ { H \times W \times C }$ is fed into three parallel multi-scale paths: + +
MethodsSIDD[47]DND [48]MethodsPolyU [40]Nam [49]
PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑
DnCNN-B[10]23.660.58332.430.790RDN[12]37.940.94638.160.956
CBDNet [50]33.280.86838.060.942FFDNet+ [19]38.170.95138.810.957
RIDNet [44]38.710.91439.260.953TWSC [51]38.680.95838.960.962
AINDNet [52]39.150.95539.530.956CBDNet [50]38.740.96139.080.969
VDN[53]39.230.95539.380.952RIDNet [44]38.860.96239.200.973
CycleISP [22]39.520.95739.560.956VDN [53]39.040.96539.680.976
MPRNet [54]39.710.95839.800.954MPRNet [54]39.070.96939.410.974
MIRNet [55]39.720.95939.880.956MIRNet [55]39.180.97339.570.979
RIDNet*(Ours)39.250.95639.550.955RIDNet*(Ours)39.540.97139.690.975
MPRNet*(Ours)40.060.96040.180.961MPRNet*(Ours)40.480.98240.720.984
MIRNet*(Ours)40.070.96040.250.962MIRNet*(Ours)40.550.98340.780.986
+ +Table 1: Comparison on four benchmarks. \* denotes denoisers finetuned with images generated by PNGAN. + +$$ +\mathbf { F } _ { M _ { k } ^ { j } } ^ { 1 } = F C A ( \mathbf { F } _ { M _ { k } ^ { j } } ) , \mathbf { F } _ { M _ { k } ^ { j } } ^ { 2 } = f _ { u p } ^ { 2 } ( F C A ( f _ { s i d } ^ { 2 } ( \mathbf { F } _ { M _ { k } ^ { j } } ) ) ) , \mathbf { F } _ { M _ { k } ^ { j } } ^ { 4 } = f _ { u p } ^ { 4 } ( F C A ( f _ { s i d } ^ { 4 } ( \mathbf { F } _ { M _ { k } ^ { j } } ) ) ) , +$$ + +where $F C A$ denotes Fast Channel Attention. $f _ { u p } ^ { 2 }$ denotes a conv layer after bilinear interpolation upsampling, 2 is the scale factor. $f _ { u p } ^ { 4 }$ is similarly defined. $f _ { s i d } ^ { 2 }$ means Shift-Invariant Downsample [46], 2 is also the scale factor. $f _ { s i d } ^ { 4 }$ is similarly defined. Subsequently, the output feature is derived by: + +$$ +M _ { k } ^ { j } ( { \bf F } _ { M _ { k } ^ { j } } ) = { \bf F } _ { M _ { k } ^ { j } } + f ( [ { \bf F } _ { M _ { k } ^ { j } } ^ { 1 } , { \bf F } _ { M _ { k } ^ { j } } ^ { 2 } , { \bf F } _ { M _ { k } ^ { j } } ^ { 4 } ] ) , +$$ + +where $f$ represents the last conv layer, $[ \cdot , \cdot , \cdot ]$ denotes the concatenating operation. The architecture of FCA is shown in Fig. 3 (d). We define the input feature as $\mathbf { F } _ { d }$ , then FCA can be formulated as: + +$$ +F C A ( \mathbf { F } _ { d } ) = \mathbf { F } _ { d } \cdot \big ( 1 + \sigma \big ( f _ { 1 D C } ( G A P ( \mathbf { F } _ { d } ) ) \big ) \big ) , +$$ + +where $\sigma$ represents the Sigmoid activation function, $G A P$ means global average pooling along the spatial wise, $f _ { 1 D C }$ denotes 1-Dimension Convolution. In this work, we set $t = 3$ , $n = 2$ , and $C = 6 4$ . + +# 2.4 Overall Training Objective + +In addition to the aforementioned losses, we employ a perceptual loss function that assesses a solution with respect to perceptually relevant characteristics (e.g., the structural contents and detailed textures): + +$$ +\begin{array} { r } { \mathcal { L } _ { p } = \left| \left| V G G ( \mathbf { I } _ { f d } ) - V G G ( \mathbf { I } _ { r d } ) \right| \right| _ { 2 } ^ { 2 } , \mathbf { I } _ { f d } = D _ { d } ( \mathbf { I } _ { f n } ) , \mathbf { I } _ { r d } = D _ { d } ( \mathbf { I } _ { r n } ) , } \end{array} +$$ + +where $V G G$ denotes the last feature map of VGG16 [56]. Eventually, the training objective is: + +$$ +\mathcal { L } = \mathcal { L } _ { 1 } + \lambda _ { p } \cdot \mathcal { L } _ { p } + \lambda _ { R a } \cdot ( \mathcal { L } _ { D } + \mathcal { L } _ { G } ) , +$$ + +where $\lambda _ { p }$ and $\lambda _ { R a }$ are two hyper-parameters controlling the importance balance. The proposed PNGAN framework is end-to-end trained by minimizing $\mathcal { L }$ . Note that the parameters in $D _ { d }$ and VGG16 are fixed. Each mini-batch training procedure is divided into two steps: (i) Fix $D$ and train $G$ . (ii) Fix $G$ and train $D$ . This pixel-level adversarial training scheme promotes $D$ the ability to distinguish fake noisy images from real noisy images and allows $G$ to learn to create the solutions that are highly similar to real camera noisy images and thus difficult to be classified by $D$ . + +# 3 Experiment + +# 3.1 Experiment Setup + +Datasets. We first use SIDD [47] train set to train $D _ { d }$ . Then we fix $D _ { d }$ to train $G$ on the same set. Subsequently, $G$ uses clean images from DIV2K [57], Flickr2K [58], BSD68 [59], Kodak24 [60], and Urban100 [61] to generate realistic noisy-clean image pairs. We use the generated data and SIDD train set jointly to finetune real denoisers and evaluate them on four real denoising benchmarks: SIDD [47], DND [48], PolyU [40], and Nam [49]. The images in SIDD [47] are collected using five smartphone cameras in 10 static scenes. There are 320 image pairs for training and 1,280 image patch pairs for validation. DND [48] composes 50 noisy-clean image pairs captured by 4 consumer + +![](images/1e86de1e279adf639e620e96667f698ef1bc2bbbb98ac88a6e9e011a83d52de8.jpg) +Figure 4: Domain discrepancy comparisons. We use the metric, Maximum Mean Discrepancy (MMD) to measure the domain discrepancy between synthetic and real noisy datasets, PNGAN generating and real noisy datasets. Under both setting1 and 2, the discrepancy decreases significantly when PNGAN is applied. + +
MethodsSIDD [47]S1DF2K[57,58]
S1 +PNGANS2S2+PNGANRealS1 + PNGANS2S2+PNGAN
RIDNet22.5537.92 (+15.37)36.1338.71 (+2.58)38.6922.5532.10(+9.55)33.9838.14 (+4.16)
MPRNet22.8638.52 (+15.66)36.5239.53 (+3.01)39.4522.8532.82(+9.97)34.1938.61 (+4.42)
MIRNet22.8338.76 (+15.93)36.5539.57 (+3.02)39.5823.0832.34(+9.26)34.2638.72 (+4.46)
+ +Table 2: Training denoisers with different data from scratch. PSNR is reported. $^ { \mathrm { ~ S 1 , 2 = } }$ synthetic setting1,2. + +cameras. 1,000 patches at size $5 1 2 \times 5 1 2$ are cropped from the collected images. PolyU [40] consists of 40 real camera noisy images. Nam [49] is composed of real noisy images of 11 static scenes. + +Implementation Details. We set the hyper-parameter $\lambda _ { p } = 6 \times 1 0 ^ { - 3 }$ , $\lambda _ { R a } = 8 \times 1 0 ^ { - 4 }$ . For synthetic setting1, we set the noise intensity, $\sigma _ { n } = 5 0$ . For synthetic setting2, we directly exploit CycleISP to generate the synthetic noisy input. All the sub-modules $( D _ { d } , G$ , and $D$ ) are trained with the Adam [62] optimizer $\beta _ { 1 } = 0 . 9$ and $\beta _ { 1 } = 0 . 9 9 9 9 )$ for $7 \times 1 0 ^ { 5 }$ iterations. The initial learning rate is set to $2 \times 1 \bar { 0 } ^ { - 4 }$ . The cosine annealing strategy [63] is employed to steadily decrease the learning rate from the initial value to $1 0 ^ { - 6 }$ during the training procedure. Patches at size $1 2 8 \times 1 2 8$ cropped from training images are fed into the models. The batch size is set as 8. The horizontal and vertical flips are performed for data augmentation. All the models are trained on RTX8000 GPUs. In the finetuning phase, the learning rate is set to $1 \times 1 0 ^ { - 6 }$ , other settings remain unchanged. + +# 3.2 Quantitative Results + +Domain Discrepancy Validation. We use the widely applied metric, Maximum Mean Discrepancy (MMD) [64] to measure the domain discrepancy between synthetic and real-world noisy images, PNGAN generating, and real noisy images on four real noisy benchmarks. For DND, we derive a pseudo clean version by denoising the real noisy counterparts with a pre-trained MIRNet [55]. Then we use the pseudo clean version to synthesize noisy images. The results are depicted as a histogram in Fig. 4. For setting1, the domain discrepancy decreases by $74 \%$ , $7 5 \%$ , $44 \%$ , and $43 \%$ on SIDD, DND, PolyU, and Nam when PNGAN is exploited. For setting2, the discrepancy decreases by $64 \%$ , $67 \%$ , $46 \%$ , and $44 \%$ . These results demonstrate that PNGAN can narrow the discrepancy between synthetic and real noisy datasets. Please refer to the supplementary for detailed calculation process. + +Comparison with SOTA Methods. We use the generated noisy-clean image pairs (setting2) to finetune a series of denoisers. We compare our models with SOTA algorithms on four real denoising datasets: SIDD, DND, PolyU, and Nam. The results are reported in Tab. 1. \* denotes denoisers finetuned with image pairs generated by PNGAN. We have the following observations: (i) Our denoisers outperform SOTA methods by a large margin. Specifically, MPRNet\* and MIRNet\* exceed the recent best method MIRNet by 0.34 and $0 . 3 5 \mathrm { d B }$ on SIDD, 0.30 and 0.37 dB on DND. RIDNet\*, MPRNet\*, and MIRNet\* surpass the best performers by 0.36, 1.30, and 1.37 dB on PolyU and 0.01, 1.04, and 1.10 dB on Nam. (ii) Compared with the counterparts that are not finetuned, our models achieve a significant promotion. In particular, RIDNet\* is 0.54, 0.29, 0.68, and $0 . 4 9 \mathrm { d B }$ higher than RIDNet on SIDD, DND, PolyU, and Nam. MPRNet\* achieves 0.35, 0.38, 1.41, and 1.31 dB gain than MPRNet on SIDD, DND, PolyU, and Nam. MIRNet\* is improved by 0.35, 0.37, 1.37, and 1.21 dB. This evidence clearly suggests the high similarity between PNGAN generating and real noisy images. Denoisers adapted with our fake image pairs generalize better across different benchmarks. + +Train from Scratch. For more strong comparisons, we use the fake noisy images generated from clean SIDD train and DF2K (DIV2K+Flicker2K) respectively to train denoisers from scratch. The PSNR results evaluated on SIDD test are listed in Tab. 2. All models are trained with the same experiment schedule except the training data. It can be observed: (i) On SIDD train, when PNGAN is applied to setting1, denoisers are promoted by $\sim 1 5 . 6 5$ dB and only $\sim 0 . 8 4$ dB lower than those + +![](images/f209d8c64470dae44b21785b843771996ba06d36a206d10d5ce3e25053032886.jpg) + +Figure 5: Visual comparisons of noisy images on SIDD, DND, PolyU, and Nam. Please zoom in. + +
MethodsPNGAN ComponentGenerator Architecture
Baseline1+Dd+D+LpBaseline2+ Multi-scale+ SID+FCA
RIDNet14.5435.37 (+20.83)37.49(+2.12) 37.92(+0.43)35.6237.01(+1.39)37.23 (+0.22)37.92 (+0.69)
MPRNet14.2536.26 (+22.01)38.27(+2.01) 38.52(+0.25)36.2837.47(+1.19)37.86 (+0.39)38.52 (+0.66)
MIRNet13.5736.15 (+22.58)38.28(+2.13) 38.76(+0.48)36.3937.66(+1.27)37.89 (+0.23)38.76 (+0.87)
+ +Table 3: Ablation study of PNGAN component and the noise generator architecture. PSNR is reported. + +trained with real data (SIDD train set). While applying PNGAN to setting2 (CycleISP), denoisers are improved by $\sim 2 . 8 7$ dB. Surprisingly, in this case, denoisers achieve almost the same performance as those trained with real data. The relative error is $0 . 2 \%$ . (ii) To validate the generality of PNGAN, we also adopt synthetic DF2K noisy-clean image pairs to train denoisers. As shown in the right part of Tab. 2, when PNGAN is applied to setting1, denoisers are promoted by $\sim 9 . 5 9$ dB. While applying PNGAN to setting2, denoisers are improved by $\sim 4 . 3 5 ~ \mathrm { d B }$ and only $\sim 0 . 7 5$ dB lower than those trained with SIDD real train set. These results convincingly demonstrate: (i) The generated noise is highly similar to the real noise especially when PNGAN is applied to synthetic setting2. (ii) PNGAN can significantly narrow the domain discrepancy between synthetic and real-world noise. + +# 3.3 Qualitative Results + +Visual Examinations of Noisy Images. To intuitively evaluate the generated noisy images, we provide visual comparisons of noisy images on the four real noisy datasets, as shown in Fig. 5. Note that the clean image of DND is pseudo, denoised from its noisy version by a MIRNet. The left part depicts noisy images from SIDD, DND, PolyU, and Nam (top to down). The right part exhibits the patches cropped by the yellow bboxes, from left to right: clean, synthetic setting1, setting2 (CycleISP), PNGAN generating, and real noisy images. As can be seen from the zoom-in patches: (i) Noisy images synthesized by setting1 is signal-independent. The distribution and intensity remain unchanged across diverse scenes, indicating the characteristics of AWGN fundamentally differ from those of the real noise. (ii) Noisy images generated by PNGAN are closer to the real noise than those synthesized by setting2 visually. Noise synthesized by setting2 shows randomness that is obviously inconsistent with the real noise in terms of intensity and distribution. While PNGAN can model spatio-chromatically correlated and non-Gaussian noise more accurately. (iii) Even if passing through the same camera pipeline, different shooting conditions lead to the diversity of real noise. It’s unreasonable for the noise synthesized by CycleISP to show nearly uniform fitting to different input images. In contrast, PNGAN can adaptively simulate more sophisticated and photo-realistic models. This adaptability allows PNGAN to show robust performance across different real noisy datasets. + +Visual Comparison of Denoised Images. We compare the visual results of denoisers before and after being finetuned (denoted with \*) with the generated data in Fig. 4. We observe that models finetuned with the generated data are more effective in real noise removal. Furthermore, they are capable of preserving the structural content, textural details, and spatial smoothness of the homogeneous regions. In contrast, original models either yield over-smooth images sacrificing fine textural details and structural content or introduce redundant blotchy texture and chroma artifacts. + +# 3.4 Ablation Study + +Break-down Ablations. We perform break-down ablations to evaluate the effects of PNGAN components and SMNet architecture. We select setting1 to synthesize the noisy input from SIDD train set. Then we use the generated data only to train the denoisers from scratch and evaluate them on SIDD test. The PSNR results are reported in Tab. 3. (i) Firstly, $G$ is set as SMNet to validate the effects of PNGAN components. We start from Baseline1, no discriminator is used and the $\mathcal { L } _ { 1 }$ loss is + +![](images/f6d75ee03ec6067d2f9aa52e21200e934dc47ff620fce2476744f1cbb443fb7c.jpg) +Figure 6: Visual results of denoisers before and after being finetuned with fake data. Please zoom in. + +![](images/fc34288e794ee45ce52754399b02b426e3ae8073c3b515db897f489f87669e4e.jpg) +Figure 7: Parameter analysis of $\lambda _ { p }$ , $\lambda _ { R a }$ , and $\sigma _ { n }$ + +
qSIDD [47]PolyU[40]Nam [49]Total
PSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIM
None38.710.91438.860.96239.200.97338.760.929
039.320.95738.010.94938.340.95838.920.955
20%39.290.95738.450.95938.870.97039.030.958
40%39.280.95639.020.96639.260.97339.200.959
60%39.260.95639.540.97139.690.97539.350.961
80%39.230.95539.560.97239.720.97639.330.960
100%39.210.95539.570.97239.730.97639.330.960
+ +Table 4: Analysis of the finetuning data ratio $q$ + +directly performed between $\mathbf { I } _ { f n }$ and ${ \mathbf I } _ { r n }$ in Eq. (2). Denoisers trained with the generated data collapse dramatically, implying the naive strategy mentioned in Sec. 2.2 is unfeasible. When $D _ { d }$ is applied, the denoisers are promoted by 21.81 dB on average. In addition, the PSNR and SSIM between the denoised counterparts of generated and real noisy images are $3 9 . 1 4 ~ \mathrm { d B }$ and 0.928 on average respectively. This evidence indicates that $D _ { d }$ successfully conducts the image domain alignment as mentioned in Sec. 2.2. Subsequently, we use an image-level $D$ with stride conv layers to classify whether the whole generated image is real. Nonetheless, the performance of denoisers remains almost unchanged. After deploying $D$ , the models are improved by ${ \sim } 2 . 0 9$ dB, suggesting that the pixel-level noise model is more in line with real noise scenes and benefits generating more realistic noisy images. When ${ \mathcal { L } } _ { p }$ is used, the denoisers gain a slight improvement by about 0.39 dB, indicating ${ \mathcal { L } } _ { p }$ facilitates yielding more vivid results. (ii) Secondly, we only change the architecture of $G$ to study the effects of its components. We start from Baseline2 that doesn’t exploit multi-scale feature fusion, SID, and FCA. When we add two different scale branches and use bilinear interpolation to downsample and upsample, denoisers trained with the generated images are promoted by about $1 . 2 8 \ : \mathrm { d B }$ . After applying SID and FCA, the denoisers further gain 0.28 and $0 . 7 4 \mathrm { d B }$ improvement on average. These results convincingly demonstrate the superiority of the proposed SMNet in real-world noise fitting. + +Parameter Analysis. We adopt RIDNet as the baseline to perform parameter analysis. We firstly validate the effects of $\lambda _ { p }$ , $\lambda _ { R a }$ in Eq. (13), and the noise intensity of setting1, i.e., $\sigma _ { n }$ . We change the parameters, train $G$ , use $G$ to generate realistic noisy images from clean images of SIDD train set, train RIDNet with the generated data, and evaluate its performance on SIDD test set. When analyzing one parameter, we fix the others at their optimal values. The PSNR results are shown in Fig. 7. The optimal setting is $\lambda _ { p } = 6 \times 1 0 ^ { - 3 }$ , $\lambda _ { R a } = 8 \times 1 0 ^ { - 4 }$ , and $\sigma _ { n } = 4 0$ or 50. Secondly, we evaluate the effect of the ratio of finetuning data. We denote the ratio of extended training data (setting2) to SIDD real noisy training data as $q$ . We change the value of $q$ , finetuned the original RIDNet, and test on three real denoising datasets: SIDD, PolyU, and Nam. The results are listed in Tab. 4. When $q = 0$ , all the finetuning data comes from SIDD train set, RIDNet achieves the best performance on SIDD. However, its performance on PolyU and Nam degrades drastically due to the domain discrepancy between different real noisy datasets. We gradually increase the value of $q$ to study its effects. The average performance on the three datasets yields the maximum when $q = 6 0 \%$ . + +# 4 Conclusion + +Too much research focuses on designing a CNN architecture for real noise removal. In contrast, this work investigates how to generate more realistic noisy images so as to boom the denoising performance. We first formulate a noise model that treats each noisy pixel as a random variable. Then we propose a novel framework PNGAN to perform the image and noise domain alignment. 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Smola, “A kernel two-sample test,” in JMLR, 2012. 7 \ No newline at end of file diff --git a/parse/train/Wua2zjxJdYo/Wua2zjxJdYo_content_list.json b/parse/train/Wua2zjxJdYo/Wua2zjxJdYo_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b0344cff7b1267a5ae18322310160b6049ea8422 --- /dev/null +++ b/parse/train/Wua2zjxJdYo/Wua2zjxJdYo_content_list.json @@ -0,0 +1,1153 @@ +[ + { + "type": "text", + "text": "Learning to Generate Realistic Noisy Images via Pixel-level Noise-aware Adversarial Training ", + "text_level": 1, + "bbox": [ + 209, + 122, + 790, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yuanhao Cai 1,2, Xiaowan $\\mathbf { H } \\mathbf { u } ^ { \\mathrm { ~ 1 , 2 ~ } }$ , Haoqian Wang $^ { 1 , 2 , * }$ , Yulun Zhang 3, Hanspeter Pfister 4, Donglai Wei 5 ", + "bbox": [ + 302, + 224, + 702, + 255 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Shenzhen International Graduate School, Tsinghua University, 2 Shenzhen Institute of Future Media Technology, 3 ETH Zürich, 4 Harvard University, 5 Boston College ", + "bbox": [ + 289, + 256, + 707, + 297 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 319, + 535, + 335 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Existing deep learning real denoising methods require a large amount of noisyclean image pairs for supervision. Nonetheless, capturing a real noisy-clean dataset is an unacceptable expensive and cumbersome procedure. To alleviate this problem, this work investigates how to generate realistic noisy images. Firstly, we formulate a simple yet reasonable noise model that treats each real noisy pixel as a random variable. This model splits the noisy image generation problem into two subproblems: image domain alignment and noise domain alignment. Subsequently, we propose a novel framework, namely Pixel-level Noise-aware Generative Adversarial Network (PNGAN). PNGAN employs a pre-trained real denoiser to map the fake and real noisy images into a nearly noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a pixel-level adversarial training to conduct noise domain alignment. Additionally, for better noise fitting, we present an efficient architecture Simple Multi-scale Network (SMNet) as the generator. Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers trained with the generated noisy images achieve state-of-the-art (SOTA) results on four real denoising benchmarks. ", + "bbox": [ + 233, + 340, + 766, + 575 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 599, + 312, + 617 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Image denoising is an important yet challenging problem in low-level vision. It aims to restore a clean image from its noisy counterpart. Traditional approaches concentrate on designing a rational maximum a posteriori (MAP) model, containing regularization and fidelity terms, from a Bayesian perspective [1]. Some image priors like low-rankness [2, 3, 4], sparsity [5], and non-local similarity [6, 7] are exploited to customize a better rational MAP model. However, these hand-crafted methods are inferior in representing capacity. With the development of deep learning, image denoising has witnessed significant progress. Deep convolutional neural network (CNN) applies a powerful learning model to eliminate noise and has achieved promising performance [8, 9, 10, 11, 12, 13, 14]. These deep CNN denoisers rely on a large-scale dataset of real-world noisy-clean image pairs. Nonetheless, collecting even small datasets is extremely tedious and labor-intensive. The process of acquiring real-world noisy-clean image pairs is to take hundreds of noisy images of the same scene and average them to get the clean image. To get more image pairs, researchers try to synthesize noisy images. ", + "bbox": [ + 174, + 621, + 825, + 786 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In particular, there are two common settings for synthesizing noisy images. As shown in Fig. 1 (a1), setting1 directly adds the additive white Gaussian noise (AWGN) with the clean RGB image. For a long time, single image denoising [15, 16, 17, 18, 19, 10] is performed with setting1. Nevertheless, fundamentally different from AWGN, real camera noise is generally more sophisticated and signaldependent[20, 21]. The noise produced by photon sensing is further affected by the in-camera signal processing (ISP) pipeline (e.g., Gama correction, compression, and demosaicing). Models trained with setting1 are easily over-fitted to AWGN and fail in real noise removal. Setting2 is based on ISP-modeling CNN [22] and Poisson-Gaussian [21, 23] noise model that modeling photon sensing with Poisson and remaining stationary disturbances with Gaussian has been adopted in RAW denoising. As shown in Fig. 1 (a2), setting2 adds a Poisson-Gaussian noise with the clean RAW image and then passes the result through a pre-trained RAW2RGB CNN to obtain the RGB noisy counterpart. Notably, when the clean RAW image is unavailable, a pre-trained RGB2RAW CNN is utilized to transform the clean RGB image to its RAW counterpart [22]. However, setting2 has the following drawbacks: (i) The noise is assumed to obey a hand-crafted probability distribution. However, because of the randomness and complexity of real camera noise, it’s difficult to customize a hand-crafted probability distribution to model all the characteristics of real noise. (ii) The ISP pipeline is very sophisticated and hard to be completely modeled. The RAW2RGB branch only learns the mapping from the clean RAW domain to the clean RGB space. However, the mapping from the Poisson-Gaussian noisy RAW domain to the real noisy RGB space can not be ensured. (iii) The ISP pipelines of different devices vary significantly, which results in the poor generality and robustness of ISP modeling CNNs. Thus, whether noisy images are synthesized with setting1 or 2, there still remains a discrepancy between synthetic and real noisy datasets. We notice that GAN utilizes the internal information of the input image and external information from other images when modeling image priors. Hence, we propose to use GAN to adaptively learn the real noise distribution. ", + "bbox": [ + 174, + 794, + 825, + 876 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 340 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GAN is firstly introduced in [24] and has been proven successful in image synthesis [25, 26, 27] and translation [26, 27]. Subsequently, GAN is applied to image restoration and enhancement, e.g., super resolution [28, 29, 30], style transfer [27, 31], enlighten [32, 33], deraining [34], dehazing [35], image inpainting [36, 37], image editing [38, 39], and mobile photo enhancement [40, 41]. Although GAN is widely applied in low-level vision tasks, few works are dedicated to investigating the realistic noise generation problem [42]. Chen et al. [43] propose a simple GAN that takes Gaussian noise as input to generate noisy patches. However, as in general, this GAN is image-level, i.e., it treats images as samples and attempts to approximate the probability distribution of real-world noisy images. This image-level GAN neglects that each pixel of a real noisy image is a random variable and the real noise is spatio-chromatically correlated, thus results in coarse learning of the real noise distribution. ", + "bbox": [ + 174, + 345, + 825, + 484 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To alleviate the above problems, this work focuses on learning how to generate realistic noisy images so as to augment the training data for real denoisers. To begin with, we propose a simple yet reasonable noise model that treats each pixel of a real noisy image as a random variable. This noise model splits the noise generation problem into two sub-problems: image domain alignment and noise domain alignment. Subsequently, to tackle these two sub-problems, we propose a novel Pixel-level Noise-aware Generative Adversarial Network (PNGAN). During the training procedure of PNGAN, we employ a pre-trained real denoiser to map the generated and real noisy images into a nearly noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a pixel-level adversarial training that encourages the generator to adaptively simulate the real noise distribution so as to conduct the noise domain alignment. In addition, for better real noise fitting, we present a lightweight yet efficient CNN architecture, Simple Multi-scale Network (SMNet) as the generator. SMNet repeatedly aggregates multi-scale features to capture rich auto-correlation, which provides more sufficient spatial representations for noise simulating. Different from general image-level GAN, our discriminator is pixel-level. The discriminator outputs a score map. Each position on the score map indicates how realistic the corresponding noisy pixel is. With this pixellevel noise-aware adversarial training, the generator is encouraged to create solutions that are highly similar to real noisy images and thus difficult to be distinguished. ", + "bbox": [ + 174, + 491, + 825, + 726 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In conclusion, our contributions can be summarized into four points: ", + "bbox": [ + 173, + 732, + 622, + 746 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(1) We formulate a simple yet reasonable noise model. This model treats each noisy pixel as a random variable and then splits the noisy image generation into two parts: image and noise domain alignment. ", + "bbox": [ + 173, + 752, + 825, + 781 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(2) We propose a novel framework, PNGAN. It establishes an effective pixel-level adversarial training to encourage the generator to favor solutions that reside on the manifold of real noisy images. ", + "bbox": [ + 173, + 786, + 823, + 815 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(3) We customize an efficient CNN architecture, SMNet learning rich multi-scale auto-correlation for better noise fitting. SMNet serves as the generator in PNGAN costing only 0.8M parameters. ", + "bbox": [ + 173, + 820, + 821, + 849 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(4) Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers finetuned with the generated noisy images achieve SOTA results on four real denoising benchmarks. ", + "bbox": [ + 176, + 856, + 825, + 898 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/09f874f387c331b9a197a98ffe2b51b7acdc68dca22e5b8d83d8a65e04c2d5c0.jpg", + "image_caption": [ + "Figure 1: The pipeline of using PNGAN to perform data augmentation. It is divided into: (a) synthesizing phase, (b) training phase, and (c) finetuning phase. Please refer to the text (Sec. 2) for more detailed descriptions. " + ], + "image_footnote": [], + "bbox": [ + 176, + 88, + 815, + 277 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Proposed Method ", + "text_level": 1, + "bbox": [ + 174, + 329, + 352, + 345 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As shown in Fig. 1, the pipeline of using PNGAN to perform data augmentation consists of three phases. (a) is the synthesizing phase. (a1) and (a2) are two common synthetic settings. In this phase, we produce the synthetic noisy image from its clean RGB or RAW counterpart. (b) is the training phase of PNGAN. The generator $G$ adopts the synthetic image as input. Which synthetic setting is selected is controlled by the switch. By using a pre-trained real denoiser $D _ { d }$ , PNGAN establishes a pixel-level noise-aware adversarial training between the generator $G$ and discriminator $D$ so as to simultaneously conduct image and noise domain alignment. $D _ { d }$ is set as RIDNet [44] in this work. (c) is the finetuning phase. Firstly, in (c1), the generator creates extended fake noisy-clean image pairs. Secondly, in (c2), the fake and real data are jointly utilized to finetune a series of real denoisers. ", + "bbox": [ + 173, + 356, + 826, + 482 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Pixel-level Noise Modelling ", + "text_level": 1, + "bbox": [ + 174, + 496, + 400, + 511 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Real camera noise is sophisticated and signal-dependent. Specifically, in the real camera system, the RAW noise produced by photon sensing comes from multiple sources (e.g., short noise, thermal noise, and dark current noise) and is further affected by the ISP pipeline. Besides, illumination changes and camera movement inevitably lead to spatial pixel misalignment and color or brightness deviation. Hence, hand-designed noise models based on mathematical assumptions are difficult to accurately and completely describe the properties of real noise. Different from previous methods, we don’t base our noise model on any mathematical assumptions. Instead, we use CNN to implicitly simulate the characteristics of real noise. We begin by noting that when taking multiple noisy images of the same scene, the noise intensity of the same pixel varies a lot. Simultaneously, affected by the ISP pipeline, the real noise is spatio-chromatically correlated. Thus, the correlation between different pixels of the same real noisy image should be considered. In light of these facts, we treat each pixel of a real noisy image as a random variable and formulate a simple yet reasonable noise model: ", + "bbox": [ + 173, + 522, + 826, + 689 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dbe88430ac2f919c6cf4acf9a7d845c445835202c472919c037d2a58226d4558.jpg", + "text": "$$\n\\mathbf { I } _ { r n } [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] + \\mathbf { N } [ i ] , \\quad D _ { d } ( \\mathbf { I } _ { r n } ) [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] , \\quad 1 \\leq i \\leq H \\times W ,\n$$", + "text_format": "latex", + "bbox": [ + 263, + 694, + 735, + 713 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\hat { \\mathbf { I } } _ { c l e a n } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ is the predicted clean counterpart of ${ \\mathbf I } _ { r n }$ , it’s denoised by $D _ { d }$ . Each $\\mathbf { N } [ i ]$ is a random noise variable with unknown probability distribution. Therefore, each $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ can also be viewed as a distribution-unknown random variable. Now we aim to design a framework to generate a fake noisy image $\\mathbf { I } _ { f n } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ such that the probability distribution of $\\mathbf { I } _ { f n } [ i ]$ and $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ is as close as possible. Please note that the mapping learned by $D _ { d }$ is precisely from ${ \\mathbf I } _ { r n }$ to $\\hat { \\mathbf { I } } _ { c l e a n }$ . If the constant in Eq. (1) is set as the clean image $\\mathbf { I } _ { c l e a n } \\in \\mathbb { R } ^ { \\bar { H } \\times W \\times 3 }$ , the subsequent domain alignment will introduce unnecessary errors and eventually lead to inaccurate results. ", + "bbox": [ + 173, + 720, + 825, + 823 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Pixel-level Noise-aware Adversarial Training ", + "text_level": 1, + "bbox": [ + 174, + 833, + 526, + 848 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our goal is to generate realistic noisy images. According to the noise model in Eq. (1), we split this problem into two sub-problems: (i) Image domain alignment aims to align $\\hat { \\mathbf { I } } _ { c l e a n } [ i ]$ . (ii) Noise domain alignment targets at modeling the distribution of $\\mathbf { N } [ i ]$ . To handle the sub-problems, PNGAN establishes a novel pixel-level noise-aware adversarial training between $G$ and $D$ in Fig. 1 (b). ", + "bbox": [ + 174, + 853, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Image Domain Alignment. A very naive strategy to construct both image and noise domain alignment is to directly minimize the distance of $\\mathbf { I } _ { f n }$ and ${ \\mathbf I } _ { r n }$ . However, due to the intrinsic randomness, complexity, and irregularity of real noise, directly deploying $\\mathcal { L } _ { 1 }$ loss between $\\mathbf { I } _ { f n }$ and ${ \\mathbf I } _ { r n }$ is unreasonable and drastically damages the quality of $\\mathbf { I } _ { f n }$ . Besides, as analyzed in Sec. 2.1, each pixel of ${ \\mathbf I } _ { r n }$ is a distribution-unknown random variable. This indicates that such a naive strategy challenges the training and may easily cause the non-convergence issue. Therefore, the noise interference should be eliminated while constructing the image domain alignment. To this end, we feed $\\mathbf { I } _ { f n }$ and ${ \\mathbf I } _ { r n }$ into $D _ { d }$ to obtain their denoised versions and then perform $\\mathcal { L } _ { 1 }$ loss between $\\mathbf { I } _ { f n }$ and ${ \\mathbf I } _ { r n }$ : ", + "bbox": [ + 173, + 90, + 826, + 203 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0e5895af820834ff08d72a109b9f076ed004531068649b5c4a7fc8efd987e550.jpg", + "text": "$$\n\\mathcal { L } _ { 1 } = \\sum _ { i = 1 } ^ { H \\times W } \\left| \\left| D _ { d } ( \\mathbf { I } _ { f n } ) [ i ] - D _ { d } ( \\mathbf { I } _ { r n } ) [ i ] \\right| \\right| _ { 1 } = \\sum _ { i = 1 } ^ { H \\times W } \\left| \\left| D _ { d } ( \\mathbf { I } _ { f n } ) [ i ] - \\hat { \\mathbf { I } } _ { c l e a n } [ i ] \\right| \\right| _ { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 205, + 756, + 250 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "By using $D _ { d }$ , we can transfer ${ \\mathbf I } _ { r n }$ and $\\mathbf { I } _ { f n }$ into a nearly noise-free solution space. The value of $\\hat { \\mathbf { I } } _ { c l e a n }$ is relatively stable. Therefore, minimizing $\\mathcal { L } _ { 1 }$ can encourage $G$ to favor solutions that after being denoised by $D _ { d }$ converge to $\\hat { \\mathbf { I } } _ { c l e a n }$ . In this way, the image domain alignment is constructed. ", + "bbox": [ + 173, + 251, + 825, + 299 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Noise Domain Alignment. Becasue of the complexity and variability of real noise, it’s hard to completely seperate $\\mathbf { N } [ i ]$ from $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ in Eq (1). Fortunately, we note that on the basis of constructing the image domain alignment of $\\hat { \\mathbf { I } } _ { c l e a n } [ i ]$ , the noise domain alignment of $\\mathbf { N } [ i ]$ is equivalent to the distribution estimation of $\\mathbf { I } _ { \\mathit { r n } } [ \\bar { i } ]$ . Additionally, as the real noise is signaldependent, the alignment between $\\mathbf { I } _ { f n } [ i ]$ and $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ is more beneficial to capture the correlation between noise and scene. We denote the ", + "bbox": [ + 173, + 318, + 485, + 472 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/51ccb4cb837145780a480b4df93e5a343951a841cb8ba4022dc8e241605f5765.jpg", + "image_caption": [ + "Figure 2: Architecture of discriminator. " + ], + "image_footnote": [], + "bbox": [ + 500, + 321, + 821, + 439 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "distribution of $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ as $P _ { d a t a } ( x _ { i } )$ , some real noisy pixel samples of $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ as $\\{ x _ { i } ^ { 1 } , x _ { i } ^ { 2 } , . . . , x _ { i } ^ { m } \\}$ such that $x _ { i } ^ { k } \\sim P _ { d a t a } ( x _ { i } )$ , and the distribution of $\\mathbf { I } _ { f n } [ i ]$ as $P _ { G } ( x _ { i } ; \\theta _ { G } )$ . Here $\\theta _ { G }$ is the parameter of $G$ . Then we formulate the noise domain aligment into a maximum likelihood estimation problem: ", + "bbox": [ + 174, + 472, + 828, + 516 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/35fdbb243a7e39762c9789b1ce3d51a288f2e7cadb80eb55ef7187e2ca6ec2ab.jpg", + "text": "$$\n\\theta _ { G } ^ { * } = a r g \\operatorname * { m a x } _ { \\theta _ { G } } \\sum _ { i = 1 } ^ { H \\times W } \\sum _ { k = 1 } ^ { m } \\log P _ { G } ( x _ { i } ^ { k } ; \\theta _ { G } ) = a r g m a x \\operatorname { \\mathbb { E } } _ { i } \\left[ \\mathbb { E } _ { x _ { i } ^ { k } } \\left[ \\log P _ { G } ( x _ { i } ^ { k } ; \\theta _ { G } ) \\right] \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 217, + 518, + 779, + 563 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbb { E }$ means taking the average value. To approach this upper bound as close as possible, we present $D$ and establish the pixel-level adversarial traininig between $G$ and $D$ . The architecture of $D$ is shown in Fig. 2. $D$ consists of 4 convolutional $( c o n v )$ layers and utilizes LeakyReLU activation $( \\alpha = 0 . 2$ ). General discriminator treats a image as a sample and outputs a score indicating how realistic the image is. Instead, $D$ is a pixel-level classifier. $D$ adopts the fake and real noisy images as input in a mini-batch and outputs a score map $\\mathbf { P } \\in \\mathbb { R } ^ { H \\times W }$ for each image. Specifically, the information of $\\mathbf { P } [ i ] \\in [ 0 , 1 ]$ is the probability value indicating how realistic $P _ { G } ( x _ { i } ; \\theta _ { G } )$ is. $G$ aims to generate more realistic $\\mathbf { I } _ { f n } [ i ]$ to fool $D$ while $D$ targets at distinguishing $\\mathbf { I } _ { f n } [ i ]$ from $\\mathbf { I } _ { \\mathit { r n } } [ i ]$ . According to Eq .(3), we formulate the adversarial training between $G$ and $D$ as a min-max problem: ", + "bbox": [ + 173, + 564, + 826, + 690 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/171bcbadedc90e33a056334cd9e5b70c31c8bf07c43fc59732e24181d15526b3.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { G } } \\operatorname* { m } _ { \\theta _ { D } } \\mathbb { E } _ { i } \\left[ \\mathbb { E } _ { \\mathbf { I } _ { r n } } \\left[ \\mathrm { l o g } ( D ( \\mathbf { I } _ { r n } ; \\theta _ { D } ) [ i ] ) \\right] \\right] + \\mathbb { E } _ { i } \\left[ \\mathbb { E } _ { \\mathbf { I } _ { f n } } \\left[ \\mathrm { l o g } ( 1 - D ( \\mathbf { I } _ { f n } ; \\theta _ { D } ) [ i ] ) \\right] \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 230, + 691, + 766, + 717 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbb { E } _ { \\mathbf { I } _ { r n } }$ and $\\mathbb { E } _ { \\mathbf { I } _ { f n } }$ respectively represent the operation of taking the average for all fake and real data in the mini-batch. As analyzed in [45], to make GANs analogous to divergence minimization and produce sensible predictions based on the a priori knowledge that half of the samples in the mini-batch are fake, we utilize the recently proposed relativistic discriminator [45] as follow: ", + "bbox": [ + 173, + 719, + 825, + 775 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/683671f20ee5af98d05486537c7b4ac23addf37f03bc6724e71e1752ff4567a5.jpg", + "text": "$$\n\\begin{array} { r l } & { D ( \\mathbf { I } _ { r n } ; \\theta _ { D } ) = \\sigma ( C _ { D } ( \\mathbf { I } _ { r n } ) ) , \\quad D _ { R a } ( \\mathbf { I } _ { r n } , \\mathbf { I } _ { f n } ) = \\sigma ( C _ { D } ( \\mathbf { I } _ { r n } ) - \\mathbb { E } _ { \\mathbf { I } _ { f n } } ( C _ { D } ( \\mathbf { I } _ { f n } ) ) ) , } \\\\ & { D ( \\mathbf { I } _ { f n } ; \\theta _ { D } ) = \\sigma ( C _ { D } ( \\mathbf { I } _ { f n } ) ) , \\quad D _ { R a } ( \\mathbf { I } _ { f n } , \\mathbf { I } _ { r n } ) = \\sigma ( C _ { D } ( \\mathbf { I } _ { f n } ) - \\mathbb { E } _ { \\mathbf { I } _ { r n } } ( C _ { D } ( \\mathbf { I } _ { r n } ) ) ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 777, + 764, + 816 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D _ { R a }$ denotes the relativistic discriminator, $\\sigma$ means the Sigmoid activation, and $C _ { D }$ represents the non-transformed discriminator output. $D _ { R a }$ estimates the probability that real data is more realistic than fake data and also directs the generator to create a fake image that is more realistic than real images. The loss functions of $D$ and $G$ are then defined in a symmetrical form: ", + "bbox": [ + 173, + 818, + 823, + 873 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/12e12f88b6b92be2d72b03d36347263c07545712dd34aa3a95527c11e37a138f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { D } = - \\mathbb { E } _ { i } [ \\mathbb { E } _ { { \\mathbf { I } } _ { r n } } \\big [ \\mathrm { l o g } ( D _ { R a } ( { \\mathbf { I } } _ { r n } , { \\mathbf { I } } _ { f n } ) [ i ] ) \\big ] + \\mathbb { E } _ { { \\mathbf { I } } _ { f n } } \\big [ \\mathrm { l o g } ( 1 - D _ { R a } ( { \\mathbf { I } } _ { f n } , { \\mathbf { I } } _ { r n } ) [ i ] ) \\big ] \\big ] , } \\\\ & { \\mathcal { L } _ { G } = - \\mathbb { E } _ { i } [ \\mathbb { E } _ { { \\mathbf { I } } _ { r n } } \\big [ \\mathrm { l o g } ( 1 - D _ { R a } ( { \\mathbf { I } } _ { r n } , { \\mathbf { I } } _ { f n } ) [ i ] ) \\big ] + \\mathbb { E } _ { { \\mathbf { I } } _ { f n } } \\big [ \\mathrm { l o g } ( D _ { R a } ( { \\mathbf { I } } _ { f n } , { \\mathbf { I } } _ { r n } ) [ i ] ) \\big ] ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 876, + 761, + 919 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/24b711af4a5f63491e29b6e034035d38283bbbffaf19904c2afc7a1fe897c1b6.jpg", + "image_caption": [ + "Figure 3: Details of the generator. (a) is the architecture of SMNet. (b) depicts the components of SRG. (c) shows the details of MAB. MAB is equipped with FCA, which is illustrated in (d). " + ], + "image_footnote": [], + "bbox": [ + 178, + 89, + 815, + 224 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "During the training procedure, we fix $D$ to train $G$ and fix $G$ to train $D$ iteratively. Minimizing $\\mathcal { L } _ { G }$ and $\\mathcal { L } _ { D }$ alternately allows us to train a generative model $G$ with the goal of fooling the pixel-level discriminator $D$ that is trained to distinguish fake noisy images from real noisy images. This pixellevel noise-aware adversarial training scheme encourages $G$ to favor perceptually natural solutions that reside on the manifold of real noisy images so as to construct the noise domain alignment. ", + "bbox": [ + 174, + 265, + 825, + 334 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 Noisy Image Generating ", + "text_level": 1, + "bbox": [ + 174, + 343, + 383, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Sec. 2.1, we denote the probability distribution of $\\mathbf { I } _ { f n } [ i ]$ as $P _ { G } ( x _ { i } ; \\theta _ { G } )$ . Now we customize a light-weight yet efficient CNN architecture, SMNet as $G$ to generate $P _ { G } ( x _ { i } ; \\theta _ { G } )$ . In this section, we firstly introduce the input setting of $G$ and subsequently detail the architecture of SMNet. ", + "bbox": [ + 174, + 364, + 825, + 406 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input Setting. We aim to generate a realistic noisy image from its clean counterpart. A naive setting is to directly adopt the clean image as the input to generate the noisy image. However, this naive setting is not in line with the fact. When we repeatedly feed the same clean image to a pre-trained $G$ , $G$ outputs completely the same noisy images. In contrast, when taking multiple pictures in the real world, the real noisy images vary a lot in the intensity of each pixel. This is caused by many factors (e.g., photon sensing noise, ISP pipelines, and illumination conditions). Hence, the naive input setting containing no distribution is unreasonable. We review that the general GANs sample from an initial random distribution (usually Gaussian) to generate a fake image. Hence, the input of $G$ should contain a random distribution so as to generate multiple noisy images of the same scene. We note that the two common synthetic settings meet this condition. Therefore, we utilize the two common settings to produce the synthetic image and then adopt the synthetic image as the input of $G$ Subsequently, we propose a light-weight yet efficient architecture, SMNet for better real noise fitting. ", + "bbox": [ + 174, + 412, + 825, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "SMNet Architecture. The architecture of SMNet is shown in Fig. 3 (a). SMNet involves $t$ Simple Residual Groups (SRG) and each SRG contains $n$ Multi-scale Attention Blocks (MAB). The synthetic input $\\mathbf { I } _ { s y n } \\in \\dot { \\mathbb { R } } ^ { H \\times W \\times 3 }$ continuously undergoes a conv layer $f _ { 1 }$ , $t$ SRGs, and a conv layer $f _ { 2 }$ , then adds with a long identity mapping for efficient residual learning to eventually generate the fake noisy counterpart $\\mathbf { I } _ { f n } \\in \\mathbb { R } ^ { H \\times W \\times \\dot { 3 } }$ . This process can be formulated as: ", + "bbox": [ + 173, + 585, + 825, + 655 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/069647ce1bbde87b2623ccb494d7fbd812afb4db119d1b0aed7d9d4502fa25ec.jpg", + "text": "$$\n\\mathbf { I } _ { f n } = \\mathbf { I } _ { s y n } + f _ { 2 } ( S _ { t } ( \\mathbf { F } _ { S _ { t } } ) ) , ~ \\mathbf { F } _ { S _ { j + 1 } } = S _ { j } ( \\mathbf { F } _ { S _ { j } } ) , ~ \\mathbf { F } _ { S _ { 1 } } = f _ { 1 } ( \\mathbf { I } _ { f n } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 659, + 715, + 678 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $S _ { j }$ denotes the $j _ { t h }$ SRG, $1 \\leq j \\leq t - 1$ . The components of SRG are depicted in Fig. 3 (b). We define the input feature of the $j _ { t h }$ SRG as $\\mathbf { F } _ { S _ { j } } \\in \\mathbb { R } ^ { H \\times W \\times C }$ and its channel as $C$ . $\\mathbf { F } _ { S _ { j } }$ continuously undergoes a conv layer, $n$ MABs, and a conv layer to add with an identity mapping: ", + "bbox": [ + 174, + 680, + 825, + 724 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c5b7abf8dbad70d78cac6802ec6026212e4617d92566dc729bd00da13bf4499b.jpg", + "text": "$$\n\\mathbf { F } _ { S _ { j + 1 } } = \\mathbf { F } _ { S _ { j } } + M _ { n } ^ { j } ( \\mathbf { F } _ { M _ { n } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { k + 1 } ^ { j } } = M _ { k } ^ { j } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { 1 } ^ { j } } = \\mathbf { F } _ { S _ { j } } ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 728, + 717, + 752 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $M _ { k } ^ { j }$ denotes the $k _ { t h }$ MAB of the $j _ { t h }$ SRG, $1 \\leq k \\leq n - 1$ . MAB is the basic building block and the most significant component of SMNet. The details of MAB are depicted in Fig. 3 (c). We customize MAB with the following motivations: (i) Multi-scale feature fusion can increase the receptive field and multi-resolution contextual information can cover rich auto-correlation, which provides more sufficient spatial representations for noise fitting. (ii) The noise level decreases as the scale increases and nonlinear sampling operations can increase the richness of the mapping in the potential space of real noise. Therefore, we exploit parallel multi-resolution branch aggregation from top to bottom and bottom to top to facilitate the learning of complex real noise. (iii) Specifically, during the feature downsampling, general downsample operation damages the image information, resulting in pixel discontinuity and jagged artifact. To alleviate these issues, we exploit ShiftInvariant Downsample [46] that copes with the discontinuity by using continuous pooling and filtering operation, preserving rich cross-correlation information between original and downsampled images. (iv) To efficiently capture continuous channel correlation and avoid information loss, we use the 1D channel attention module, Fast Channel Attention (FCA) instead of the general 2D convolution attention module. The input feature, $\\mathbf { F } _ { M _ { k } ^ { j } } \\in \\mathbb { R } ^ { H \\times W \\times C }$ is fed into three parallel multi-scale paths: ", + "bbox": [ + 173, + 757, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/546b8c677dc49d325c44eda39bdf0d665e3b73b4b396982bb5b8753c09eee4ae.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Comparison on four benchmarks. \\* denotes denoisers finetuned with images generated by PNGAN. " + ], + "table_body": "
MethodsSIDD[47]DND [48]MethodsPolyU [40]Nam [49]
PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑
DnCNN-B[10]23.660.58332.430.790RDN[12]37.940.94638.160.956
CBDNet [50]33.280.86838.060.942FFDNet+ [19]38.170.95138.810.957
RIDNet [44]38.710.91439.260.953TWSC [51]38.680.95838.960.962
AINDNet [52]39.150.95539.530.956CBDNet [50]38.740.96139.080.969
VDN[53]39.230.95539.380.952RIDNet [44]38.860.96239.200.973
CycleISP [22]39.520.95739.560.956VDN [53]39.040.96539.680.976
MPRNet [54]39.710.95839.800.954MPRNet [54]39.070.96939.410.974
MIRNet [55]39.720.95939.880.956MIRNet [55]39.180.97339.570.979
RIDNet*(Ours)39.250.95639.550.955RIDNet*(Ours)39.540.97139.690.975
MPRNet*(Ours)40.060.96040.180.961MPRNet*(Ours)40.480.98240.720.984
MIRNet*(Ours)40.070.96040.250.962MIRNet*(Ours)40.550.98340.780.986
", + "bbox": [ + 173, + 89, + 825, + 243 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 272, + 825, + 332 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d8812e238931f775aec826b5b35a9ff073436a3479f38b62179c4c2c90adcbd2.jpg", + "text": "$$\n\\mathbf { F } _ { M _ { k } ^ { j } } ^ { 1 } = F C A ( \\mathbf { F } _ { M _ { k } ^ { j } } ) , \\mathbf { F } _ { M _ { k } ^ { j } } ^ { 2 } = f _ { u p } ^ { 2 } ( F C A ( f _ { s i d } ^ { 2 } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) ) ) , \\mathbf { F } _ { M _ { k } ^ { j } } ^ { 4 } = f _ { u p } ^ { 4 } ( F C A ( f _ { s i d } ^ { 4 } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 184, + 339, + 790, + 363 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $F C A$ denotes Fast Channel Attention. $f _ { u p } ^ { 2 }$ denotes a conv layer after bilinear interpolation upsampling, 2 is the scale factor. $f _ { u p } ^ { 4 }$ is similarly defined. $f _ { s i d } ^ { 2 }$ means Shift-Invariant Downsample [46], 2 is also the scale factor. $f _ { s i d } ^ { 4 }$ is similarly defined. Subsequently, the output feature is derived by: ", + "bbox": [ + 174, + 369, + 826, + 419 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ee5e80a066375b4407e727a95299e0ee5e9766d31d5b3a445682b5aac7c8294f.jpg", + "text": "$$\nM _ { k } ^ { j } ( { \\bf F } _ { M _ { k } ^ { j } } ) = { \\bf F } _ { M _ { k } ^ { j } } + f ( [ { \\bf F } _ { M _ { k } ^ { j } } ^ { 1 } , { \\bf F } _ { M _ { k } ^ { j } } ^ { 2 } , { \\bf F } _ { M _ { k } ^ { j } } ^ { 4 } ] ) ,\n$$", + "text_format": "latex", + "bbox": [ + 351, + 426, + 645, + 450 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $f$ represents the last conv layer, $[ \\cdot , \\cdot , \\cdot ]$ denotes the concatenating operation. The architecture of FCA is shown in Fig. 3 (d). We define the input feature as $\\mathbf { F } _ { d }$ , then FCA can be formulated as: ", + "bbox": [ + 173, + 457, + 823, + 486 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/469e619d5b1cf1e699bcf95e36f2bda94fb54c969c72938e0a1b62f47b8431c3.jpg", + "text": "$$\nF C A ( \\mathbf { F } _ { d } ) = \\mathbf { F } _ { d } \\cdot \\big ( 1 + \\sigma \\big ( f _ { 1 D C } ( G A P ( \\mathbf { F } _ { d } ) ) \\big ) \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 491, + 656, + 511 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\sigma$ represents the Sigmoid activation function, $G A P$ means global average pooling along the spatial wise, $f _ { 1 D C }$ denotes 1-Dimension Convolution. In this work, we set $t = 3$ , $n = 2$ , and $C = 6 4$ . ", + "bbox": [ + 173, + 516, + 823, + 545 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2.4 Overall Training Objective ", + "text_level": 1, + "bbox": [ + 174, + 553, + 401, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In addition to the aforementioned losses, we employ a perceptual loss function that assesses a solution with respect to perceptually relevant characteristics (e.g., the structural contents and detailed textures): ", + "bbox": [ + 173, + 571, + 825, + 601 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0e3375d8a7fdfd967303cf8a05dadf39341d41f514611e29b5b02ba3a4cd8e5e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { p } = \\left| \\left| V G G ( \\mathbf { I } _ { f d } ) - V G G ( \\mathbf { I } _ { r d } ) \\right| \\right| _ { 2 } ^ { 2 } , \\mathbf { I } _ { f d } = D _ { d } ( \\mathbf { I } _ { f n } ) , \\mathbf { I } _ { r d } = D _ { d } ( \\mathbf { I } _ { r n } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 607, + 735, + 630 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $V G G$ denotes the last feature map of VGG16 [56]. Eventually, the training objective is: ", + "bbox": [ + 173, + 633, + 789, + 650 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2e43ce87f7045384769711d55504ff3ea8d3c8cf30ea07c2bda90f00834cc834.jpg", + "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { 1 } + \\lambda _ { p } \\cdot \\mathcal { L } _ { p } + \\lambda _ { R a } \\cdot ( \\mathcal { L } _ { D } + \\mathcal { L } _ { G } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 656, + 630, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\lambda _ { p }$ and $\\lambda _ { R a }$ are two hyper-parameters controlling the importance balance. The proposed PNGAN framework is end-to-end trained by minimizing $\\mathcal { L }$ . Note that the parameters in $D _ { d }$ and VGG16 are fixed. Each mini-batch training procedure is divided into two steps: (i) Fix $D$ and train $G$ . (ii) Fix $G$ and train $D$ . This pixel-level adversarial training scheme promotes $D$ the ability to distinguish fake noisy images from real noisy images and allows $G$ to learn to create the solutions that are highly similar to real camera noisy images and thus difficult to be classified by $D$ . ", + "bbox": [ + 173, + 679, + 825, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 Experiment ", + "text_level": 1, + "bbox": [ + 174, + 772, + 305, + 789 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 Experiment Setup ", + "text_level": 1, + "bbox": [ + 174, + 796, + 339, + 810 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets. We first use SIDD [47] train set to train $D _ { d }$ . Then we fix $D _ { d }$ to train $G$ on the same set. Subsequently, $G$ uses clean images from DIV2K [57], Flickr2K [58], BSD68 [59], Kodak24 [60], and Urban100 [61] to generate realistic noisy-clean image pairs. We use the generated data and SIDD train set jointly to finetune real denoisers and evaluate them on four real denoising benchmarks: SIDD [47], DND [48], PolyU [40], and Nam [49]. The images in SIDD [47] are collected using five smartphone cameras in 10 static scenes. There are 320 image pairs for training and 1,280 image patch pairs for validation. DND [48] composes 50 noisy-clean image pairs captured by 4 consumer ", + "bbox": [ + 174, + 814, + 826, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/1e86de1e279adf639e620e96667f698ef1bc2bbbb98ac88a6e9e011a83d52de8.jpg", + "image_caption": [ + "Figure 4: Domain discrepancy comparisons. We use the metric, Maximum Mean Discrepancy (MMD) to measure the domain discrepancy between synthetic and real noisy datasets, PNGAN generating and real noisy datasets. Under both setting1 and 2, the discrepancy decreases significantly when PNGAN is applied. " + ], + "image_footnote": [], + "bbox": [ + 173, + 88, + 820, + 196 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/9665359c84e08272d6c3cfb857e4c2a27f16946ce646659a5682c63c725fca79.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodsSIDD [47]S1DF2K[57,58]
S1 +PNGANS2S2+PNGANRealS1 + PNGANS2S2+PNGAN
RIDNet22.5537.92 (+15.37)36.1338.71 (+2.58)38.6922.5532.10(+9.55)33.9838.14 (+4.16)
MPRNet22.8638.52 (+15.66)36.5239.53 (+3.01)39.4522.8532.82(+9.97)34.1938.61 (+4.42)
MIRNet22.8338.76 (+15.93)36.5539.57 (+3.02)39.5823.0832.34(+9.26)34.2638.72 (+4.46)
", + "bbox": [ + 174, + 242, + 823, + 311 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2: Training denoisers with different data from scratch. PSNR is reported. $^ { \\mathrm { ~ S 1 , 2 = } }$ synthetic setting1,2. ", + "bbox": [ + 178, + 318, + 818, + 332 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "cameras. 1,000 patches at size $5 1 2 \\times 5 1 2$ are cropped from the collected images. PolyU [40] consists of 40 real camera noisy images. Nam [49] is composed of real noisy images of 11 static scenes. ", + "bbox": [ + 176, + 338, + 821, + 367 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implementation Details. We set the hyper-parameter $\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }$ , $\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }$ . For synthetic setting1, we set the noise intensity, $\\sigma _ { n } = 5 0$ . For synthetic setting2, we directly exploit CycleISP to generate the synthetic noisy input. All the sub-modules $( D _ { d } , G$ , and $D$ ) are trained with the Adam [62] optimizer $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 1 } = 0 . 9 9 9 9 )$ for $7 \\times 1 0 ^ { 5 }$ iterations. The initial learning rate is set to $2 \\times 1 \\bar { 0 } ^ { - 4 }$ . The cosine annealing strategy [63] is employed to steadily decrease the learning rate from the initial value to $1 0 ^ { - 6 }$ during the training procedure. Patches at size $1 2 8 \\times 1 2 8$ cropped from training images are fed into the models. The batch size is set as 8. The horizontal and vertical flips are performed for data augmentation. All the models are trained on RTX8000 GPUs. In the finetuning phase, the learning rate is set to $1 \\times 1 0 ^ { - 6 }$ , other settings remain unchanged. ", + "bbox": [ + 173, + 372, + 825, + 498 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 Quantitative Results ", + "text_level": 1, + "bbox": [ + 174, + 506, + 356, + 520 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Domain Discrepancy Validation. We use the widely applied metric, Maximum Mean Discrepancy (MMD) [64] to measure the domain discrepancy between synthetic and real-world noisy images, PNGAN generating, and real noisy images on four real noisy benchmarks. For DND, we derive a pseudo clean version by denoising the real noisy counterparts with a pre-trained MIRNet [55]. Then we use the pseudo clean version to synthesize noisy images. The results are depicted as a histogram in Fig. 4. For setting1, the domain discrepancy decreases by $74 \\%$ , $7 5 \\%$ , $44 \\%$ , and $43 \\%$ on SIDD, DND, PolyU, and Nam when PNGAN is exploited. For setting2, the discrepancy decreases by $64 \\%$ , $67 \\%$ , $46 \\%$ , and $44 \\%$ . These results demonstrate that PNGAN can narrow the discrepancy between synthetic and real noisy datasets. Please refer to the supplementary for detailed calculation process. ", + "bbox": [ + 173, + 525, + 825, + 650 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison with SOTA Methods. We use the generated noisy-clean image pairs (setting2) to finetune a series of denoisers. We compare our models with SOTA algorithms on four real denoising datasets: SIDD, DND, PolyU, and Nam. The results are reported in Tab. 1. \\* denotes denoisers finetuned with image pairs generated by PNGAN. We have the following observations: (i) Our denoisers outperform SOTA methods by a large margin. Specifically, MPRNet\\* and MIRNet\\* exceed the recent best method MIRNet by 0.34 and $0 . 3 5 \\mathrm { d B }$ on SIDD, 0.30 and 0.37 dB on DND. RIDNet\\*, MPRNet\\*, and MIRNet\\* surpass the best performers by 0.36, 1.30, and 1.37 dB on PolyU and 0.01, 1.04, and 1.10 dB on Nam. (ii) Compared with the counterparts that are not finetuned, our models achieve a significant promotion. In particular, RIDNet\\* is 0.54, 0.29, 0.68, and $0 . 4 9 \\mathrm { d B }$ higher than RIDNet on SIDD, DND, PolyU, and Nam. MPRNet\\* achieves 0.35, 0.38, 1.41, and 1.31 dB gain than MPRNet on SIDD, DND, PolyU, and Nam. MIRNet\\* is improved by 0.35, 0.37, 1.37, and 1.21 dB. This evidence clearly suggests the high similarity between PNGAN generating and real noisy images. Denoisers adapted with our fake image pairs generalize better across different benchmarks. ", + "bbox": [ + 173, + 655, + 826, + 835 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Train from Scratch. For more strong comparisons, we use the fake noisy images generated from clean SIDD train and DF2K (DIV2K+Flicker2K) respectively to train denoisers from scratch. The PSNR results evaluated on SIDD test are listed in Tab. 2. All models are trained with the same experiment schedule except the training data. It can be observed: (i) On SIDD train, when PNGAN is applied to setting1, denoisers are promoted by $\\sim 1 5 . 6 5$ dB and only $\\sim 0 . 8 4$ dB lower than those ", + "bbox": [ + 174, + 842, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f209d8c64470dae44b21785b843771996ba06d36a206d10d5ce3e25053032886.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 178, + 88, + 820, + 250 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/1513bed74d11c207af43181473cd814a19645c1a153db337a6586f22c7b564f5.jpg", + "table_caption": [ + "Figure 5: Visual comparisons of noisy images on SIDD, DND, PolyU, and Nam. Please zoom in. " + ], + "table_footnote": [], + "table_body": "
MethodsPNGAN ComponentGenerator Architecture
Baseline1+Dd+D+LpBaseline2+ Multi-scale+ SID+FCA
RIDNet14.5435.37 (+20.83)37.49(+2.12) 37.92(+0.43)35.6237.01(+1.39)37.23 (+0.22)37.92 (+0.69)
MPRNet14.2536.26 (+22.01)38.27(+2.01) 38.52(+0.25)36.2837.47(+1.19)37.86 (+0.39)38.52 (+0.66)
MIRNet13.5736.15 (+22.58)38.28(+2.13) 38.76(+0.48)36.3937.66(+1.27)37.89 (+0.23)38.76 (+0.87)
", + "bbox": [ + 174, + 275, + 825, + 339 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Ablation study of PNGAN component and the noise generator architecture. PSNR is reported. ", + "bbox": [ + 176, + 343, + 805, + 356 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "trained with real data (SIDD train set). While applying PNGAN to setting2 (CycleISP), denoisers are improved by $\\sim 2 . 8 7$ dB. Surprisingly, in this case, denoisers achieve almost the same performance as those trained with real data. The relative error is $0 . 2 \\%$ . (ii) To validate the generality of PNGAN, we also adopt synthetic DF2K noisy-clean image pairs to train denoisers. As shown in the right part of Tab. 2, when PNGAN is applied to setting1, denoisers are promoted by $\\sim 9 . 5 9$ dB. While applying PNGAN to setting2, denoisers are improved by $\\sim 4 . 3 5 ~ \\mathrm { d B }$ and only $\\sim 0 . 7 5$ dB lower than those trained with SIDD real train set. These results convincingly demonstrate: (i) The generated noise is highly similar to the real noise especially when PNGAN is applied to synthetic setting2. (ii) PNGAN can significantly narrow the domain discrepancy between synthetic and real-world noise. ", + "bbox": [ + 173, + 359, + 825, + 484 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 Qualitative Results ", + "text_level": 1, + "bbox": [ + 174, + 491, + 346, + 506 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Visual Examinations of Noisy Images. To intuitively evaluate the generated noisy images, we provide visual comparisons of noisy images on the four real noisy datasets, as shown in Fig. 5. Note that the clean image of DND is pseudo, denoised from its noisy version by a MIRNet. The left part depicts noisy images from SIDD, DND, PolyU, and Nam (top to down). The right part exhibits the patches cropped by the yellow bboxes, from left to right: clean, synthetic setting1, setting2 (CycleISP), PNGAN generating, and real noisy images. As can be seen from the zoom-in patches: (i) Noisy images synthesized by setting1 is signal-independent. The distribution and intensity remain unchanged across diverse scenes, indicating the characteristics of AWGN fundamentally differ from those of the real noise. (ii) Noisy images generated by PNGAN are closer to the real noise than those synthesized by setting2 visually. Noise synthesized by setting2 shows randomness that is obviously inconsistent with the real noise in terms of intensity and distribution. While PNGAN can model spatio-chromatically correlated and non-Gaussian noise more accurately. (iii) Even if passing through the same camera pipeline, different shooting conditions lead to the diversity of real noise. It’s unreasonable for the noise synthesized by CycleISP to show nearly uniform fitting to different input images. In contrast, PNGAN can adaptively simulate more sophisticated and photo-realistic models. This adaptability allows PNGAN to show robust performance across different real noisy datasets. ", + "bbox": [ + 174, + 510, + 825, + 729 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Visual Comparison of Denoised Images. We compare the visual results of denoisers before and after being finetuned (denoted with \\*) with the generated data in Fig. 4. We observe that models finetuned with the generated data are more effective in real noise removal. Furthermore, they are capable of preserving the structural content, textural details, and spatial smoothness of the homogeneous regions. In contrast, original models either yield over-smooth images sacrificing fine textural details and structural content or introduce redundant blotchy texture and chroma artifacts. ", + "bbox": [ + 174, + 736, + 825, + 820 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.4 Ablation Study ", + "text_level": 1, + "bbox": [ + 174, + 825, + 318, + 839 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Break-down Ablations. We perform break-down ablations to evaluate the effects of PNGAN components and SMNet architecture. We select setting1 to synthesize the noisy input from SIDD train set. Then we use the generated data only to train the denoisers from scratch and evaluate them on SIDD test. The PSNR results are reported in Tab. 3. (i) Firstly, $G$ is set as SMNet to validate the effects of PNGAN components. We start from Baseline1, no discriminator is used and the $\\mathcal { L } _ { 1 }$ loss is ", + "bbox": [ + 174, + 842, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f6d75ee03ec6067d2f9aa52e21200e934dc47ff620fce2476744f1cbb443fb7c.jpg", + "image_caption": [ + "Figure 6: Visual results of denoisers before and after being finetuned with fake data. Please zoom in. " + ], + "image_footnote": [], + "bbox": [ + 178, + 88, + 820, + 251 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/fc34288e794ee45ce52754399b02b426e3ae8073c3b515db897f489f87669e4e.jpg", + "image_caption": [ + "Figure 7: Parameter analysis of $\\lambda _ { p }$ , $\\lambda _ { R a }$ , and $\\sigma _ { n }$ " + ], + "image_footnote": [], + "bbox": [ + 181, + 275, + 495, + 359 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/224c1afe790c6c6af0483e47255231e18abcc541eea012e640dc817546bcca90.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
qSIDD [47]PolyU[40]Nam [49]Total
PSNRSSIMPSNRSSIMPSNRSSIMPSNRSSIM
None38.710.91438.860.96239.200.97338.760.929
039.320.95738.010.94938.340.95838.920.955
20%39.290.95738.450.95938.870.97039.030.958
40%39.280.95639.020.96639.260.97339.200.959
60%39.260.95639.540.97139.690.97539.350.961
80%39.230.95539.560.97239.720.97639.330.960
100%39.210.95539.570.97239.730.97639.330.960
", + "bbox": [ + 503, + 277, + 812, + 361 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4: Analysis of the finetuning data ratio $q$ ", + "bbox": [ + 517, + 366, + 800, + 378 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "directly performed between $\\mathbf { I } _ { f n }$ and ${ \\mathbf I } _ { r n }$ in Eq. (2). Denoisers trained with the generated data collapse dramatically, implying the naive strategy mentioned in Sec. 2.2 is unfeasible. When $D _ { d }$ is applied, the denoisers are promoted by 21.81 dB on average. In addition, the PSNR and SSIM between the denoised counterparts of generated and real noisy images are $3 9 . 1 4 ~ \\mathrm { d B }$ and 0.928 on average respectively. This evidence indicates that $D _ { d }$ successfully conducts the image domain alignment as mentioned in Sec. 2.2. Subsequently, we use an image-level $D$ with stride conv layers to classify whether the whole generated image is real. Nonetheless, the performance of denoisers remains almost unchanged. After deploying $D$ , the models are improved by ${ \\sim } 2 . 0 9$ dB, suggesting that the pixel-level noise model is more in line with real noise scenes and benefits generating more realistic noisy images. When ${ \\mathcal { L } } _ { p }$ is used, the denoisers gain a slight improvement by about 0.39 dB, indicating ${ \\mathcal { L } } _ { p }$ facilitates yielding more vivid results. (ii) Secondly, we only change the architecture of $G$ to study the effects of its components. We start from Baseline2 that doesn’t exploit multi-scale feature fusion, SID, and FCA. When we add two different scale branches and use bilinear interpolation to downsample and upsample, denoisers trained with the generated images are promoted by about $1 . 2 8 \\ : \\mathrm { d B }$ . After applying SID and FCA, the denoisers further gain 0.28 and $0 . 7 4 \\mathrm { d B }$ improvement on average. These results convincingly demonstrate the superiority of the proposed SMNet in real-world noise fitting. ", + "bbox": [ + 174, + 383, + 825, + 603 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Parameter Analysis. We adopt RIDNet as the baseline to perform parameter analysis. We firstly validate the effects of $\\lambda _ { p }$ , $\\lambda _ { R a }$ in Eq. (13), and the noise intensity of setting1, i.e., $\\sigma _ { n }$ . We change the parameters, train $G$ , use $G$ to generate realistic noisy images from clean images of SIDD train set, train RIDNet with the generated data, and evaluate its performance on SIDD test set. When analyzing one parameter, we fix the others at their optimal values. The PSNR results are shown in Fig. 7. The optimal setting is $\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }$ , $\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }$ , and $\\sigma _ { n } = 4 0$ or 50. Secondly, we evaluate the effect of the ratio of finetuning data. We denote the ratio of extended training data (setting2) to SIDD real noisy training data as $q$ . We change the value of $q$ , finetuned the original RIDNet, and test on three real denoising datasets: SIDD, PolyU, and Nam. The results are listed in Tab. 4. When $q = 0$ , all the finetuning data comes from SIDD train set, RIDNet achieves the best performance on SIDD. However, its performance on PolyU and Nam degrades drastically due to the domain discrepancy between different real noisy datasets. We gradually increase the value of $q$ to study its effects. The average performance on the three datasets yields the maximum when $q = 6 0 \\%$ . ", + "bbox": [ + 173, + 609, + 825, + 790 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 795, + 299, + 810 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Too much research focuses on designing a CNN architecture for real noise removal. In contrast, this work investigates how to generate more realistic noisy images so as to boom the denoising performance. We first formulate a noise model that treats each noisy pixel as a random variable. Then we propose a novel framework PNGAN to perform the image and noise domain alignment. For better noise fitting, we customize an efficient architecture, SMNet as the generator. Experiments show that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution. Denoisers finetuned with the generated data outperform SOTA methods on real denoising datasets. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 89, + 330, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work is jointly supported by the NSFC fund (61831014), in part by the Shenzhen Science and Technology Project under Grant (ZDYBH201900000002, JCYJ20180508152042002, CJGJZD20200617102601004). ", + "bbox": [ + 174, + 121, + 826, + 162 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 183, + 266, + 199 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] PP， ng， [2] S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” in CVPR, 2014. 1 [3] F. Zhu, G. Chen, and P.-A. Heng, “From noise modeling to blind image denoising,” in CVPR, 2016. 1 [4] J. Xu, L. Zhang, and D. 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Karen Simonyan, “Very deep convolutional networks for large-scale image recognition,” in ICLR, 2015. 6 \n[57] R. Timofte, E. Agustsson, L. Van Gool, M.-H. Yang, and L. Zhang, “Ntire 2017 challenge on single image super-resolution: Methods and results,” in CVPRW, 2017. 6, 7 \n[58] B. Lim, S. Son, H. Kim, S. Nah, and K. Mu Lee, “Enhanced deep residual networks for single image super-resolution,” in CVPRW, 2017. 6, 7 \n[59] S. Roth and M. J. Black, “Fields of experts: A framework for learning image priors,” in CVPR, 2015. 6 \n[60] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, and e. a. Michael Bernstein, “Imagenet large scale visual recognition challenge,” in IJCV, 2015. 6 \n[61] J.-B. Huang, A. Singh, and N. Ahuja, “Single image super-resolution from transformed selfexemplars,” in CVPR, 2015. 6 \n[62] D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in ICLR, 2015. 7 \n[63] I. Loshchilov and F. Hutter, “Sgdr: Stochastic gradient descent with warm restarts,” in ICLR, 2017. 7 \n[64] A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. 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Firstly, we formulate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 315, + 469, + 326 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 469, + 326 + ], + "score": 1.0, + "content": "a simple yet reasonable noise model that treats each real noisy pixel as a random", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 325, + 471, + 338 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 471, + 338 + ], + "score": 1.0, + "content": "variable. This model splits the noisy image generation problem into two sub-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 335, + 471, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 471, + 349 + ], + "score": 1.0, + "content": "problems: image domain alignment and noise domain alignment. Subsequently,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 346, + 471, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 471, + 359 + ], + "score": 1.0, + "content": "we propose a novel framework, namely Pixel-level Noise-aware Generative Adver-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "score": 1.0, + "content": "sarial Network (PNGAN). PNGAN employs a pre-trained real denoiser to map the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 367, + 470, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 382 + ], + "score": 1.0, + "content": "fake and real noisy images into a nearly noise-free solution space to perform image", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 380, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 142, + 380, + 469, + 391 + ], + "score": 1.0, + "content": "domain alignment. Simultaneously, PNGAN establishes a pixel-level adversarial", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 389, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 470, + 403 + ], + "score": 1.0, + "content": "training to conduct noise domain alignment. Additionally, for better noise fitting,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 402, + 469, + 414 + ], + "spans": [ + { + "bbox": [ + 141, + 402, + 469, + 414 + ], + "score": 1.0, + "content": "we present an efficient architecture Simple Multi-scale Network (SMNet) as the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 412, + 469, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 469, + 425 + ], + "score": 1.0, + "content": "generator. Qualitative validation shows that noise generated by PNGAN is highly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 423, + 469, + 435 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 469, + 435 + ], + "score": 1.0, + "content": "similar to real noise in terms of intensity and distribution. Quantitative experiments", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 433, + 469, + 447 + ], + "spans": [ + { + "bbox": [ + 141, + 433, + 469, + 447 + ], + "score": 1.0, + "content": "demonstrate that a series of denoisers trained with the generated noisy images", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 444, + 442, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 442, + 457 + ], + "score": 1.0, + "content": "achieve state-of-the-art (SOTA) results on four real denoising benchmarks.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 270, + 471, + 457 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 191, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 192, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 192, + 491 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "Image denoising is an important yet challenging problem in low-level vision. It aims to restore a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "clean image from its noisy counterpart. Traditional approaches concentrate on designing a rational", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "maximum a posteriori (MAP) model, containing regularization and fidelity terms, from a Bayesian", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 525, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 506, + 537 + ], + "score": 1.0, + "content": "perspective [1]. Some image priors like low-rankness [2, 3, 4], sparsity [5], and non-local similarity [6,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "7] are exploited to customize a better rational MAP model. However, these hand-crafted methods", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "are inferior in representing capacity. With the development of deep learning, image denoising has", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 571 + ], + "score": 1.0, + "content": "witnessed significant progress. Deep convolutional neural network (CNN) applies a powerful learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "model to eliminate noise and has achieved promising performance [8, 9, 10, 11, 12, 13, 14]. These", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "deep CNN denoisers rely on a large-scale dataset of real-world noisy-clean image pairs. Nonetheless,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 589, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 604 + ], + "score": 1.0, + "content": "collecting even small datasets is extremely tedious and labor-intensive. The process of acquiring", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "real-world noisy-clean image pairs is to take hundreds of noisy images of the same scene and average", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 612, + 496, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 496, + 625 + ], + "score": 1.0, + "content": "them to get the clean image. To get more image pairs, researchers try to synthesize noisy images.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 492, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "In particular, there are two common settings for synthesizing noisy images. As shown in Fig. 1 (a1),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "setting1 directly adds the additive white Gaussian noise (AWGN) with the clean RGB image. For a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "long time, single image denoising [15, 16, 17, 18, 19, 10] is performed with setting1. Nevertheless,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "score": 1.0, + "content": "fundamentally different from AWGN, real camera noise is generally more sophisticated and signal-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "dependent[20, 21]. The noise produced by photon sensing is further affected by the in-camera", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 505, + 695 + ], + "score": 1.0, + "content": "signal processing (ISP) pipeline (e.g., Gama correction, compression, and demosaicing). Models", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "trained with setting1 are easily over-fitted to AWGN and fail in real noise removal. Setting2 is", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "based on ISP-modeling CNN [22] and Poisson-Gaussian [21, 23] noise model that modeling photon", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "sensing with Poisson and remaining stationary disturbances with Gaussian has been adopted in RAW", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "denoising. As shown in Fig. 1 (a2), setting2 adds a Poisson-Gaussian noise with the clean RAW", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "image and then passes the result through a pre-trained RAW2RGB CNN to obtain the RGB noisy", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "counterpart. Notably, when the clean RAW image is unavailable, a pre-trained RGB2RAW CNN", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "is utilized to transform the clean RGB image to its RAW counterpart [22]. However, setting2 has", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "the following drawbacks: (i) The noise is assumed to obey a hand-crafted probability distribution.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "However, because of the randomness and complexity of real camera noise, it’s difficult to customize", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "a hand-crafted probability distribution to model all the characteristics of real noise. (ii) The ISP", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "pipeline is very sophisticated and hard to be completely modeled. The RAW2RGB branch only learns", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "the mapping from the clean RAW domain to the clean RGB space. However, the mapping from the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "Poisson-Gaussian noisy RAW domain to the real noisy RGB space can not be ensured. (iii) The ISP", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "pipelines of different devices vary significantly, which results in the poor generality and robustness", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "of ISP modeling CNNs. Thus, whether noisy images are synthesized with setting1 or 2, there still", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "remains a discrepancy between synthetic and real noisy datasets. We notice that GAN utilizes the", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "internal information of the input image and external information from other images when modeling", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 258, + 473, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 473, + 271 + ], + "score": 1.0, + "content": "image priors. Hence, we propose to use GAN to adaptively learn the real noise distribution.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 628, + 506, + 695 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "trained with setting1 are easily over-fitted to AWGN and fail in real noise removal. Setting2 is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "based on ISP-modeling CNN [22] and Poisson-Gaussian [21, 23] noise model that modeling photon", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "sensing with Poisson and remaining stationary disturbances with Gaussian has been adopted in RAW", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "denoising. As shown in Fig. 1 (a2), setting2 adds a Poisson-Gaussian noise with the clean RAW", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "image and then passes the result through a pre-trained RAW2RGB CNN to obtain the RGB noisy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "counterpart. Notably, when the clean RAW image is unavailable, a pre-trained RGB2RAW CNN", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "is utilized to transform the clean RGB image to its RAW counterpart [22]. However, setting2 has", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "the following drawbacks: (i) The noise is assumed to obey a hand-crafted probability distribution.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "However, because of the randomness and complexity of real camera noise, it’s difficult to customize", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "a hand-crafted probability distribution to model all the characteristics of real noise. (ii) The ISP", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "pipeline is very sophisticated and hard to be completely modeled. The RAW2RGB branch only learns", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "the mapping from the clean RAW domain to the clean RGB space. However, the mapping from the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "Poisson-Gaussian noisy RAW domain to the real noisy RGB space can not be ensured. (iii) The ISP", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "pipelines of different devices vary significantly, which results in the poor generality and robustness", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "of ISP modeling CNNs. Thus, whether noisy images are synthesized with setting1 or 2, there still", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "remains a discrepancy between synthetic and real noisy datasets. We notice that GAN utilizes the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 261 + ], + "score": 1.0, + "content": "internal information of the input image and external information from other images when modeling", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 258, + 473, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 473, + 271 + ], + "score": 1.0, + "content": "image priors. Hence, we propose to use GAN to adaptively learn the real noise distribution.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 274, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 504, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 504, + 286 + ], + "score": 1.0, + "content": "GAN is firstly introduced in [24] and has been proven successful in image synthesis [25, 26, 27]", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 284, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 507, + 299 + ], + "score": 1.0, + "content": "and translation [26, 27]. Subsequently, GAN is applied to image restoration and enhancement, e.g.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "score": 1.0, + "content": "super resolution [28, 29, 30], style transfer [27, 31], enlighten [32, 33], deraining [34], dehazing [35],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "image inpainting [36, 37], image editing [38, 39], and mobile photo enhancement [40, 41]. Although", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "GAN is widely applied in low-level vision tasks, few works are dedicated to investigating the realistic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "noise generation problem [42]. Chen et al. [43] propose a simple GAN that takes Gaussian noise as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "input to generate noisy patches. However, as in general, this GAN is image-level, i.e., it treats images", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 351, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 362 + ], + "score": 1.0, + "content": "as samples and attempts to approximate the probability distribution of real-world noisy images. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "image-level GAN neglects that each pixel of a real noisy image is a random variable and the real", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "noise is spatio-chromatically correlated, thus results in coarse learning of the real noise distribution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "To alleviate the above problems, this work focuses on learning how to generate realistic noisy images", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "score": 1.0, + "content": "so as to augment the training data for real denoisers. To begin with, we propose a simple yet", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "reasonable noise model that treats each pixel of a real noisy image as a random variable. This noise", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "model splits the noise generation problem into two sub-problems: image domain alignment and noise", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "domain alignment. Subsequently, to tackle these two sub-problems, we propose a novel Pixel-level", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "Noise-aware Generative Adversarial Network (PNGAN). During the training procedure of PNGAN,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "we employ a pre-trained real denoiser to map the generated and real noisy images into a nearly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "pixel-level adversarial training that encourages the generator to adaptively simulate the real noise", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 485, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 501 + ], + "score": 1.0, + "content": "distribution so as to conduct the noise domain alignment. In addition, for better real noise fitting,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "we present a lightweight yet efficient CNN architecture, Simple Multi-scale Network (SMNet) as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "the generator. SMNet repeatedly aggregates multi-scale features to capture rich auto-correlation,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "which provides more sufficient spatial representations for noise simulating. Different from general", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "image-level GAN, our discriminator is pixel-level. The discriminator outputs a score map. Each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "position on the score map indicates how realistic the corresponding noisy pixel is. With this pixel-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "level noise-aware adversarial training, the generator is encouraged to create solutions that are highly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 563, + 369, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 369, + 576 + ], + "score": 1.0, + "content": "similar to real noisy images and thus difficult to be distinguished.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 381, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 381, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 381, + 592 + ], + "score": 1.0, + "content": "In conclusion, our contributions can be summarized into four points:", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "(1) We formulate a simple yet reasonable noise model. This model treats each noisy pixel as a random", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "variable and then splits the noisy image generation into two parts: image and noise domain alignment.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 504, + 646 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 637 + ], + "score": 1.0, + "content": "(2) We propose a novel framework, PNGAN. It establishes an effective pixel-level adversarial training", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 634, + 482, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 482, + 648 + ], + "score": 1.0, + "content": "to encourage the generator to favor solutions that reside on the manifold of real noisy images.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 503, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "(3) We customize an efficient CNN architecture, SMNet learning rich multi-scale auto-correlation for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 662, + 479, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 479, + 675 + ], + "score": 1.0, + "content": "better noise fitting. SMNet serves as the generator in PNGAN costing only 0.8M parameters.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "(4) Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "finetuned with the generated noisy images achieve SOTA results on four real denoising benchmarks.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 270 + ], + "lines": [], + "index": 8.5, + "bbox_fs": [ + 105, + 73, + 506, + 271 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 274, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 504, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 504, + 286 + ], + "score": 1.0, + "content": "GAN is firstly introduced in [24] and has been proven successful in image synthesis [25, 26, 27]", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 284, + 507, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 507, + 299 + ], + "score": 1.0, + "content": "and translation [26, 27]. Subsequently, GAN is applied to image restoration and enhancement, e.g.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "score": 1.0, + "content": "super resolution [28, 29, 30], style transfer [27, 31], enlighten [32, 33], deraining [34], dehazing [35],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "image inpainting [36, 37], image editing [38, 39], and mobile photo enhancement [40, 41]. Although", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "GAN is widely applied in low-level vision tasks, few works are dedicated to investigating the realistic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "noise generation problem [42]. Chen et al. [43] propose a simple GAN that takes Gaussian noise as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "input to generate noisy patches. However, as in general, this GAN is image-level, i.e., it treats images", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 351, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 362 + ], + "score": 1.0, + "content": "as samples and attempts to approximate the probability distribution of real-world noisy images. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "image-level GAN neglects that each pixel of a real noisy image is a random variable and the real", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "noise is spatio-chromatically correlated, thus results in coarse learning of the real noise distribution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 273, + 507, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "To alleviate the above problems, this work focuses on learning how to generate realistic noisy images", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "score": 1.0, + "content": "so as to augment the training data for real denoisers. To begin with, we propose a simple yet", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "reasonable noise model that treats each pixel of a real noisy image as a random variable. This noise", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "model splits the noise generation problem into two sub-problems: image domain alignment and noise", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "domain alignment. Subsequently, to tackle these two sub-problems, we propose a novel Pixel-level", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "Noise-aware Generative Adversarial Network (PNGAN). During the training procedure of PNGAN,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "we employ a pre-trained real denoiser to map the generated and real noisy images into a nearly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "noise-free solution space to perform image domain alignment. Simultaneously, PNGAN establishes a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "pixel-level adversarial training that encourages the generator to adaptively simulate the real noise", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 485, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 501 + ], + "score": 1.0, + "content": "distribution so as to conduct the noise domain alignment. In addition, for better real noise fitting,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "we present a lightweight yet efficient CNN architecture, Simple Multi-scale Network (SMNet) as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "the generator. SMNet repeatedly aggregates multi-scale features to capture rich auto-correlation,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "which provides more sufficient spatial representations for noise simulating. Different from general", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "image-level GAN, our discriminator is pixel-level. The discriminator outputs a score map. Each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "position on the score map indicates how realistic the corresponding noisy pixel is. With this pixel-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "level noise-aware adversarial training, the generator is encouraged to create solutions that are highly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 563, + 369, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 369, + 576 + ], + "score": 1.0, + "content": "similar to real noisy images and thus difficult to be distinguished.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 389, + 507, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 381, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 381, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 381, + 592 + ], + "score": 1.0, + "content": "In conclusion, our contributions can be summarized into four points:", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45, + "bbox_fs": [ + 106, + 579, + 381, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "(1) We formulate a simple yet reasonable noise model. This model treats each noisy pixel as a random", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "variable and then splits the noisy image generation into two parts: image and noise domain alignment.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 106, + 596, + 506, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 504, + 646 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 637 + ], + "score": 1.0, + "content": "(2) We propose a novel framework, PNGAN. It establishes an effective pixel-level adversarial training", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 634, + 482, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 482, + 648 + ], + "score": 1.0, + "content": "to encourage the generator to favor solutions that reside on the manifold of real noisy images.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 622, + 505, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 503, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "(3) We customize an efficient CNN architecture, SMNet learning rich multi-scale auto-correlation for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 662, + 479, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 479, + 675 + ], + "score": 1.0, + "content": "better noise fitting. SMNet serves as the generator in PNGAN costing only 0.8M parameters.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 650, + 505, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "(4) Qualitative validation shows that noise generated by PNGAN is highly similar to real noise in", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "terms of intensity and distribution. Quantitative experiments demonstrate that a series of denoisers", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "finetuned with the generated noisy images achieve SOTA results on four real denoising benchmarks.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 678, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 70, + 499, + 220 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 70, + 499, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 499, + 220 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 499, + 220 + ], + "score": 0.969, + "type": "image", + "image_path": "09f874f387c331b9a197a98ffe2b51b7acdc68dca22e5b8d83d8a65e04c2d5c0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 70, + 499, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 120.0, + 499, + 170.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 170.0, + 499, + 220.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 226, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "Figure 1: The pipeline of using PNGAN to perform data augmentation. It is divided into: (a) synthesizing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "phase, (b) training phase, and (c) finetuning phase. Please refer to the text (Sec. 2) for more detailed descriptions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 261, + 216, + 274 + ], + "lines": [ + { + "bbox": [ + 104, + 259, + 217, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 217, + 277 + ], + "score": 1.0, + "content": "2 Proposed Method", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 506, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 506, + 296 + ], + "score": 1.0, + "content": "As shown in Fig. 1, the pipeline of using PNGAN to perform data augmentation consists of three", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "phases. (a) is the synthesizing phase. (a1) and (a2) are two common synthetic settings. In this phase,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 303, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 506, + 320 + ], + "score": 1.0, + "content": "we produce the synthetic noisy image from its clean RGB or RAW counterpart. (b) is the training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 314, + 241, + 329 + ], + "score": 1.0, + "content": "phase of PNGAN. The generator", + "type": "text" + }, + { + "bbox": [ + 241, + 316, + 250, + 326 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "adopts the synthetic image as input. Which synthetic setting is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 397, + 339 + ], + "score": 1.0, + "content": "selected is controlled by the switch. By using a pre-trained real denoiser", + "type": "text" + }, + { + "bbox": [ + 398, + 327, + 411, + 338 + ], + "score": 0.88, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 326, + 506, + 339 + ], + "score": 1.0, + "content": ", PNGAN establishes a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 376, + 349 + ], + "score": 1.0, + "content": "pixel-level noise-aware adversarial training between the generator", + "type": "text" + }, + { + "bbox": [ + 377, + 338, + 386, + 347 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 337, + 461, + 349 + ], + "score": 1.0, + "content": "and discriminator", + "type": "text" + }, + { + "bbox": [ + 461, + 338, + 471, + 347 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "so as to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 351, + 361 + ], + "score": 1.0, + "content": "simultaneously conduct image and noise domain alignment.", + "type": "text" + }, + { + "bbox": [ + 351, + 348, + 365, + 360 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 348, + 507, + 361 + ], + "score": 1.0, + "content": "is set as RIDNet [44] in this work.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "(c) is the finetuning phase. Firstly, in (c1), the generator creates extended fake noisy-clean image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 507, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 507, + 383 + ], + "score": 1.0, + "content": "pairs. Secondly, in (c2), the fake and real data are jointly utilized to finetune a series of real denoisers.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 245, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 247, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 247, + 408 + ], + "score": 1.0, + "content": "2.1 Pixel-level Noise Modelling", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Real camera noise is sophisticated and signal-dependent. Specifically, in the real camera system, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "RAW noise produced by photon sensing comes from multiple sources (e.g., short noise, thermal noise,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "and dark current noise) and is further affected by the ISP pipeline. Besides, illumination changes and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "camera movement inevitably lead to spatial pixel misalignment and color or brightness deviation.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "Hence, hand-designed noise models based on mathematical assumptions are difficult to accurately", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 470, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 480 + ], + "score": 1.0, + "content": "and completely describe the properties of real noise. Different from previous methods, we don’t base", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "our noise model on any mathematical assumptions. Instead, we use CNN to implicitly simulate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "characteristics of real noise. We begin by noting that when taking multiple noisy images of the same", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "scene, the noise intensity of the same pixel varies a lot. Simultaneously, affected by the ISP pipeline,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "score": 1.0, + "content": "the real noise is spatio-chromatically correlated. Thus, the correlation between different pixels of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "same real noisy image should be considered. In light of these facts, we treat each pixel of a real noisy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 426, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 426, + 546 + ], + "score": 1.0, + "content": "image as a random variable and formulate a simple yet reasonable noise model:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 550, + 450, + 565 + ], + "lines": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "spans": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "score": 0.89, + "content": "\\mathbf { I } _ { r n } [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] + \\mathbf { N } [ i ] , \\quad D _ { d } ( \\mathbf { I } _ { r n } ) [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] , \\quad 1 \\leq i \\leq H \\times W ,", + "type": "interline_equation", + "image_path": "dbe88430ac2f919c6cf4acf9a7d845c445835202c472919c037d2a58226d4558.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 569, + 507, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 132, + 587 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 570, + 209, + 584 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { I } } _ { c l e a n } \\in \\mathbb { R } ^ { H \\times W \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 569, + 353, + 587 + ], + "score": 1.0, + "content": "is the predicted clean counterpart of", + "type": "text" + }, + { + "bbox": [ + 353, + 573, + 367, + 583 + ], + "score": 0.93, + "content": "{ \\mathbf I } _ { r n }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 569, + 435, + 587 + ], + "score": 1.0, + "content": ", it’s denoised by", + "type": "text" + }, + { + "bbox": [ + 435, + 573, + 449, + 583 + ], + "score": 0.87, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 569, + 475, + 587 + ], + "score": 1.0, + "content": ". Each", + "type": "text" + }, + { + "bbox": [ + 476, + 572, + 495, + 584 + ], + "score": 0.88, + "content": "\\mathbf { N } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 569, + 507, + 587 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 504, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 432, + 596 + ], + "score": 1.0, + "content": "a random noise variable with unknown probability distribution. 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Now we aim to design a framework to generate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 602, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 602, + 185, + 619 + ], + "score": 1.0, + "content": "a fake noisy image", + "type": "text" + }, + { + "bbox": [ + 186, + 605, + 253, + 618 + ], + "score": 0.91, + "content": "\\mathbf { I } _ { f n } \\in \\mathbb { R } ^ { H \\times W \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 602, + 416, + 619 + ], + "score": 1.0, + "content": "such that the probability distribution of", + "type": "text" + }, + { + "bbox": [ + 416, + 605, + 441, + 618 + ], + "score": 0.91, + "content": "\\mathbf { I } _ { f n } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 602, + 459, + 619 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 605, + 483, + 617 + ], + "score": 0.9, + "content": "\\mathbf { I } _ { \\mathit { r n } } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 602, + 507, + 619 + ], + "score": 1.0, + "content": "is as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 617, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 340, + 632 + ], + "score": 1.0, + "content": "close as possible. 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It is divided into: (a) synthesizing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "phase, (b) training phase, and (c) finetuning phase. Please refer to the text (Sec. 2) for more detailed descriptions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 261, + 216, + 274 + ], + "lines": [ + { + "bbox": [ + 104, + 259, + 217, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 217, + 277 + ], + "score": 1.0, + "content": "2 Proposed Method", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 506, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 506, + 296 + ], + "score": 1.0, + "content": "As shown in Fig. 1, the pipeline of using PNGAN to perform data augmentation consists of three", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "phases. (a) is the synthesizing phase. (a1) and (a2) are two common synthetic settings. In this phase,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 303, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 506, + 320 + ], + "score": 1.0, + "content": "we produce the synthetic noisy image from its clean RGB or RAW counterpart. (b) is the training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 314, + 241, + 329 + ], + "score": 1.0, + "content": "phase of PNGAN. The generator", + "type": "text" + }, + { + "bbox": [ + 241, + 316, + 250, + 326 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "adopts the synthetic image as input. Which synthetic setting is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 397, + 339 + ], + "score": 1.0, + "content": "selected is controlled by the switch. By using a pre-trained real denoiser", + "type": "text" + }, + { + "bbox": [ + 398, + 327, + 411, + 338 + ], + "score": 0.88, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 326, + 506, + 339 + ], + "score": 1.0, + "content": ", PNGAN establishes a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 376, + 349 + ], + "score": 1.0, + "content": "pixel-level noise-aware adversarial training between the generator", + "type": "text" + }, + { + "bbox": [ + 377, + 338, + 386, + 347 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 337, + 461, + 349 + ], + "score": 1.0, + "content": "and discriminator", + "type": "text" + }, + { + "bbox": [ + 461, + 338, + 471, + 347 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "so as to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 351, + 361 + ], + "score": 1.0, + "content": "simultaneously conduct image and noise domain alignment.", + "type": "text" + }, + { + "bbox": [ + 351, + 348, + 365, + 360 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 348, + 507, + 361 + ], + "score": 1.0, + "content": "is set as RIDNet [44] in this work.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "(c) is the finetuning phase. Firstly, in (c1), the generator creates extended fake noisy-clean image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 507, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 507, + 383 + ], + "score": 1.0, + "content": "pairs. Secondly, in (c2), the fake and real data are jointly utilized to finetune a series of real denoisers.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 283, + 507, + 383 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 245, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 247, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 247, + 408 + ], + "score": 1.0, + "content": "2.1 Pixel-level Noise Modelling", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "Real camera noise is sophisticated and signal-dependent. Specifically, in the real camera system, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "RAW noise produced by photon sensing comes from multiple sources (e.g., short noise, thermal noise,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "and dark current noise) and is further affected by the ISP pipeline. Besides, illumination changes and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "camera movement inevitably lead to spatial pixel misalignment and color or brightness deviation.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "Hence, hand-designed noise models based on mathematical assumptions are difficult to accurately", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 470, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 480 + ], + "score": 1.0, + "content": "and completely describe the properties of real noise. Different from previous methods, we don’t base", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "our noise model on any mathematical assumptions. Instead, we use CNN to implicitly simulate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "characteristics of real noise. We begin by noting that when taking multiple noisy images of the same", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "scene, the noise intensity of the same pixel varies a lot. Simultaneously, affected by the ISP pipeline,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 506, + 525 + ], + "score": 1.0, + "content": "the real noise is spatio-chromatically correlated. Thus, the correlation between different pixels of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "same real noisy image should be considered. In light of these facts, we treat each pixel of a real noisy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 426, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 426, + 546 + ], + "score": 1.0, + "content": "image as a random variable and formulate a simple yet reasonable noise model:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 414, + 506, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 550, + 450, + 565 + ], + "lines": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "spans": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "score": 0.89, + "content": "\\mathbf { I } _ { r n } [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] + \\mathbf { N } [ i ] , \\quad D _ { d } ( \\mathbf { I } _ { r n } ) [ i ] = \\hat { \\mathbf { I } } _ { c l e a n } [ i ] , \\quad 1 \\leq i \\leq H \\times W ,", + "type": "interline_equation", + "image_path": "dbe88430ac2f919c6cf4acf9a7d845c445835202c472919c037d2a58226d4558.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 161, + 550, + 450, + 565 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 569, + 507, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 132, + 587 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 570, + 209, + 584 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { I } } _ { c l e a n } \\in \\mathbb { R } ^ { H \\times W \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 569, + 353, + 587 + ], + "score": 1.0, + "content": "is the predicted clean counterpart of", + "type": "text" + }, + { + "bbox": [ + 353, + 573, + 367, + 583 + ], + "score": 0.93, + "content": "{ \\mathbf I } _ { r n }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 569, + 435, + 587 + ], + "score": 1.0, + "content": ", it’s denoised by", + "type": "text" + }, + { + "bbox": [ + 435, + 573, + 449, + 583 + ], + "score": 0.87, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 569, + 475, + 587 + ], + "score": 1.0, + "content": ". 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According to the noise model in Eq. (1), we split", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 425, + 703 + ], + "score": 1.0, + "content": "this problem into two sub-problems: (i) Image domain alignment aims to align", + "type": "text" + }, + { + "bbox": [ + 425, + 687, + 460, + 701 + ], + "score": 0.91, + "content": "\\hat { \\mathbf { I } } _ { c l e a n } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 688, + 506, + 703 + ], + "score": 1.0, + "content": ". (ii) Noise", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 331, + 713 + ], + "score": 1.0, + "content": "domain alignment targets at modeling the distribution of", + "type": "text" + }, + { + "bbox": [ + 331, + 701, + 350, + 712 + ], + "score": 0.89, + "content": "\\mathbf { N } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 699, + 505, + 713 + ], + "score": 1.0, + "content": ". To handle the sub-problems, PNGAN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 484, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 393, + 723 + ], + "score": 1.0, + "content": "establishes a novel pixel-level noise-aware adversarial training between", + "type": "text" + }, + { + "bbox": [ + 394, + 711, + 403, + 721 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 711, + 421, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 711, + 430, + 721 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 711, + 484, + 723 + ], + "score": 1.0, + "content": "in Fig. 1 (b).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 676, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "Image Domain Alignment. A very naive strategy to construct both image and noise domain align-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 282, + 96 + ], + "score": 1.0, + "content": "ment is to directly minimize the distance of", + "type": "text" + }, + { + "bbox": [ + 282, + 84, + 298, + 96 + ], + "score": 0.91, + "content": "\\mathbf { I } _ { f n }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 84, + 316, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 84, + 331, + 95 + ], + "score": 0.91, + "content": "{ \\mathbf I } _ { r n }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 84, + 507, + 96 + ], + "score": 1.0, + "content": ". 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Besides, as analyzed in Sec. 2.1, each pixel of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 121, + 127 + ], + "score": 0.88, + "content": "{ \\mathbf I } _ { r n }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "is a distribution-unknown random variable. This indicates that such a naive strategy challenges", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "the training and may easily cause the non-convergence issue. 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Now we customize a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 323, + 313 + ], + "score": 1.0, + "content": "light-weight yet efficient CNN architecture, SMNet as", + "type": "text" + }, + { + "bbox": [ + 323, + 301, + 332, + 310 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 300, + 379, + 313 + ], + "score": 1.0, + "content": "to generate", + "type": "text" + }, + { + "bbox": [ + 380, + 301, + 426, + 312 + ], + "score": 0.92, + "content": "P _ { G } ( x _ { i } ; \\theta _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 300, + 505, + 313 + ], + "score": 1.0, + "content": ". In this section, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 465, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 250, + 324 + ], + "score": 1.0, + "content": "firstly introduce the input setting of", + "type": "text" + }, + { + "bbox": [ + 250, + 312, + 259, + 321 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 311, + 465, + 324 + ], + "score": 1.0, + "content": "and subsequently detail the architecture of SMNet.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "Input Setting. We aim to generate a realistic noisy image from its clean counterpart. A naive setting", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "is to directly adopt the clean image as the input to generate the noisy image. However, this naive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "setting is not in line with the fact. When we repeatedly feed the same clean image to a pre-trained", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 107, + 361, + 115, + 371 + ], + "score": 0.51, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 361, + 119, + 373 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 119, + 361, + 129, + 371 + ], + "score": 0.63, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "outputs completely the same noisy images. In contrast, when taking multiple pictures in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "real world, the real noisy images vary a lot in the intensity of each pixel. This is caused by many", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "factors (e.g., photon sensing noise, ISP pipelines, and illumination conditions). Hence, the naive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "input setting containing no distribution is unreasonable. We review that the general GANs sample", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "from an initial random distribution (usually Gaussian) to generate a fake image. Hence, the input of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 414, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 107, + 415, + 116, + 425 + ], + "score": 0.75, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 414, + 507, + 428 + ], + "score": 1.0, + "content": "should contain a random distribution so as to generate multiple noisy images of the same scene.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "We note that the two common synthetic settings meet this condition. Therefore, we utilize the two", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 437, + 503, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 494, + 450 + ], + "score": 1.0, + "content": "common settings to produce the synthetic image and then adopt the synthetic image as the input of", + "type": "text" + }, + { + "bbox": [ + 495, + 438, + 503, + 447 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 507, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 507, + 461 + ], + "score": 1.0, + "content": "Subsequently, we propose a light-weight yet efficient architecture, SMNet for better real noise fitting.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 468, + 476 + ], + "score": 1.0, + "content": "SMNet Architecture. The architecture of SMNet is shown in Fig. 3 (a). SMNet involves", + "type": "text" + }, + { + "bbox": [ + 468, + 465, + 473, + 474 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "Simple", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 293, + 487 + ], + "score": 1.0, + "content": "Residual Groups (SRG) and each SRG contains", + "type": "text" + }, + { + "bbox": [ + 294, + 477, + 301, + 485 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "Multi-scale Attention Blocks (MAB). The synthetic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 482, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 103, + 482, + 130, + 500 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 130, + 485, + 201, + 498 + ], + "score": 0.91, + "content": "\\mathbf { I } _ { s y n } \\in \\dot { \\mathbb { R } } ^ { H \\times W \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 482, + 352, + 500 + ], + "score": 1.0, + "content": "continuously undergoes a conv layer", + "type": "text" + }, + { + "bbox": [ + 353, + 486, + 363, + 497 + ], + "score": 0.83, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 482, + 367, + 500 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 367, + 487, + 372, + 496 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 482, + 470, + 500 + ], + "score": 1.0, + "content": "SRGs, and a conv layer", + "type": "text" + }, + { + "bbox": [ + 471, + 486, + 481, + 497 + ], + "score": 0.87, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 482, + 507, + 500 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "score": 1.0, + "content": "adds with a long identity mapping for efficient residual learning to eventually generate the fake noisy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 504, + 369, + 522 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 154, + 522 + ], + "score": 1.0, + "content": "counterpart", + "type": "text" + }, + { + "bbox": [ + 155, + 507, + 222, + 520 + ], + "score": 0.92, + "content": "\\mathbf { I } _ { f n } \\in \\mathbb { R } ^ { H \\times W \\times \\dot { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 504, + 369, + 522 + ], + "score": 1.0, + "content": ". This process can be formulated as:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 522, + 438, + 537 + ], + "lines": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "spans": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "score": 0.87, + "content": "\\mathbf { I } _ { f n } = \\mathbf { I } _ { s y n } + f _ { 2 } ( S _ { t } ( \\mathbf { F } _ { S _ { t } } ) ) , ~ \\mathbf { F } _ { S _ { j + 1 } } = S _ { j } ( \\mathbf { F } _ { S _ { j } } ) , ~ \\mathbf { F } _ { S _ { 1 } } = f _ { 1 } ( \\mathbf { I } _ { f n } ) ,", + "type": "interline_equation", + "image_path": "069647ce1bbde87b2623ccb494d7fbd812afb4db119d1b0aed7d9d4502fa25ec.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 552 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 540, + 144, + 552 + ], + "score": 0.89, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 539, + 191, + 552 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 191, + 540, + 204, + 551 + ], + "score": 0.86, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 539, + 229, + 552 + ], + "score": 1.0, + "content": "SRG,", + "type": "text" + }, + { + "bbox": [ + 230, + 540, + 286, + 551 + ], + "score": 0.89, + "content": "1 \\leq j \\leq t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "score": 1.0, + "content": ". The components of SRG are depicted in Fig. 3 (b). We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 547, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 226, + 566 + ], + "score": 1.0, + "content": "define the input feature of the", + "type": "text" + }, + { + "bbox": [ + 226, + 552, + 240, + 563 + ], + "score": 0.87, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 547, + 273, + 566 + ], + "score": 1.0, + "content": "SRG as", + "type": "text" + }, + { + "bbox": [ + 274, + 550, + 345, + 564 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { S _ { j } } \\in \\mathbb { R } ^ { H \\times W \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 547, + 419, + 566 + ], + "score": 1.0, + "content": "and its channel as", + "type": "text" + }, + { + "bbox": [ + 419, + 552, + 428, + 562 + ], + "score": 0.57, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 547, + 432, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 433, + 552, + 450, + 564 + ], + "score": 0.69, + "content": "\\mathbf { F } _ { S _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 547, + 506, + 566 + ], + "score": 1.0, + "content": "continuously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 561, + 442, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 202, + 576 + ], + "score": 1.0, + "content": "undergoes a conv layer,", + "type": "text" + }, + { + "bbox": [ + 202, + 564, + 209, + 572 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 561, + 442, + 576 + ], + "score": 1.0, + "content": "MABs, and a conv layer to add with an identity mapping:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 577, + 439, + 596 + ], + "lines": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "spans": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "score": 0.9, + "content": "\\mathbf { F } _ { S _ { j + 1 } } = \\mathbf { F } _ { S _ { j } } + M _ { n } ^ { j } ( \\mathbf { F } _ { M _ { n } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { k + 1 } ^ { j } } = M _ { k } ^ { j } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { 1 } ^ { j } } = \\mathbf { F } _ { S _ { j } } ,", + "type": "interline_equation", + "image_path": "c5b7abf8dbad70d78cac6802ec6026212e4617d92566dc729bd00da13bf4499b.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 134, + 615 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 600, + 150, + 614 + ], + "score": 0.92, + "content": "M _ { k } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 203, + 615 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 203, + 602, + 218, + 613 + ], + "score": 0.89, + "content": "k _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 600, + 273, + 615 + ], + "score": 1.0, + "content": "MAB of the", + "type": "text" + }, + { + "bbox": [ + 274, + 602, + 287, + 613 + ], + "score": 0.88, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 600, + 315, + 615 + ], + "score": 1.0, + "content": "SRG,", + "type": "text" + }, + { + "bbox": [ + 316, + 601, + 385, + 613 + ], + "score": 0.91, + "content": "1 \\leq k \\leq n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 600, + 506, + 615 + ], + "score": 1.0, + "content": ". MAB is the basic building", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "block and the most significant component of SMNet. The details of MAB are depicted in Fig. 3 (c).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "We customize MAB with the following motivations: (i) Multi-scale feature fusion can increase the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "receptive field and multi-resolution contextual information can cover rich auto-correlation, which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "provides more sufficient spatial representations for noise fitting. (ii) The noise level decreases as the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "scale increases and nonlinear sampling operations can increase the richness of the mapping in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "potential space of real noise. Therefore, we exploit parallel multi-resolution branch aggregation from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "top to bottom and bottom to top to facilitate the learning of complex real noise. (iii) Specifically,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "during the feature downsampling, general downsample operation damages the image information,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "resulting in pixel discontinuity and jagged artifact. 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(a) is the architecture of SMNet. (b) depicts the components of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 192, + 475, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 475, + 204 + ], + "score": 1.0, + "content": "SRG. (c) shows the details of MAB. MAB is equipped with FCA, which is illustrated in (d).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 258, + 224 + ], + "score": 1.0, + "content": "During the training procedure, we fix", + "type": "text" + }, + { + "bbox": [ + 258, + 211, + 268, + 220 + ], + "score": 0.72, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 208, + 300, + 224 + ], + "score": 1.0, + "content": "to train", + "type": "text" + }, + { + "bbox": [ + 301, + 211, + 310, + 221 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 208, + 341, + 224 + ], + "score": 1.0, + "content": "and fix", + "type": "text" + }, + { + "bbox": [ + 341, + 210, + 351, + 221 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 208, + 382, + 224 + ], + "score": 1.0, + "content": "to train", + "type": "text" + }, + { + "bbox": [ + 383, + 211, + 393, + 220 + ], + "score": 0.72, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 208, + 489, + 224 + ], + "score": 1.0, + "content": "iteratively. Minimizing", + "type": "text" + }, + { + "bbox": [ + 489, + 210, + 504, + 222 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { G }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 123, + 233 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 222, + 139, + 232 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 221, + 336, + 233 + ], + "score": 1.0, + "content": "alternately allows us to train a generative model", + "type": "text" + }, + { + "bbox": [ + 336, + 222, + 345, + 231 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "with the goal of fooling the pixel-level", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 162, + 245 + ], + "score": 1.0, + "content": "discriminator", + "type": "text" + }, + { + "bbox": [ + 163, + 232, + 172, + 242 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "that is trained to distinguish fake noisy images from real noisy images. This pixel-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 339, + 256 + ], + "score": 1.0, + "content": "level noise-aware adversarial training scheme encourages", + "type": "text" + }, + { + "bbox": [ + 340, + 243, + 349, + 253 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 242, + 506, + 256 + ], + "score": 1.0, + "content": "to favor perceptually natural solutions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 485, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 485, + 267 + ], + "score": 1.0, + "content": "that reside on the manifold of real noisy images so as to construct the noise domain alignment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 208, + 506, + 267 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 272, + 235, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 236, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 236, + 288 + ], + "score": 1.0, + "content": "2.3 Noisy Image Generating", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 325, + 302 + ], + "score": 1.0, + "content": "In Sec. 2.1, we denote the probability distribution of", + "type": "text" + }, + { + "bbox": [ + 326, + 289, + 350, + 302 + ], + "score": 0.92, + "content": "\\mathbf { I } _ { f n } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 289, + 363, + 302 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 364, + 289, + 410, + 301 + ], + "score": 0.91, + "content": "P _ { G } ( x _ { i } ; \\theta _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 289, + 506, + 302 + ], + "score": 1.0, + "content": ". Now we customize a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 323, + 313 + ], + "score": 1.0, + "content": "light-weight yet efficient CNN architecture, SMNet as", + "type": "text" + }, + { + "bbox": [ + 323, + 301, + 332, + 310 + ], + "score": 0.79, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 300, + 379, + 313 + ], + "score": 1.0, + "content": "to generate", + "type": "text" + }, + { + "bbox": [ + 380, + 301, + 426, + 312 + ], + "score": 0.92, + "content": "P _ { G } ( x _ { i } ; \\theta _ { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 300, + 505, + 313 + ], + "score": 1.0, + "content": ". In this section, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 311, + 465, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 250, + 324 + ], + "score": 1.0, + "content": "firstly introduce the input setting of", + "type": "text" + }, + { + "bbox": [ + 250, + 312, + 259, + 321 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 311, + 465, + 324 + ], + "score": 1.0, + "content": "and subsequently detail the architecture of SMNet.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 289, + 506, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "Input Setting. We aim to generate a realistic noisy image from its clean counterpart. A naive setting", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "is to directly adopt the clean image as the input to generate the noisy image. However, this naive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "setting is not in line with the fact. When we repeatedly feed the same clean image to a pre-trained", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 107, + 361, + 115, + 371 + ], + "score": 0.51, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 361, + 119, + 373 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 119, + 361, + 129, + 371 + ], + "score": 0.63, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "outputs completely the same noisy images. In contrast, when taking multiple pictures in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "real world, the real noisy images vary a lot in the intensity of each pixel. This is caused by many", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "factors (e.g., photon sensing noise, ISP pipelines, and illumination conditions). Hence, the naive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "input setting containing no distribution is unreasonable. We review that the general GANs sample", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "from an initial random distribution (usually Gaussian) to generate a fake image. Hence, the input of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 414, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 107, + 415, + 116, + 425 + ], + "score": 0.75, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 414, + 507, + 428 + ], + "score": 1.0, + "content": "should contain a random distribution so as to generate multiple noisy images of the same scene.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "We note that the two common synthetic settings meet this condition. Therefore, we utilize the two", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 437, + 503, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 494, + 450 + ], + "score": 1.0, + "content": "common settings to produce the synthetic image and then adopt the synthetic image as the input of", + "type": "text" + }, + { + "bbox": [ + 495, + 438, + 503, + 447 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 507, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 507, + 461 + ], + "score": 1.0, + "content": "Subsequently, we propose a light-weight yet efficient architecture, SMNet for better real noise fitting.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 327, + 507, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 464, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 468, + 476 + ], + "score": 1.0, + "content": "SMNet Architecture. The architecture of SMNet is shown in Fig. 3 (a). SMNet involves", + "type": "text" + }, + { + "bbox": [ + 468, + 465, + 473, + 474 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "Simple", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 293, + 487 + ], + "score": 1.0, + "content": "Residual Groups (SRG) and each SRG contains", + "type": "text" + }, + { + "bbox": [ + 294, + 477, + 301, + 485 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "Multi-scale Attention Blocks (MAB). The synthetic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 482, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 103, + 482, + 130, + 500 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 130, + 485, + 201, + 498 + ], + "score": 0.91, + "content": "\\mathbf { I } _ { s y n } \\in \\dot { \\mathbb { R } } ^ { H \\times W \\times 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 482, + 352, + 500 + ], + "score": 1.0, + "content": "continuously undergoes a conv layer", + "type": "text" + }, + { + "bbox": [ + 353, + 486, + 363, + 497 + ], + "score": 0.83, + "content": "f _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 482, + 367, + 500 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 367, + 487, + 372, + 496 + ], + "score": 0.6, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 482, + 470, + 500 + ], + "score": 1.0, + "content": "SRGs, and a conv layer", + "type": "text" + }, + { + "bbox": [ + 471, + 486, + 481, + 497 + ], + "score": 0.87, + "content": "f _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 482, + 507, + 500 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "score": 1.0, + "content": "adds with a long identity mapping for efficient residual learning to eventually generate the fake noisy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 504, + 369, + 522 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 154, + 522 + ], + "score": 1.0, + "content": "counterpart", + "type": "text" + }, + { + "bbox": [ + 155, + 507, + 222, + 520 + ], + "score": 0.92, + "content": "\\mathbf { I } _ { f n } \\in \\mathbb { R } ^ { H \\times W \\times \\dot { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 504, + 369, + 522 + ], + "score": 1.0, + "content": ". This process can be formulated as:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 103, + 463, + 507, + 522 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 522, + 438, + 537 + ], + "lines": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "spans": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "score": 0.87, + "content": "\\mathbf { I } _ { f n } = \\mathbf { I } _ { s y n } + f _ { 2 } ( S _ { t } ( \\mathbf { F } _ { S _ { t } } ) ) , ~ \\mathbf { F } _ { S _ { j + 1 } } = S _ { j } ( \\mathbf { F } _ { S _ { j } } ) , ~ \\mathbf { F } _ { S _ { 1 } } = f _ { 1 } ( \\mathbf { I } _ { f n } ) ,", + "type": "interline_equation", + "image_path": "069647ce1bbde87b2623ccb494d7fbd812afb4db119d1b0aed7d9d4502fa25ec.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 172, + 522, + 438, + 537 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 552 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 540, + 144, + 552 + ], + "score": 0.89, + "content": "S _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 539, + 191, + 552 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 191, + 540, + 204, + 551 + ], + "score": 0.86, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 539, + 229, + 552 + ], + "score": 1.0, + "content": "SRG,", + "type": "text" + }, + { + "bbox": [ + 230, + 540, + 286, + 551 + ], + "score": 0.89, + "content": "1 \\leq j \\leq t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 539, + 505, + 552 + ], + "score": 1.0, + "content": ". The components of SRG are depicted in Fig. 3 (b). We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 547, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 226, + 566 + ], + "score": 1.0, + "content": "define the input feature of the", + "type": "text" + }, + { + "bbox": [ + 226, + 552, + 240, + 563 + ], + "score": 0.87, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 547, + 273, + 566 + ], + "score": 1.0, + "content": "SRG as", + "type": "text" + }, + { + "bbox": [ + 274, + 550, + 345, + 564 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { S _ { j } } \\in \\mathbb { R } ^ { H \\times W \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 547, + 419, + 566 + ], + "score": 1.0, + "content": "and its channel as", + "type": "text" + }, + { + "bbox": [ + 419, + 552, + 428, + 562 + ], + "score": 0.57, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 547, + 432, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 433, + 552, + 450, + 564 + ], + "score": 0.69, + "content": "\\mathbf { F } _ { S _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 547, + 506, + 566 + ], + "score": 1.0, + "content": "continuously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 561, + 442, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 202, + 576 + ], + "score": 1.0, + "content": "undergoes a conv layer,", + "type": "text" + }, + { + "bbox": [ + 202, + 564, + 209, + 572 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 561, + 442, + 576 + ], + "score": 1.0, + "content": "MABs, and a conv layer to add with an identity mapping:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 539, + 506, + 576 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 577, + 439, + 596 + ], + "lines": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "spans": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "score": 0.9, + "content": "\\mathbf { F } _ { S _ { j + 1 } } = \\mathbf { F } _ { S _ { j } } + M _ { n } ^ { j } ( \\mathbf { F } _ { M _ { n } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { k + 1 } ^ { j } } = M _ { k } ^ { j } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) , ~ \\mathbf { F } _ { M _ { 1 } ^ { j } } = \\mathbf { F } _ { S _ { j } } ,", + "type": "interline_equation", + "image_path": "c5b7abf8dbad70d78cac6802ec6026212e4617d92566dc729bd00da13bf4499b.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 172, + 577, + 439, + 596 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 134, + 615 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 600, + 150, + 614 + ], + "score": 0.92, + "content": "M _ { k } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 203, + 615 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 203, + 602, + 218, + 613 + ], + "score": 0.89, + "content": "k _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 600, + 273, + 615 + ], + "score": 1.0, + "content": "MAB of the", + "type": "text" + }, + { + "bbox": [ + 274, + 602, + 287, + 613 + ], + "score": 0.88, + "content": "j _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 600, + 315, + 615 + ], + "score": 1.0, + "content": "SRG,", + "type": "text" + }, + { + "bbox": [ + 316, + 601, + 385, + 613 + ], + "score": 0.91, + "content": "1 \\leq k \\leq n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 600, + 506, + 615 + ], + "score": 1.0, + "content": ". MAB is the basic building", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "block and the most significant component of SMNet. The details of MAB are depicted in Fig. 3 (c).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "We customize MAB with the following motivations: (i) Multi-scale feature fusion can increase the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "receptive field and multi-resolution contextual information can cover rich auto-correlation, which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "provides more sufficient spatial representations for noise fitting. (ii) The noise level decreases as the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "scale increases and nonlinear sampling operations can increase the richness of the mapping in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "potential space of real noise. Therefore, we exploit parallel multi-resolution branch aggregation from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "top to bottom and bottom to top to facilitate the learning of complex real noise. (iii) Specifically,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "during the feature downsampling, general downsample operation damages the image information,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "resulting in pixel discontinuity and jagged artifact. To alleviate these issues, we exploit Shift-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "Invariant Downsample [46] that copes with the discontinuity by using continuous pooling and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "filtering operation, preserving rich cross-correlation information between original and downsampled", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "images. (iv) To efficiently capture continuous channel correlation and avoid information loss, we use", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "the 1D channel attention module, Fast Channel Attention (FCA) instead of the general 2D convolution", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 102, + 247, + 502, + 264 + ], + "spans": [ + { + "bbox": [ + 102, + 247, + 253, + 263 + ], + "score": 1.0, + "content": "attention module. 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MethodsSIDD[47]DND [48]MethodsPolyU [40]Nam [49]
PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑
DnCNN-B[10]23.660.58332.430.790RDN[12]37.940.94638.160.956
CBDNet [50]33.280.86838.060.942FFDNet+ [19]38.170.95138.810.957
RIDNet [44]38.710.91439.260.953TWSC [51]38.680.95838.960.962
AINDNet [52]39.150.95539.530.956CBDNet [50]38.740.96139.080.969
VDN[53]39.230.95539.380.952RIDNet [44]38.860.96239.200.973
CycleISP [22]39.520.95739.560.956VDN [53]39.040.96539.680.976
MPRNet [54]39.710.95839.800.954MPRNet [54]39.070.96939.410.974
MIRNet [55]39.720.95939.880.956MIRNet [55]39.180.97339.570.979
RIDNet*(Ours)39.250.95639.550.955RIDNet*(Ours)39.540.97139.690.975
MPRNet*(Ours)40.060.96040.180.961MPRNet*(Ours)40.480.98240.720.984
MIRNet*(Ours)40.070.96040.250.962MIRNet*(Ours)40.550.98340.780.986
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The input feature,", + "type": "text" + }, + { + "bbox": [ + 253, + 248, + 327, + 264 + ], + "score": 0.93, + "content": "\\mathbf { F } _ { M _ { k } ^ { j } } \\in \\mathbb { R } ^ { H \\times W \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 247, + 502, + 263 + ], + "score": 1.0, + "content": "is fed into three parallel multi-scale paths:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 269, + 484, + 288 + ], + "lines": [ + { + "bbox": [ + 113, + 269, + 484, + 288 + ], + "spans": [ + { + "bbox": [ + 113, + 269, + 484, + 288 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { M _ { k } ^ { j } } ^ { 1 } = F C A ( \\mathbf { F } _ { M _ { k } ^ { j } } ) , \\mathbf { F } _ { M _ { k } ^ { j } } ^ { 2 } = f _ { u p } ^ { 2 } ( F C A ( f _ { s i d } ^ { 2 } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) ) ) , \\mathbf { F } _ { M _ { k } ^ { j } } ^ { 4 } = f _ { u p } ^ { 4 } ( F C A ( f _ { s i d } ^ { 4 } ( \\mathbf { F } _ { M _ { k } ^ { j } } ) ) ) ,", + "type": "interline_equation", + "image_path": "d8812e238931f775aec826b5b35a9ff073436a3479f38b62179c4c2c90adcbd2.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 113, + 269, + 484, + 288 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 506, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 507, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 132, + 308 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 294, + 157, + 304 + ], + "score": 0.81, + "content": "F C A", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 291, + 287, + 308 + ], + "score": 1.0, + "content": "denotes Fast Channel Attention.", + "type": "text" + }, + { + "bbox": [ + 288, + 294, + 302, + 308 + ], + "score": 0.91, + "content": "f _ { u p } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 291, + 507, + 308 + ], + "score": 1.0, + "content": "denotes a conv layer after bilinear interpolation up-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 303, + 509, + 325 + ], + "spans": [ + { + "bbox": [ + 103, + 303, + 228, + 325 + ], + "score": 1.0, + "content": "sampling, 2 is the scale factor.", + "type": "text" + }, + { + "bbox": [ + 228, + 306, + 243, + 321 + ], + "score": 0.91, + "content": "f _ { u p } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 303, + 325, + 325 + ], + "score": 1.0, + "content": "is similarly defined.", + "type": "text" + }, + { + "bbox": [ + 325, + 306, + 343, + 320 + ], + "score": 0.91, + "content": "f _ { s i d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 303, + 509, + 325 + ], + "score": 1.0, + "content": "means Shift-Invariant Downsample [46],", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 317, + 496, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 207, + 335 + ], + "score": 1.0, + "content": "2 is also the scale factor.", + "type": "text" + }, + { + "bbox": [ + 207, + 320, + 224, + 333 + ], + "score": 0.91, + "content": "f _ { s i d } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 317, + 496, + 335 + ], + "score": 1.0, + "content": "is similarly defined. Subsequently, the output feature is derived by:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 338, + 395, + 357 + ], + "lines": [ + { + "bbox": [ + 215, + 338, + 395, + 357 + ], + "spans": [ + { + "bbox": [ + 215, + 338, + 395, + 357 + ], + "score": 0.93, + "content": "M _ { k } ^ { j } ( { \\bf F } _ { M _ { k } ^ { j } } ) = { \\bf F } _ { M _ { k } ^ { j } } + f ( [ { \\bf F } _ { M _ { k } ^ { j } } ^ { 1 } , { \\bf F } _ { M _ { k } ^ { j } } ^ { 2 } , { \\bf F } _ { M _ { k } ^ { j } } ^ { 4 } ] ) ,", + "type": "interline_equation", + "image_path": "ee5e80a066375b4407e727a95299e0ee5e9766d31d5b3a445682b5aac7c8294f.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 215, + 338, + 395, + 357 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 504, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 132, + 375 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 363, + 140, + 374 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 362, + 258, + 375 + ], + "score": 1.0, + "content": "represents the last conv layer,", + "type": "text" + }, + { + "bbox": [ + 258, + 362, + 281, + 374 + ], + "score": 0.9, + "content": "[ \\cdot , \\cdot , \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "denotes the concatenating operation. The architecture of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 373, + 486, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 339, + 386 + ], + "score": 1.0, + "content": "FCA is shown in Fig. 3 (d). We define the input feature as", + "type": "text" + }, + { + "bbox": [ + 340, + 374, + 353, + 384 + ], + "score": 0.9, + "content": "\\mathbf { F } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 373, + 486, + 386 + ], + "score": 1.0, + "content": ", then FCA can be formulated as:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 389, + 402, + 405 + ], + "lines": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "spans": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "score": 0.92, + "content": "F C A ( \\mathbf { F } _ { d } ) = \\mathbf { F } _ { d } \\cdot \\big ( 1 + \\sigma \\big ( f _ { 1 D C } ( G A P ( \\mathbf { F } _ { d } ) ) \\big ) \\big ) ,", + "type": "interline_equation", + "image_path": "469e619d5b1cf1e699bcf95e36f2bda94fb54c969c72938e0a1b62f47b8431c3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 504, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 133, + 422 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 412, + 140, + 419 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 409, + 317, + 422 + ], + "score": 1.0, + "content": "represents the Sigmoid activation function,", + "type": "text" + }, + { + "bbox": [ + 317, + 410, + 342, + 420 + ], + "score": 0.85, + "content": "G A P", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "means global average pooling along the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 420, + 502, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 158, + 432 + ], + "score": 1.0, + "content": "spatial wise,", + "type": "text" + }, + { + "bbox": [ + 158, + 421, + 181, + 432 + ], + "score": 0.91, + "content": "f _ { 1 D C }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 420, + 402, + 432 + ], + "score": 1.0, + "content": "denotes 1-Dimension Convolution. In this work, we set", + "type": "text" + }, + { + "bbox": [ + 402, + 421, + 422, + 430 + ], + "score": 0.87, + "content": "t = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 420, + 425, + 432 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 426, + 421, + 448, + 430 + ], + "score": 0.9, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 420, + 469, + 432 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 469, + 420, + 498, + 430 + ], + "score": 0.9, + "content": "C = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 420, + 502, + 432 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 438, + 246, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 246, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 246, + 453 + ], + "score": 1.0, + "content": "2.4 Overall Training Objective", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "In addition to the aforementioned losses, we employ a perceptual loss function that assesses a solution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "with respect to perceptually relevant characteristics (e.g., the structural contents and detailed textures):", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 481, + 450, + 499 + ], + "lines": [ + { + "bbox": [ + 160, + 481, + 450, + 499 + ], + "spans": [ + { + "bbox": [ + 160, + 481, + 450, + 499 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { L } _ { p } = \\left| \\left| V G G ( \\mathbf { I } _ { f d } ) - V G G ( \\mathbf { I } _ { r d } ) \\right| \\right| _ { 2 } ^ { 2 } , \\mathbf { I } _ { f d } = D _ { d } ( \\mathbf { I } _ { f n } ) , \\mathbf { I } _ { r d } = D _ { d } ( \\mathbf { I } _ { r n } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "0e3375d8a7fdfd967303cf8a05dadf39341d41f514611e29b5b02ba3a4cd8e5e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 160, + 481, + 450, + 499 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 483, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 486, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 133, + 517 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 503, + 158, + 513 + ], + "score": 0.37, + "content": "V G G", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 500, + 486, + 517 + ], + "score": 1.0, + "content": "denotes the last feature map of VGG16 [56]. Eventually, the training objective is:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 520, + 386, + 534 + ], + "lines": [ + { + "bbox": [ + 224, + 520, + 386, + 534 + ], + "spans": [ + { + "bbox": [ + 224, + 520, + 386, + 534 + ], + "score": 0.93, + "content": "\\mathcal { L } = \\mathcal { L } _ { 1 } + \\lambda _ { p } \\cdot \\mathcal { L } _ { p } + \\lambda _ { R a } \\cdot ( \\mathcal { L } _ { D } + \\mathcal { L } _ { G } ) ,", + "type": "interline_equation", + "image_path": "2e43ce87f7045384769711d55504ff3ea8d3c8cf30ea07c2bda90f00834cc834.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 224, + 520, + 386, + 534 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 134, + 552 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 540, + 146, + 551 + ], + "score": 0.88, + "content": "\\lambda _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 538, + 165, + 552 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 540, + 184, + 550 + ], + "score": 0.9, + "content": "\\lambda _ { R a }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "are two hyper-parameters controlling the importance balance. The proposed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 343, + 562 + ], + "score": 1.0, + "content": "PNGAN framework is end-to-end trained by minimizing", + "type": "text" + }, + { + "bbox": [ + 343, + 550, + 351, + 560 + ], + "score": 0.76, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 549, + 472, + 562 + ], + "score": 1.0, + "content": ". Note that the parameters in", + "type": "text" + }, + { + "bbox": [ + 472, + 550, + 486, + 561 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 456, + 573 + ], + "score": 1.0, + "content": "VGG16 are fixed. Each mini-batch training procedure is divided into two steps: (i) Fix", + "type": "text" + }, + { + "bbox": [ + 456, + 561, + 466, + 571 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "and train", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 572, + 116, + 582 + ], + "score": 0.74, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 571, + 152, + 584 + ], + "score": 1.0, + "content": ". 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This pixel-level adversarial training scheme promotes", + "type": "text" + }, + { + "bbox": [ + 439, + 572, + 449, + 582 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "the ability to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 370, + 595 + ], + "score": 1.0, + "content": "distinguish fake noisy images from real noisy images and allows", + "type": "text" + }, + { + "bbox": [ + 371, + 583, + 380, + 592 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "to learn to create the solutions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 592, + 468, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 454, + 606 + ], + "score": 1.0, + "content": "that are highly similar to real camera noisy images and thus difficult to be classified by", + "type": "text" + }, + { + "bbox": [ + 455, + 594, + 464, + 603 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 592, + 468, + 606 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 187, + 625 + ], + "lines": [ + { + "bbox": [ + 103, + 609, + 189, + 629 + ], + "spans": [ + { + "bbox": [ + 103, + 609, + 189, + 629 + ], + "score": 1.0, + "content": "3 Experiment", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 631, + 208, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 209, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 209, + 646 + ], + "score": 1.0, + "content": "3.1 Experiment Setup", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 316, + 657 + ], + "score": 1.0, + "content": "Datasets. We first use SIDD [47] train set to train", + "type": "text" + }, + { + "bbox": [ + 316, + 646, + 330, + 656 + ], + "score": 0.88, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 645, + 384, + 657 + ], + "score": 1.0, + "content": ". Then we fix", + "type": "text" + }, + { + "bbox": [ + 385, + 646, + 399, + 656 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 645, + 430, + 657 + ], + "score": 1.0, + "content": "to train", + "type": "text" + }, + { + "bbox": [ + 431, + 646, + 440, + 655 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "on the same set.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 165, + 669 + ], + "score": 1.0, + "content": "Subsequently,", + "type": "text" + }, + { + "bbox": [ + 166, + 657, + 175, + 666 + ], + "score": 0.75, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "uses clean images from DIV2K [57], Flickr2K [58], BSD68 [59], Kodak24 [60],", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "and Urban100 [61] to generate realistic noisy-clean image pairs. We use the generated data and SIDD", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 678, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 689 + ], + "score": 1.0, + "content": "train set jointly to finetune real denoisers and evaluate them on four real denoising benchmarks:", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "SIDD [47], DND [48], PolyU [40], and Nam [49]. The images in SIDD [47] are collected using five", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "smartphone cameras in 10 static scenes. 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DND [48] composes 50 noisy-clean image pairs captured by 4 consumer", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 71, + 505, + 193 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 71, + 505, + 193 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 71, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 71, + 505, + 193 + ], + "score": 0.984, + "html": "
MethodsSIDD[47]DND [48]MethodsPolyU [40]Nam [49]
PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑
DnCNN-B[10]23.660.58332.430.790RDN[12]37.940.94638.160.956
CBDNet [50]33.280.86838.060.942FFDNet+ [19]38.170.95138.810.957
RIDNet [44]38.710.91439.260.953TWSC [51]38.680.95838.960.962
AINDNet [52]39.150.95539.530.956CBDNet [50]38.740.96139.080.969
VDN[53]39.230.95539.380.952RIDNet [44]38.860.96239.200.973
CycleISP [22]39.520.95739.560.956VDN [53]39.040.96539.680.976
MPRNet [54]39.710.95839.800.954MPRNet [54]39.070.96939.410.974
MIRNet [55]39.720.95939.880.956MIRNet [55]39.180.97339.570.979
RIDNet*(Ours)39.250.95639.550.955RIDNet*(Ours)39.540.97139.690.975
MPRNet*(Ours)40.060.96040.180.961MPRNet*(Ours)40.480.98240.720.984
MIRNet*(Ours)40.070.96040.250.962MIRNet*(Ours)40.550.98340.780.986
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We define the input feature as", + "type": "text" + }, + { + "bbox": [ + 340, + 374, + 353, + 384 + ], + "score": 0.9, + "content": "\\mathbf { F } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 373, + 486, + 386 + ], + "score": 1.0, + "content": ", then FCA can be formulated as:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 106, + 362, + 506, + 386 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 389, + 402, + 405 + ], + "lines": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "spans": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "score": 0.92, + "content": "F C A ( \\mathbf { F } _ { d } ) = \\mathbf { F } _ { d } \\cdot \\big ( 1 + \\sigma \\big ( f _ { 1 D C } ( G A P ( \\mathbf { F } _ { d } ) ) \\big ) \\big ) ,", + "type": "interline_equation", + "image_path": "469e619d5b1cf1e699bcf95e36f2bda94fb54c969c72938e0a1b62f47b8431c3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 208, + 389, + 402, + 405 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 504, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 133, + 422 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 412, + 140, + 419 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 409, + 317, + 422 + ], + "score": 1.0, + "content": "represents the Sigmoid activation function,", + "type": "text" + }, + { + "bbox": [ + 317, + 410, + 342, + 420 + ], + "score": 0.85, + "content": "G A P", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "means global average pooling along the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 420, + 502, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 158, + 432 + ], + "score": 1.0, + "content": "spatial wise,", + "type": "text" + }, + { + "bbox": [ + 158, + 421, + 181, + 432 + ], + "score": 0.91, + "content": "f _ { 1 D C }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 420, + 402, + 432 + ], + "score": 1.0, + "content": "denotes 1-Dimension Convolution. 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The proposed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 343, + 562 + ], + "score": 1.0, + "content": "PNGAN framework is end-to-end trained by minimizing", + "type": "text" + }, + { + "bbox": [ + 343, + 550, + 351, + 560 + ], + "score": 0.76, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 549, + 472, + 562 + ], + "score": 1.0, + "content": ". Note that the parameters in", + "type": "text" + }, + { + "bbox": [ + 472, + 550, + 486, + 561 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 456, + 573 + ], + "score": 1.0, + "content": "VGG16 are fixed. Each mini-batch training procedure is divided into two steps: (i) Fix", + "type": "text" + }, + { + "bbox": [ + 456, + 561, + 466, + 571 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "and train", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 107, + 572, + 116, + 582 + ], + "score": 0.74, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 571, + 152, + 584 + ], + "score": 1.0, + "content": ". 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This pixel-level adversarial training scheme promotes", + "type": "text" + }, + { + "bbox": [ + 439, + 572, + 449, + 582 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "the ability to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 370, + 595 + ], + "score": 1.0, + "content": "distinguish fake noisy images from real noisy images and allows", + "type": "text" + }, + { + "bbox": [ + 371, + 583, + 380, + 592 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "to learn to create the solutions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 592, + 468, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 454, + 606 + ], + "score": 1.0, + "content": "that are highly similar to real camera noisy images and thus difficult to be classified by", + "type": "text" + }, + { + "bbox": [ + 455, + 594, + 464, + 603 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 592, + 468, + 606 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 538, + 506, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 187, + 625 + ], + "lines": [ + { + "bbox": [ + 103, + 609, + 189, + 629 + ], + "spans": [ + { + "bbox": [ + 103, + 609, + 189, + 629 + ], + "score": 1.0, + "content": "3 Experiment", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 631, + 208, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 209, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 209, + 646 + ], + "score": 1.0, + "content": "3.1 Experiment Setup", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 316, + 657 + ], + "score": 1.0, + "content": "Datasets. We first use SIDD [47] train set to train", + "type": "text" + }, + { + "bbox": [ + 316, + 646, + 330, + 656 + ], + "score": 0.88, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 645, + 384, + 657 + ], + "score": 1.0, + "content": ". Then we fix", + "type": "text" + }, + { + "bbox": [ + 385, + 646, + 399, + 656 + ], + "score": 0.89, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 645, + 430, + 657 + ], + "score": 1.0, + "content": "to train", + "type": "text" + }, + { + "bbox": [ + 431, + 646, + 440, + 655 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "on the same set.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 655, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 165, + 669 + ], + "score": 1.0, + "content": "Subsequently,", + "type": "text" + }, + { + "bbox": [ + 166, + 657, + 175, + 666 + ], + "score": 0.75, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 655, + 507, + 669 + ], + "score": 1.0, + "content": "uses clean images from DIV2K [57], Flickr2K [58], BSD68 [59], Kodak24 [60],", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "and Urban100 [61] to generate realistic noisy-clean image pairs. 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We use the metric, Maximum Mean Discrepancy (MMD) to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "measure the domain discrepancy between synthetic and real noisy datasets, PNGAN generating and real noisy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 473, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 473, + 189 + ], + "score": 1.0, + "content": "datasets. Under both setting1 and 2, the discrepancy decreases significantly when PNGAN is applied.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 107, + 192, + 504, + 247 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 192, + 504, + 247 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 192, + 504, + 247 + ], + "spans": [ + { + "bbox": [ + 107, + 192, + 504, + 247 + ], + "score": 0.973, + "html": "
MethodsSIDD [47]S1DF2K[57,58]
S1 +PNGANS2S2+PNGANRealS1 + PNGANS2S2+PNGAN
RIDNet22.5537.92 (+15.37)36.1338.71 (+2.58)38.6922.5532.10(+9.55)33.9838.14 (+4.16)
MPRNet22.8638.52 (+15.66)36.5239.53 (+3.01)39.4522.8532.82(+9.97)34.1938.61 (+4.42)
MIRNet22.8338.76 (+15.93)36.5539.57 (+3.02)39.5823.0832.34(+9.26)34.2638.72 (+4.46)
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PSNR is reported.", + "type": "text" + }, + { + "bbox": [ + 401, + 253, + 426, + 263 + ], + "score": 0.79, + "content": "^ { \\mathrm { ~ S 1 , 2 = } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 250, + 500, + 265 + ], + "score": 1.0, + "content": "synthetic setting1,2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 268, + 503, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 228, + 281 + ], + "score": 1.0, + "content": "cameras. 1,000 patches at size", + "type": "text" + }, + { + "bbox": [ + 228, + 269, + 267, + 279 + ], + "score": 0.88, + "content": "5 1 2 \\times 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "are cropped from the collected images. PolyU [40] consists", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 490, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 490, + 292 + ], + "score": 1.0, + "content": "of 40 real camera noisy images. Nam [49] is composed of real noisy images of 11 static scenes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 324, + 308 + ], + "score": 1.0, + "content": "Implementation Details. We set the hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 325, + 295, + 380, + 308 + ], + "score": 0.91, + "content": "\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 295, + 383, + 308 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 384, + 295, + 446, + 307 + ], + "score": 0.92, + "content": "\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 295, + 505, + 308 + ], + "score": 1.0, + "content": ". For synthetic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 252, + 319 + ], + "score": 1.0, + "content": "setting1, we set the noise intensity,", + "type": "text" + }, + { + "bbox": [ + 252, + 307, + 286, + 318 + ], + "score": 0.91, + "content": "\\sigma _ { n } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 306, + 505, + 319 + ], + "score": 1.0, + "content": ". For synthetic setting2, we directly exploit CycleISP", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 352, + 330 + ], + "score": 1.0, + "content": "to generate the synthetic noisy input. All the sub-modules", + "type": "text" + }, + { + "bbox": [ + 353, + 318, + 384, + 329 + ], + "score": 0.76, + "content": "( D _ { d } , G", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 317, + 406, + 330 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 407, + 318, + 416, + 328 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 317, + 506, + 330 + ], + "score": 1.0, + "content": ") are trained with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 198, + 342 + ], + "score": 1.0, + "content": "Adam [62] optimizer", + "type": "text" + }, + { + "bbox": [ + 199, + 329, + 238, + 340 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 327, + 257, + 342 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 329, + 313, + 340 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = 0 . 9 9 9 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 327, + 330, + 342 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 330, + 328, + 359, + 339 + ], + "score": 0.95, + "content": "7 \\times 1 0 ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "iterations. The initial learning rate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 338, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 140, + 353 + ], + "score": 1.0, + "content": "is set to", + "type": "text" + }, + { + "bbox": [ + 140, + 339, + 174, + 350 + ], + "score": 0.91, + "content": "2 \\times 1 \\bar { 0 } ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 338, + 506, + 353 + ], + "score": 1.0, + "content": ". The cosine annealing strategy [63] is employed to steadily decrease the learning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 222, + 363 + ], + "score": 1.0, + "content": "rate from the initial value to", + "type": "text" + }, + { + "bbox": [ + 222, + 350, + 244, + 361 + ], + "score": 0.9, + "content": "1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 350, + 430, + 363 + ], + "score": 1.0, + "content": "during the training procedure. Patches at size", + "type": "text" + }, + { + "bbox": [ + 430, + 351, + 469, + 361 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "cropped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "from training images are fed into the models. The batch size is set as 8. The horizontal and vertical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "flips are performed for data augmentation. All the models are trained on RTX8000 GPUs. In the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 450, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 277, + 396 + ], + "score": 1.0, + "content": "finetuning phase, the learning rate is set to", + "type": "text" + }, + { + "bbox": [ + 277, + 383, + 311, + 394 + ], + "score": 0.91, + "content": "1 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 381, + 450, + 396 + ], + "score": 1.0, + "content": ", other settings remain unchanged.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 401, + 218, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 218, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 218, + 414 + ], + "score": 1.0, + "content": "3.2 Quantitative Results", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Domain Discrepancy Validation. We use the widely applied metric, Maximum Mean Discrepancy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "(MMD) [64] to measure the domain discrepancy between synthetic and real-world noisy images,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "PNGAN generating, and real noisy images on four real noisy benchmarks. For DND, we derive a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "pseudo clean version by denoising the real noisy counterparts with a pre-trained MIRNet [55]. Then", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "we use the pseudo clean version to synthesize noisy images. The results are depicted as a histogram", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 354, + 483 + ], + "score": 1.0, + "content": "in Fig. 4. For setting1, the domain discrepancy decreases by", + "type": "text" + }, + { + "bbox": [ + 354, + 471, + 374, + 481 + ], + "score": 0.85, + "content": "74 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 470, + 378, + 483 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 471, + 398, + 481 + ], + "score": 0.84, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 470, + 402, + 483 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 471, + 422, + 481 + ], + "score": 0.85, + "content": "44 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 470, + 443, + 483 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 443, + 471, + 464, + 481 + ], + "score": 0.87, + "content": "43 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "on SIDD,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 483, + 495 + ], + "score": 1.0, + "content": "DND, PolyU, and Nam when PNGAN is exploited. For setting2, the discrepancy decreases by", + "type": "text" + }, + { + "bbox": [ + 483, + 482, + 502, + 492 + ], + "score": 0.87, + "content": "64 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 480, + 506, + 495 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 126, + 504 + ], + "score": 0.83, + "content": "67 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 491, + 129, + 506 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 129, + 493, + 150, + 503 + ], + "score": 0.85, + "content": "46 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 491, + 170, + 506 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 171, + 493, + 190, + 503 + ], + "score": 0.88, + "content": "44 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ". These results demonstrate that PNGAN can narrow the discrepancy between", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "score": 1.0, + "content": "synthetic and real noisy datasets. Please refer to the supplementary for detailed calculation process.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "Comparison with SOTA Methods. We use the generated noisy-clean image pairs (setting2) to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "score": 1.0, + "content": "finetune a series of denoisers. We compare our models with SOTA algorithms on four real denoising", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "datasets: SIDD, DND, PolyU, and Nam. The results are reported in Tab. 1. * denotes denoisers", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "finetuned with image pairs generated by PNGAN. We have the following observations: (i) Our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "denoisers outperform SOTA methods by a large margin. Specifically, MPRNet* and MIRNet* exceed", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 282, + 586 + ], + "score": 1.0, + "content": "the recent best method MIRNet by 0.34 and", + "type": "text" + }, + { + "bbox": [ + 282, + 574, + 315, + 585 + ], + "score": 0.28, + "content": "0 . 3 5 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "on SIDD, 0.30 and 0.37 dB on DND. RIDNet*,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "MPRNet*, and MIRNet* surpass the best performers by 0.36, 1.30, and 1.37 dB on PolyU and 0.01,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "1.04, and 1.10 dB on Nam. (ii) Compared with the counterparts that are not finetuned, our models", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 423, + 619 + ], + "score": 1.0, + "content": "achieve a significant promotion. In particular, RIDNet* is 0.54, 0.29, 0.68, and", + "type": "text" + }, + { + "bbox": [ + 423, + 607, + 456, + 618 + ], + "score": 0.32, + "content": "0 . 4 9 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "higher than", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "RIDNet on SIDD, DND, PolyU, and Nam. MPRNet* achieves 0.35, 0.38, 1.41, and 1.31 dB gain", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "than MPRNet on SIDD, DND, PolyU, and Nam. MIRNet* is improved by 0.35, 0.37, 1.37, and 1.21", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "score": 1.0, + "content": "dB. This evidence clearly suggests the high similarity between PNGAN generating and real noisy", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "images. Denoisers adapted with our fake image pairs generalize better across different benchmarks.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Train from Scratch. For more strong comparisons, we use the fake noisy images generated from", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "clean SIDD train and DF2K (DIV2K+Flicker2K) respectively to train denoisers from scratch. The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "PSNR results evaluated on SIDD test are listed in Tab. 2. All models are trained with the same", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "experiment schedule except the training data. It can be observed: (i) On SIDD train, when PNGAN", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 304, + 723 + ], + "score": 1.0, + "content": "is applied to setting1, denoisers are promoted by", + "type": "text" + }, + { + "bbox": [ + 304, + 711, + 339, + 721 + ], + "score": 0.6, + "content": "\\sim 1 5 . 6 5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 710, + 391, + 723 + ], + "score": 1.0, + "content": "dB and only", + "type": "text" + }, + { + "bbox": [ + 391, + 711, + 421, + 721 + ], + "score": 0.71, + "content": "\\sim 0 . 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "dB lower than those", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 70, + 502, + 156 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 70, + 502, + 156 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 70, + 502, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 502, + 156 + ], + "score": 0.96, + "type": "image", + "image_path": "1e86de1e279adf639e620e96667f698ef1bc2bbbb98ac88a6e9e011a83d52de8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 70, + 502, + 98.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 98.66666666666667, + 502, + 127.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 127.33333333333334, + 502, + 156.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 158, + 505, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "Figure 4: Domain discrepancy comparisons. We use the metric, Maximum Mean Discrepancy (MMD) to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "measure the domain discrepancy between synthetic and real noisy datasets, PNGAN generating and real noisy", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 177, + 473, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 473, + 189 + ], + "score": 1.0, + "content": "datasets. Under both setting1 and 2, the discrepancy decreases significantly when PNGAN is applied.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 107, + 192, + 504, + 247 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 192, + 504, + 247 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 192, + 504, + 247 + ], + "spans": [ + { + "bbox": [ + 107, + 192, + 504, + 247 + ], + "score": 0.973, + "html": "
MethodsSIDD [47]S1DF2K[57,58]
S1 +PNGANS2S2+PNGANRealS1 + PNGANS2S2+PNGAN
RIDNet22.5537.92 (+15.37)36.1338.71 (+2.58)38.6922.5532.10(+9.55)33.9838.14 (+4.16)
MPRNet22.8638.52 (+15.66)36.5239.53 (+3.01)39.4522.8532.82(+9.97)34.1938.61 (+4.42)
MIRNet22.8338.76 (+15.93)36.5539.57 (+3.02)39.5823.0832.34(+9.26)34.2638.72 (+4.46)
", + "type": "table", + "image_path": "9665359c84e08272d6c3cfb857e4c2a27f16946ce646659a5682c63c725fca79.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 107, + 192, + 504, + 210.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 107, + 210.33333333333334, + 504, + 228.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 107, + 228.66666666666669, + 504, + 247.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 109, + 252, + 501, + 263 + ], + "lines": [ + { + "bbox": [ + 109, + 250, + 500, + 265 + ], + "spans": [ + { + "bbox": [ + 109, + 250, + 400, + 265 + ], + "score": 1.0, + "content": "Table 2: Training denoisers with different data from scratch. PSNR is reported.", + "type": "text" + }, + { + "bbox": [ + 401, + 253, + 426, + 263 + ], + "score": 0.79, + "content": "^ { \\mathrm { ~ S 1 , 2 = } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 250, + 500, + 265 + ], + "score": 1.0, + "content": "synthetic setting1,2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 109, + 250, + 500, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 268, + 503, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 228, + 281 + ], + "score": 1.0, + "content": "cameras. 1,000 patches at size", + "type": "text" + }, + { + "bbox": [ + 228, + 269, + 267, + 279 + ], + "score": 0.88, + "content": "5 1 2 \\times 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "are cropped from the collected images. PolyU [40] consists", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 490, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 490, + 292 + ], + "score": 1.0, + "content": "of 40 real camera noisy images. Nam [49] is composed of real noisy images of 11 static scenes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 268, + 505, + 292 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 324, + 308 + ], + "score": 1.0, + "content": "Implementation Details. We set the hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 325, + 295, + 380, + 308 + ], + "score": 0.91, + "content": "\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 295, + 383, + 308 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 384, + 295, + 446, + 307 + ], + "score": 0.92, + "content": "\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 295, + 505, + 308 + ], + "score": 1.0, + "content": ". For synthetic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 252, + 319 + ], + "score": 1.0, + "content": "setting1, we set the noise intensity,", + "type": "text" + }, + { + "bbox": [ + 252, + 307, + 286, + 318 + ], + "score": 0.91, + "content": "\\sigma _ { n } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 306, + 505, + 319 + ], + "score": 1.0, + "content": ". For synthetic setting2, we directly exploit CycleISP", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 352, + 330 + ], + "score": 1.0, + "content": "to generate the synthetic noisy input. All the sub-modules", + "type": "text" + }, + { + "bbox": [ + 353, + 318, + 384, + 329 + ], + "score": 0.76, + "content": "( D _ { d } , G", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 317, + 406, + 330 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 407, + 318, + 416, + 328 + ], + "score": 0.75, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 317, + 506, + 330 + ], + "score": 1.0, + "content": ") are trained with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 198, + 342 + ], + "score": 1.0, + "content": "Adam [62] optimizer", + "type": "text" + }, + { + "bbox": [ + 199, + 329, + 238, + 340 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 327, + 257, + 342 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 329, + 313, + 340 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = 0 . 9 9 9 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 327, + 330, + 342 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 330, + 328, + 359, + 339 + ], + "score": 0.95, + "content": "7 \\times 1 0 ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "iterations. The initial learning rate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 338, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 140, + 353 + ], + "score": 1.0, + "content": "is set to", + "type": "text" + }, + { + "bbox": [ + 140, + 339, + 174, + 350 + ], + "score": 0.91, + "content": "2 \\times 1 \\bar { 0 } ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 338, + 506, + 353 + ], + "score": 1.0, + "content": ". The cosine annealing strategy [63] is employed to steadily decrease the learning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 222, + 363 + ], + "score": 1.0, + "content": "rate from the initial value to", + "type": "text" + }, + { + "bbox": [ + 222, + 350, + 244, + 361 + ], + "score": 0.9, + "content": "1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 350, + 430, + 363 + ], + "score": 1.0, + "content": "during the training procedure. Patches at size", + "type": "text" + }, + { + "bbox": [ + 430, + 351, + 469, + 361 + ], + "score": 0.9, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "cropped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "from training images are fed into the models. The batch size is set as 8. The horizontal and vertical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "flips are performed for data augmentation. All the models are trained on RTX8000 GPUs. In the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 450, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 277, + 396 + ], + "score": 1.0, + "content": "finetuning phase, the learning rate is set to", + "type": "text" + }, + { + "bbox": [ + 277, + 383, + 311, + 394 + ], + "score": 0.91, + "content": "1 \\times 1 0 ^ { - 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 381, + 450, + 396 + ], + "score": 1.0, + "content": ", other settings remain unchanged.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 295, + 506, + 396 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 401, + 218, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 218, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 218, + 414 + ], + "score": 1.0, + "content": "3.2 Quantitative Results", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Domain Discrepancy Validation. We use the widely applied metric, Maximum Mean Discrepancy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "(MMD) [64] to measure the domain discrepancy between synthetic and real-world noisy images,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "PNGAN generating, and real noisy images on four real noisy benchmarks. For DND, we derive a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "pseudo clean version by denoising the real noisy counterparts with a pre-trained MIRNet [55]. Then", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "we use the pseudo clean version to synthesize noisy images. The results are depicted as a histogram", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 354, + 483 + ], + "score": 1.0, + "content": "in Fig. 4. For setting1, the domain discrepancy decreases by", + "type": "text" + }, + { + "bbox": [ + 354, + 471, + 374, + 481 + ], + "score": 0.85, + "content": "74 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 470, + 378, + 483 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 471, + 398, + 481 + ], + "score": 0.84, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 470, + 402, + 483 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 471, + 422, + 481 + ], + "score": 0.85, + "content": "44 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 470, + 443, + 483 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 443, + 471, + 464, + 481 + ], + "score": 0.87, + "content": "43 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "on SIDD,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 483, + 495 + ], + "score": 1.0, + "content": "DND, PolyU, and Nam when PNGAN is exploited. For setting2, the discrepancy decreases by", + "type": "text" + }, + { + "bbox": [ + 483, + 482, + 502, + 492 + ], + "score": 0.87, + "content": "64 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 480, + 506, + 495 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 126, + 504 + ], + "score": 0.83, + "content": "67 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 491, + 129, + 506 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 129, + 493, + 150, + 503 + ], + "score": 0.85, + "content": "46 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 491, + 170, + 506 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 171, + 493, + 190, + 503 + ], + "score": 0.88, + "content": "44 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ". These results demonstrate that PNGAN can narrow the discrepancy between", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "score": 1.0, + "content": "synthetic and real noisy datasets. Please refer to the supplementary for detailed calculation process.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 415, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 506, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "Comparison with SOTA Methods. We use the generated noisy-clean image pairs (setting2) to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 545 + ], + "score": 1.0, + "content": "finetune a series of denoisers. We compare our models with SOTA algorithms on four real denoising", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "datasets: SIDD, DND, PolyU, and Nam. The results are reported in Tab. 1. * denotes denoisers", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "finetuned with image pairs generated by PNGAN. We have the following observations: (i) Our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "denoisers outperform SOTA methods by a large margin. Specifically, MPRNet* and MIRNet* exceed", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 282, + 586 + ], + "score": 1.0, + "content": "the recent best method MIRNet by 0.34 and", + "type": "text" + }, + { + "bbox": [ + 282, + 574, + 315, + 585 + ], + "score": 0.28, + "content": "0 . 3 5 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "on SIDD, 0.30 and 0.37 dB on DND. RIDNet*,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "MPRNet*, and MIRNet* surpass the best performers by 0.36, 1.30, and 1.37 dB on PolyU and 0.01,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "1.04, and 1.10 dB on Nam. (ii) Compared with the counterparts that are not finetuned, our models", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 423, + 619 + ], + "score": 1.0, + "content": "achieve a significant promotion. In particular, RIDNet* is 0.54, 0.29, 0.68, and", + "type": "text" + }, + { + "bbox": [ + 423, + 607, + 456, + 618 + ], + "score": 0.32, + "content": "0 . 4 9 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "higher than", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "RIDNet on SIDD, DND, PolyU, and Nam. MPRNet* achieves 0.35, 0.38, 1.41, and 1.31 dB gain", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "than MPRNet on SIDD, DND, PolyU, and Nam. MIRNet* is improved by 0.35, 0.37, 1.37, and 1.21", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 654 + ], + "score": 1.0, + "content": "dB. This evidence clearly suggests the high similarity between PNGAN generating and real noisy", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "images. Denoisers adapted with our fake image pairs generalize better across different benchmarks.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 519, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Train from Scratch. For more strong comparisons, we use the fake noisy images generated from", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "clean SIDD train and DF2K (DIV2K+Flicker2K) respectively to train denoisers from scratch. The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "PSNR results evaluated on SIDD test are listed in Tab. 2. All models are trained with the same", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "experiment schedule except the training data. It can be observed: (i) On SIDD train, when PNGAN", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 304, + 723 + ], + "score": 1.0, + "content": "is applied to setting1, denoisers are promoted by", + "type": "text" + }, + { + "bbox": [ + 304, + 711, + 339, + 721 + ], + "score": 0.6, + "content": "\\sim 1 5 . 6 5", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 710, + 391, + 723 + ], + "score": 1.0, + "content": "dB and only", + "type": "text" + }, + { + "bbox": [ + 391, + 711, + 421, + 721 + ], + "score": 0.71, + "content": "\\sim 0 . 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "dB lower than those", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 666, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 70, + 502, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 70, + 502, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 70, + 502, + 198 + ], + "spans": [ + { + "bbox": [ + 109, + 70, + 502, + 198 + ], + "score": 0.899, + "type": "image", + "image_path": "f209d8c64470dae44b21785b843771996ba06d36a206d10d5ce3e25053032886.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 70, + 502, + 112.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 112.66666666666666, + 502, + 155.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 155.33333333333331, + 502, + 197.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "table", + "bbox": [ + 107, + 218, + 505, + 269 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 127, + 204, + 483, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 203, + 484, + 216 + ], + "spans": [ + { + "bbox": [ + 127, + 203, + 484, + 216 + ], + "score": 1.0, + "content": "Figure 5: Visual comparisons of noisy images on SIDD, DND, PolyU, and Nam. Please zoom in.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 107, + 218, + 505, + 269 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 218, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 107, + 218, + 505, + 269 + ], + "score": 0.971, + "html": "
MethodsPNGAN ComponentGenerator Architecture
Baseline1+Dd+D+LpBaseline2+ Multi-scale+ SID+FCA
RIDNet14.5435.37 (+20.83)37.49(+2.12) 37.92(+0.43)35.6237.01(+1.39)37.23 (+0.22)37.92 (+0.69)
MPRNet14.2536.26 (+22.01)38.27(+2.01) 38.52(+0.25)36.2837.47(+1.19)37.86 (+0.39)38.52 (+0.66)
MIRNet13.5736.15 (+22.58)38.28(+2.13) 38.76(+0.48)36.3937.66(+1.27)37.89 (+0.23)38.76 (+0.87)
", + "type": "table", + "image_path": "1513bed74d11c207af43181473cd814a19645c1a153db337a6586f22c7b564f5.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 107, + 218, + 505, + 235.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 107, + 235.0, + 505, + 252.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 252.0, + 505, + 269.0 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 108, + 272, + 493, + 282 + ], + "lines": [ + { + "bbox": [ + 117, + 270, + 493, + 284 + ], + "spans": [ + { + "bbox": [ + 117, + 270, + 493, + 284 + ], + "score": 1.0, + "content": "Table 3: Ablation study of PNGAN component and the noise generator architecture. PSNR is reported.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "trained with real data (SIDD train set). While applying PNGAN to setting2 (CycleISP), denoisers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 172, + 309 + ], + "score": 1.0, + "content": "are improved by", + "type": "text" + }, + { + "bbox": [ + 172, + 296, + 200, + 306 + ], + "score": 0.75, + "content": "\\sim 2 . 8 7", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "dB. Surprisingly, in this case, denoisers achieve almost the same performance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 311, + 320 + ], + "score": 1.0, + "content": "as those trained with real data. The relative error is", + "type": "text" + }, + { + "bbox": [ + 312, + 307, + 334, + 317 + ], + "score": 0.89, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 307, + 506, + 320 + ], + "score": 1.0, + "content": ". (ii) To validate the generality of PNGAN,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "we also adopt synthetic DF2K noisy-clean image pairs to train denoisers. As shown in the right part", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 326, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 396, + 343 + ], + "score": 1.0, + "content": "of Tab. 2, when PNGAN is applied to setting1, denoisers are promoted by", + "type": "text" + }, + { + "bbox": [ + 396, + 329, + 425, + 339 + ], + "score": 0.83, + "content": "\\sim 9 . 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 326, + 506, + 343 + ], + "score": 1.0, + "content": "dB. While applying", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 303, + 352 + ], + "score": 1.0, + "content": "PNGAN to setting2, denoisers are improved by", + "type": "text" + }, + { + "bbox": [ + 304, + 340, + 349, + 350 + ], + "score": 0.73, + "content": "\\sim 4 . 3 5 ~ \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 339, + 388, + 352 + ], + "score": 1.0, + "content": "and only", + "type": "text" + }, + { + "bbox": [ + 389, + 340, + 419, + 350 + ], + "score": 0.62, + "content": "\\sim 0 . 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "dB lower than those", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "trained with SIDD real train set. These results convincingly demonstrate: (i) The generated noise is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "highly similar to the real noise especially when PNGAN is applied to synthetic setting2. (ii) PNGAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 373, + 462, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 462, + 385 + ], + "score": 1.0, + "content": "can significantly narrow the domain discrepancy between synthetic and real-world noise.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 389, + 212, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 212, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 212, + 402 + ], + "score": 1.0, + "content": "3.3 Qualitative Results", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "Visual Examinations of Noisy Images. To intuitively evaluate the generated noisy images, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "provide visual comparisons of noisy images on the four real noisy datasets, as shown in Fig. 5. Note", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "that the clean image of DND is pseudo, denoised from its noisy version by a MIRNet. The left part", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "depicts noisy images from SIDD, DND, PolyU, and Nam (top to down). The right part exhibits", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "the patches cropped by the yellow bboxes, from left to right: clean, synthetic setting1, setting2", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 472 + ], + "score": 1.0, + "content": "(CycleISP), PNGAN generating, and real noisy images. As can be seen from the zoom-in patches: (i)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "score": 1.0, + "content": "Noisy images synthesized by setting1 is signal-independent. The distribution and intensity remain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "unchanged across diverse scenes, indicating the characteristics of AWGN fundamentally differ from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "those of the real noise. (ii) Noisy images generated by PNGAN are closer to the real noise than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "those synthesized by setting2 visually. Noise synthesized by setting2 shows randomness that is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "obviously inconsistent with the real noise in terms of intensity and distribution. While PNGAN can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "score": 1.0, + "content": "model spatio-chromatically correlated and non-Gaussian noise more accurately. (iii) Even if passing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "through the same camera pipeline, different shooting conditions lead to the diversity of real noise. It’s", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "unreasonable for the noise synthesized by CycleISP to show nearly uniform fitting to different input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "images. In contrast, PNGAN can adaptively simulate more sophisticated and photo-realistic models.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 497, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 497, + 581 + ], + "score": 1.0, + "content": "This adaptability allows PNGAN to show robust performance across different real noisy datasets.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "Visual Comparison of Denoised Images. We compare the visual results of denoisers before and after", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "being finetuned (denoted with *) with the generated data in Fig. 4. We observe that models finetuned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "with the generated data are more effective in real noise removal. Furthermore, they are capable of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "preserving the structural content, textural details, and spatial smoothness of the homogeneous regions.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "In contrast, original models either yield over-smooth images sacrificing fine textural details and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 419, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 419, + 650 + ], + "score": 1.0, + "content": "structural content or introduce redundant blotchy texture and chroma artifacts.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 107, + 654, + 195, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 651, + 196, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 196, + 668 + ], + "score": 1.0, + "content": "3.4 Ablation Study", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Break-down Ablations. We perform break-down ablations to evaluate the effects of PNGAN", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "components and SMNet architecture. We select setting1 to synthesize the noisy input from SIDD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "train set. Then we use the generated data only to train the denoisers from scratch and evaluate them", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 375, + 712 + ], + "score": 1.0, + "content": "on SIDD test. The PSNR results are reported in Tab. 3. (i) Firstly,", + "type": "text" + }, + { + "bbox": [ + 375, + 700, + 384, + 710 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "is set as SMNet to validate the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 464, + 722 + ], + "score": 1.0, + "content": "effects of PNGAN components. 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Please zoom in.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 107, + 218, + 505, + 269 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 218, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 107, + 218, + 505, + 269 + ], + "score": 0.971, + "html": "
MethodsPNGAN ComponentGenerator Architecture
Baseline1+Dd+D+LpBaseline2+ Multi-scale+ SID+FCA
RIDNet14.5435.37 (+20.83)37.49(+2.12) 37.92(+0.43)35.6237.01(+1.39)37.23 (+0.22)37.92 (+0.69)
MPRNet14.2536.26 (+22.01)38.27(+2.01) 38.52(+0.25)36.2837.47(+1.19)37.86 (+0.39)38.52 (+0.66)
MIRNet13.5736.15 (+22.58)38.28(+2.13) 38.76(+0.48)36.3937.66(+1.27)37.89 (+0.23)38.76 (+0.87)
", + "type": "table", + "image_path": "1513bed74d11c207af43181473cd814a19645c1a153db337a6586f22c7b564f5.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 107, + 218, + 505, + 235.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 107, + 235.0, + 505, + 252.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 107, + 252.0, + 505, + 269.0 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 108, + 272, + 493, + 282 + ], + "lines": [ + { + "bbox": [ + 117, + 270, + 493, + 284 + ], + "spans": [ + { + "bbox": [ + 117, + 270, + 493, + 284 + ], + "score": 1.0, + "content": "Table 3: Ablation study of PNGAN component and the noise generator architecture. PSNR is reported.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 117, + 270, + 493, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "trained with real data (SIDD train set). While applying PNGAN to setting2 (CycleISP), denoisers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 172, + 309 + ], + "score": 1.0, + "content": "are improved by", + "type": "text" + }, + { + "bbox": [ + 172, + 296, + 200, + 306 + ], + "score": 0.75, + "content": "\\sim 2 . 8 7", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "dB. Surprisingly, in this case, denoisers achieve almost the same performance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 311, + 320 + ], + "score": 1.0, + "content": "as those trained with real data. The relative error is", + "type": "text" + }, + { + "bbox": [ + 312, + 307, + 334, + 317 + ], + "score": 0.89, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 307, + 506, + 320 + ], + "score": 1.0, + "content": ". (ii) To validate the generality of PNGAN,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "we also adopt synthetic DF2K noisy-clean image pairs to train denoisers. As shown in the right part", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 326, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 396, + 343 + ], + "score": 1.0, + "content": "of Tab. 2, when PNGAN is applied to setting1, denoisers are promoted by", + "type": "text" + }, + { + "bbox": [ + 396, + 329, + 425, + 339 + ], + "score": 0.83, + "content": "\\sim 9 . 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 326, + 506, + 343 + ], + "score": 1.0, + "content": "dB. While applying", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 303, + 352 + ], + "score": 1.0, + "content": "PNGAN to setting2, denoisers are improved by", + "type": "text" + }, + { + "bbox": [ + 304, + 340, + 349, + 350 + ], + "score": 0.73, + "content": "\\sim 4 . 3 5 ~ \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 339, + 388, + 352 + ], + "score": 1.0, + "content": "and only", + "type": "text" + }, + { + "bbox": [ + 389, + 340, + 419, + 350 + ], + "score": 0.62, + "content": "\\sim 0 . 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "dB lower than those", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "trained with SIDD real train set. These results convincingly demonstrate: (i) The generated noise is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "highly similar to the real noise especially when PNGAN is applied to synthetic setting2. (ii) PNGAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 373, + 462, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 462, + 385 + ], + "score": 1.0, + "content": "can significantly narrow the domain discrepancy between synthetic and real-world noise.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 285, + 506, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 389, + 212, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 212, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 212, + 402 + ], + "score": 1.0, + "content": "3.3 Qualitative Results", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "Visual Examinations of Noisy Images. To intuitively evaluate the generated noisy images, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "provide visual comparisons of noisy images on the four real noisy datasets, as shown in Fig. 5. Note", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "that the clean image of DND is pseudo, denoised from its noisy version by a MIRNet. The left part", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "depicts noisy images from SIDD, DND, PolyU, and Nam (top to down). The right part exhibits", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "the patches cropped by the yellow bboxes, from left to right: clean, synthetic setting1, setting2", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 472 + ], + "score": 1.0, + "content": "(CycleISP), PNGAN generating, and real noisy images. As can be seen from the zoom-in patches: (i)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "score": 1.0, + "content": "Noisy images synthesized by setting1 is signal-independent. The distribution and intensity remain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "unchanged across diverse scenes, indicating the characteristics of AWGN fundamentally differ from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "those of the real noise. (ii) Noisy images generated by PNGAN are closer to the real noise than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "those synthesized by setting2 visually. Noise synthesized by setting2 shows randomness that is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "obviously inconsistent with the real noise in terms of intensity and distribution. While PNGAN can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "score": 1.0, + "content": "model spatio-chromatically correlated and non-Gaussian noise more accurately. (iii) Even if passing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "through the same camera pipeline, different shooting conditions lead to the diversity of real noise. It’s", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "unreasonable for the noise synthesized by CycleISP to show nearly uniform fitting to different input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "images. In contrast, PNGAN can adaptively simulate more sophisticated and photo-realistic models.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 497, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 497, + 581 + ], + "score": 1.0, + "content": "This adaptability allows PNGAN to show robust performance across different real noisy datasets.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 403, + 506, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "Visual Comparison of Denoised Images. We compare the visual results of denoisers before and after", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "being finetuned (denoted with *) with the generated data in Fig. 4. We observe that models finetuned", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "with the generated data are more effective in real noise removal. Furthermore, they are capable of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "preserving the structural content, textural details, and spatial smoothness of the homogeneous regions.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "In contrast, original models either yield over-smooth images sacrificing fine textural details and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 419, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 419, + 650 + ], + "score": 1.0, + "content": "structural content or introduce redundant blotchy texture and chroma artifacts.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 583, + 506, + 650 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 654, + 195, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 651, + 196, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 196, + 668 + ], + "score": 1.0, + "content": "3.4 Ablation Study", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Break-down Ablations. We perform break-down ablations to evaluate the effects of PNGAN", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "components and SMNet architecture. We select setting1 to synthesize the noisy input from SIDD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "train set. Then we use the generated data only to train the denoisers from scratch and evaluate them", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 375, + 712 + ], + "score": 1.0, + "content": "on SIDD test. The PSNR results are reported in Tab. 3. 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80%39.230.95539.560.97239.720.97639.330.960
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These", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 465, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 504, + 480 + ], + "score": 1.0, + "content": "results convincingly demonstrate the superiority of the proposed SMNet in real-world noise fitting.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "Parameter Analysis. We adopt RIDNet as the baseline to perform parameter analysis. 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We change", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 190, + 518 + ], + "score": 1.0, + "content": "the parameters, train", + "type": "text" + }, + { + "bbox": [ + 190, + 506, + 199, + 515 + ], + "score": 0.76, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 505, + 218, + 518 + ], + "score": 1.0, + "content": ", use", + "type": "text" + }, + { + "bbox": [ + 219, + 506, + 228, + 515 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "to generate realistic noisy images from clean images of SIDD train", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "set, train RIDNet with the generated data, and evaluate its performance on SIDD test set. When", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "analyzing one parameter, we fix the others at their optimal values. The PSNR results are shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 239, + 551 + ], + "score": 1.0, + "content": "in Fig. 7. The optimal setting is", + "type": "text" + }, + { + "bbox": [ + 239, + 537, + 295, + 550 + ], + "score": 0.9, + "content": "\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 536, + 299, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 299, + 537, + 362, + 549 + ], + "score": 0.92, + "content": "\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 536, + 384, + 551 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 384, + 538, + 418, + 549 + ], + "score": 0.91, + "content": "\\sigma _ { n } = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 536, + 506, + 551 + ], + "score": 1.0, + "content": "or 50. 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When", + "type": "text" + }, + { + "bbox": [ + 163, + 582, + 185, + 593 + ], + "score": 0.9, + "content": "q = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 580, + 506, + 594 + ], + "score": 1.0, + "content": ", all the finetuning data comes from SIDD train set, RIDNet achieves the best", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "performance on SIDD. However, its performance on PolyU and Nam degrades drastically due to the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 487, + 616 + ], + "score": 1.0, + "content": "domain discrepancy between different real noisy datasets. 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The average performance on the three datasets yields the maximum when", + "type": "text" + }, + { + "bbox": [ + 468, + 614, + 503, + 626 + ], + "score": 0.91, + "content": "q = 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 183, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 185, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 185, + 645 + ], + "score": 1.0, + "content": "4 Conclusion", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "score": 1.0, + "content": "Too much research focuses on designing a CNN architecture for real noise removal. In contrast,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "this work investigates how to generate more realistic noisy images so as to boom the denoising", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "performance. We first formulate a noise model that treats each noisy pixel as a random variable. Then", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "we propose a novel framework PNGAN to perform the image and noise domain alignment. For better", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "noise fitting, we customize an efficient architecture, SMNet as the generator. Experiments show", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 701, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 506, + 712 + ], + "score": 1.0, + "content": "that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 503, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 503, + 723 + ], + "score": 1.0, + "content": "Denoisers finetuned with the generated data outperform SOTA methods on real denoising datasets.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 70, + 502, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 70, + 502, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 70, + 502, + 199 + ], + "spans": [ + { + "bbox": [ + 109, + 70, + 502, + 199 + ], + "score": 0.967, + "type": "image", + "image_path": "f6d75ee03ec6067d2f9aa52e21200e934dc47ff620fce2476744f1cbb443fb7c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 70, + 502, + 113.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 113.0, + 502, + 156.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 156.0, + 502, + 199.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 119, + 204, + 488, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 202, + 489, + 215 + ], + "spans": [ + { + "bbox": [ + 121, + 202, + 489, + 215 + ], + "score": 1.0, + "content": "Figure 6: Visual results of denoisers before and after being finetuned with fake data. 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(2). Denoisers trained with the generated data collapse", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 447, + 327 + ], + "score": 1.0, + "content": "dramatically, implying the naive strategy mentioned in Sec. 2.2 is unfeasible. When", + "type": "text" + }, + { + "bbox": [ + 447, + 314, + 461, + 325 + ], + "score": 0.88, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "is applied,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "the denoisers are promoted by 21.81 dB on average. 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This evidence indicates that", + "type": "text" + }, + { + "bbox": [ + 273, + 347, + 287, + 358 + ], + "score": 0.9, + "content": "D _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "successfully conducts the image domain alignment as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 356, + 370 + ], + "score": 1.0, + "content": "mentioned in Sec. 2.2. 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After deploying", + "type": "text" + }, + { + "bbox": [ + 219, + 380, + 228, + 389 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 379, + 342, + 393 + ], + "score": 1.0, + "content": ", the models are improved by", + "type": "text" + }, + { + "bbox": [ + 343, + 380, + 369, + 390 + ], + "score": 0.8, + "content": "{ \\sim } 2 . 0 9", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "dB, suggesting that the pixel-level", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 507, + 404 + ], + "score": 1.0, + "content": "noise model is more in line with real noise scenes and benefits generating more realistic noisy images.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 131, + 414 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 401, + 145, + 414 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 401, + 450, + 414 + ], + "score": 1.0, + "content": "is used, the denoisers gain a slight improvement by about 0.39 dB, indicating", + "type": "text" + }, + { + "bbox": [ + 451, + 401, + 463, + 414 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "facilitates", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 416, + 425 + ], + "score": 1.0, + "content": "yielding more vivid results. (ii) Secondly, we only change the architecture of", + "type": "text" + }, + { + "bbox": [ + 417, + 413, + 426, + 423 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "to study the effects", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "score": 1.0, + "content": "of its components. We start from Baseline2 that doesn’t exploit multi-scale feature fusion, SID,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "and FCA. When we add two different scale branches and use bilinear interpolation to downsample", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 444, + 458 + ], + "score": 1.0, + "content": "and upsample, denoisers trained with the generated images are promoted by about", + "type": "text" + }, + { + "bbox": [ + 444, + 445, + 477, + 456 + ], + "score": 0.28, + "content": "1 . 2 8 \\ : \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 444, + 506, + 458 + ], + "score": 1.0, + "content": ". After", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 342, + 469 + ], + "score": 1.0, + "content": "applying SID and FCA, the denoisers further gain 0.28 and", + "type": "text" + }, + { + "bbox": [ + 342, + 456, + 375, + 466 + ], + "score": 0.3, + "content": "0 . 7 4 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "improvement on average. These", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 465, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 504, + 480 + ], + "score": 1.0, + "content": "results convincingly demonstrate the superiority of the proposed SMNet in real-world noise fitting.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 303, + 507, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "Parameter Analysis. We adopt RIDNet as the baseline to perform parameter analysis. We firstly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 196, + 508 + ], + "score": 1.0, + "content": "validate the effects of", + "type": "text" + }, + { + "bbox": [ + 196, + 495, + 208, + 506 + ], + "score": 0.84, + "content": "\\lambda _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 492, + 214, + 508 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 214, + 495, + 232, + 505 + ], + "score": 0.86, + "content": "\\lambda _ { R a }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 492, + 441, + 508 + ], + "score": 1.0, + "content": "in Eq. (13), and the noise intensity of setting1, i.e.,", + "type": "text" + }, + { + "bbox": [ + 441, + 495, + 453, + 505 + ], + "score": 0.85, + "content": "\\sigma _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 492, + 505, + 508 + ], + "score": 1.0, + "content": ". We change", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 190, + 518 + ], + "score": 1.0, + "content": "the parameters, train", + "type": "text" + }, + { + "bbox": [ + 190, + 506, + 199, + 515 + ], + "score": 0.76, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 505, + 218, + 518 + ], + "score": 1.0, + "content": ", use", + "type": "text" + }, + { + "bbox": [ + 219, + 506, + 228, + 515 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "to generate realistic noisy images from clean images of SIDD train", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "set, train RIDNet with the generated data, and evaluate its performance on SIDD test set. When", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "analyzing one parameter, we fix the others at their optimal values. The PSNR results are shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 239, + 551 + ], + "score": 1.0, + "content": "in Fig. 7. The optimal setting is", + "type": "text" + }, + { + "bbox": [ + 239, + 537, + 295, + 550 + ], + "score": 0.9, + "content": "\\lambda _ { p } = 6 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 536, + 299, + 551 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 299, + 537, + 362, + 549 + ], + "score": 0.92, + "content": "\\lambda _ { R a } = 8 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 536, + 384, + 551 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 384, + 538, + 418, + 549 + ], + "score": 0.91, + "content": "\\sigma _ { n } = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 536, + 506, + 551 + ], + "score": 1.0, + "content": "or 50. Secondly, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 562 + ], + "score": 1.0, + "content": "evaluate the effect of the ratio of finetuning data. We denote the ratio of extended training data", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 294, + 572 + ], + "score": 1.0, + "content": "(setting2) to SIDD real noisy training data as", + "type": "text" + }, + { + "bbox": [ + 294, + 561, + 300, + 571 + ], + "score": 0.66, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 560, + 404, + 572 + ], + "score": 1.0, + "content": ". We change the value of", + "type": "text" + }, + { + "bbox": [ + 405, + 561, + 411, + 571 + ], + "score": 0.76, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 560, + 505, + 572 + ], + "score": 1.0, + "content": ", finetuned the original", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "RIDNet, and test on three real denoising datasets: SIDD, PolyU, and Nam. The results are listed in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 163, + 594 + ], + "score": 1.0, + "content": "Tab. 4. When", + "type": "text" + }, + { + "bbox": [ + 163, + 582, + 185, + 593 + ], + "score": 0.9, + "content": "q = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 580, + 506, + 594 + ], + "score": 1.0, + "content": ", all the finetuning data comes from SIDD train set, RIDNet achieves the best", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "performance on SIDD. However, its performance on PolyU and Nam degrades drastically due to the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 487, + 616 + ], + "score": 1.0, + "content": "domain discrepancy between different real noisy datasets. We gradually increase the value of", + "type": "text" + }, + { + "bbox": [ + 487, + 605, + 493, + 614 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 602, + 505, + 616 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 468, + 627 + ], + "score": 1.0, + "content": "study its effects. The average performance on the three datasets yields the maximum when", + "type": "text" + }, + { + "bbox": [ + 468, + 614, + 503, + 626 + ], + "score": 0.91, + "content": "q = 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 483, + 506, + 627 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 183, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 185, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 185, + 645 + ], + "score": 1.0, + "content": "4 Conclusion", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 507, + 658 + ], + "score": 1.0, + "content": "Too much research focuses on designing a CNN architecture for real noise removal. In contrast,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "this work investigates how to generate more realistic noisy images so as to boom the denoising", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "performance. We first formulate a noise model that treats each noisy pixel as a random variable. Then", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "we propose a novel framework PNGAN to perform the image and noise domain alignment. For better", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "noise fitting, we customize an efficient architecture, SMNet as the generator. Experiments show", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 701, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 506, + 712 + ], + "score": 1.0, + "content": "that noise generated by PNGAN is highly similar to real noise in terms of intensity and distribution.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 503, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 503, + 723 + ], + "score": 1.0, + "content": "Denoisers finetuned with the generated data outperform SOTA methods on real denoising datasets.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 645, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 202, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 204, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 204, + 87 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 506, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "score": 1.0, + "content": "This work is jointly supported by the NSFC fund (61831014), in part by the Shenzhen Sci-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "ence and Technology Project under Grant (ZDYBH201900000002, JCYJ20180508152042002,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 234, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 234, + 130 + ], + "score": 1.0, + "content": "CJGJZD20200617102601004).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 145, + 163, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 165, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 165, + 160 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 174, + 507, + 724 + ], + "lines": [ + { + "bbox": [ + 112, + 172, + 362, + 178 + ], + "spans": [ + { + "bbox": [ + 112, + 172, + 125, + 178 + ], + "score": 0.89, + "content": "[1]", + "type": "text" + }, + { + "bbox": [ + 175, + 172, + 186, + 178 + ], + "score": 0.828, + "content": "PP，", + "type": "text" + }, + { + "bbox": [ + 348, + 172, + 362, + 178 + ], + "score": 0.873, + "content": "ng，", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 109, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 109, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "[2] S. 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MethodsSIDD[47]DND [48]MethodsPolyU [40]Nam [49]
PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑PSNR↑SSIM↑
DnCNN-B[10]23.660.58332.430.790RDN[12]37.940.94638.160.956
CBDNet [50]33.280.86838.060.942FFDNet+ [19]38.170.95138.810.957
RIDNet [44]38.710.91439.260.953TWSC [51]38.680.95838.960.962
AINDNet [52]39.150.95539.530.956CBDNet [50]38.740.96139.080.969
VDN[53]39.230.95539.380.952RIDNet [44]38.860.96239.200.973
CycleISP [22]39.520.95739.560.956VDN [53]39.040.96539.680.976
MPRNet [54]39.710.95839.800.954MPRNet [54]39.070.96939.410.974
MIRNet [55]39.720.95939.880.956MIRNet [55]39.180.97339.570.979
RIDNet*(Ours)39.250.95639.550.955RIDNet*(Ours)39.540.97139.690.975
MPRNet*(Ours)40.060.96040.180.961MPRNet*(Ours)40.480.98240.720.984
MIRNet*(Ours)40.070.96040.250.962MIRNet*(Ours)40.550.98340.780.986
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MethodsPNGAN ComponentGenerator Architecture
Baseline1+Dd+D+LpBaseline2+ Multi-scale+ SID+FCA
RIDNet14.5435.37 (+20.83)37.49(+2.12) 37.92(+0.43)35.6237.01(+1.39)37.23 (+0.22)37.92 (+0.69)
MPRNet14.2536.26 (+22.01)38.27(+2.01) 38.52(+0.25)36.2837.47(+1.19)37.86 (+0.39)38.52 (+0.66)
MIRNet13.5736.15 (+22.58)38.28(+2.13) 38.76(+0.48)36.3937.66(+1.27)37.89 (+0.23)38.76 (+0.87)
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CleanSetting1Setting2PNGANRealCleanSeting1Setting2PNGANReal
ONDSIDD□D
DND
S5
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b/parse/train/_WnAQKse_uK/_WnAQKse_uK.md @@ -0,0 +1,272 @@ +# ViTAE: Vision Transformer Advanced by Exploring Intrinsic Inductive Bias + +Yufei Xu1∗ Qiming Zhang1∗ Jing Zhang1 Dacheng Tao2,1 + +1The University of Sydney, Australia, 2JD Explore Academy, China + +{yuxu7116,qzha2506}@uni.sydney.edu.au, jing.zhang1@sydney.edu.au, dacheng.tao@gmail.com + +# Abstract + +Transformers have shown great potential in various computer vision tasks owing to their strong capability in modeling long-range dependency using the self-attention mechanism. Nevertheless, vision transformers treat an image as 1D sequence of visual tokens, lacking an intrinsic inductive bias (IB) in modeling local visual structures and dealing with scale variance. Alternatively, they require large-scale training data and longer training schedules to learn the IB implicitly. In this paper, we propose a new Vision Transformer Advanced by Exploring intrinsic IB from convolutions, i.e., ViTAE. Technically, ViTAE has several spatial pyramid reduction modules to downsample and embed the input image into tokens with rich multi-scale context by using multiple convolutions with different dilation rates. In this way, it acquires an intrinsic scale invariance IB and is able to learn robust feature representation for objects at various scales. Moreover, in each transformer layer, ViTAE has a convolution block in parallel to the multi-head selfattention module, whose features are fused and fed into the feed-forward network. Consequently, it has the intrinsic locality IB and is able to learn local features and global dependencies collaboratively. Experiments on ImageNet as well as downstream tasks prove the superiority of ViTAE over the baseline transformer and concurrent works. Source code and pretrained models will be available at code. + +# 1 Introduction + +Transformers [79, 17, 40, 14, 46, 61] have shown a domination trend in NLP studies owing to their strong ability in modeling long-range dependencies by the self-attention mechanism [67, 81, 51]. Such success and good properties of transformers has inspired following many works that apply them in various computer vision tasks [19, 100, 97, 80, 7]. Among them, ViT [19] is the pioneering pure transformer model that embeds images into a sequence of visual tokens and models the global dependencies among them with stacked transformer blocks. Although it achieves promising performance on image classification, it requires large-scale training data and a longer training schedule. One important reason is that ViT + +![](images/4515266c49c6aff7e7d656459cb5d8b9e3662e222790882fee6a157e2ad56504.jpg) +Figure 1: Comparison of data and training efficiency of T2T-ViT-7 and ViTAE-T on ImageNet. + +lacks intrinsic inductive bias (IB) in modeling local visual structures (e.g., edges and corners) and dealing with objects at various scales like convolutions. Alternatively, ViT has to learn such IB implicitly from large-scale data. + +Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70, 62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that computes local correlation among neighbor pixels is good at extracting local features such as edges and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94], which are then aggregated into high-level features progressively by a bulk of sequential convolutions [32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not well suited to model long-range dependencies2, which is the key advantage of transformers. An interesting question comes up: Can we improve vision transformers by leveraging the good properties of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model as the teacher and consumes extra training cost. + +Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing the network structures in this paper. Current vision transformers always obtain tokens with singlescale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data. For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner. Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale neighboring contextual information and progressively structurizes the image to tokens. Motivated by the success of CNNs in dealing with scale variance, we explore a similar design in transformers, i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn scale-invariant features more efficiently from data. On the other hand, low-level local features are fundamental elements to generate high-level discriminative features. Although transformers can also learn such features at shallow layers from data, they are not skilled as convolutions by design. Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that locality is a reasonable compensation of global dependency. However, this serial structure ignores the global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the “divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and then fuse the features to account for both. In this way, we empower transformers to learn local and long-range features within each block more effectively. + +Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias $( V i T A E )$ , which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell (NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale context while NCs aim to jointly model locality and global dependencies in the token sequence. Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial resolution by $1 / 1 6$ and a series of NCs to learn discriminative features from data. ViTAE outperforms representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as well as classification accuracy and generalization on downstream tasks. + +Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e., scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature learning ability of transformers. Second, we design a new transformer architecture named ViTAE based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features effectively. Third, ViTAE outperforms representative vision transformers regarding classification accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves $7 5 . 3 \%$ and $8 2 . 0 \%$ top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively. + +# 2 Related Work + +# 2.1 CNNs with intrinsic IB + +CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream computer vision tasks. The convolution operations in CNNs extract local features from the neighbor pixels within the receptive field determined by the kernel size [42]. Following the intuition that local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For example, to effectively learn features of large objects, a large receptive field is needed by either using large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32, 68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid [8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image mainly encode the large objects while features from the large scale image respond more to small objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale context by using multiple convolutions with different receptive fields within a single layer, i.e., intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation rates in the reduction cells to encode multi-scale context into each visual token. + +# 2.2 Vision transformers with learned IB + +ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed learns the IB from amounts of data implicitly. Following works along this direction are to simplify the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75, 22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher, introducing extra computation cost during training. Recently, some works try to introduce the intrinsic IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example, [43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise structures while [47, 33] apply attention within local windows. However, these serial structure may ignore the global context during locality modeling (and vice versa). [88] establishes connection across different scales at the cost of heavy computation. Instead, we follow the “divide-and-conquer” idea and propose to model locality and global dependencies simultaneously via a parallel structure within each transformer layer. Conformer [58], the most relevant concurrent work to us, employs a unit to explore inter-block interactions between parallel convolution and transformer blocks. In contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each other within the transformer block. In addition, Conformer is not designed to have inherent scale invariance IB. + +# 3 Methodology + +# 3.1 Revisit vision transformer + +We first give a brief review of vision transformer in this part. To adapt transformers to vision tasks, ViT [19] first splits an image $x \in R ^ { H \times W \times C }$ into tokens with a reduction ratio of $p$ (i.e., $x _ { t } \in R ^ { ( ( H \times W ) / p ^ { 2 } ) \times D } )$ , where $H , W$ and $C$ denote the height, width, and channel dimensions of the input image, $D = C p ^ { 2 }$ denotes the token dimension. Then, an extra class token is concatenated to the visual tokens before adding position embeddings in an element-wise manner. The resulting tokens are fed into the following transformer layers. Each transformer layer is composed of two parts, i.e., a multi-head self-attention module (MHSA) and a feed forward network (FFN). + +![](images/12ff638562a0b78bbcef09633d771c21fc798831f54a00e148f38d5e89e3596b.jpg) +Figure 2: The structure of the proposed ViTAE. It is constructed by stacking three RCs and several NCs. Both types of cells share a simple basic structure, i.e., an MHSA module and a parallel convolutional module followed by an FFN. In particular, RC has an extra pyramid reduction module using atrous convolutions with different dilation rates to embed multi-scale context into tokens. + +MHSA Multi-head self-attention extends single-head self-attention (SHSA) by using different projection matrices for each head. Specifically, the input tokens $x _ { t }$ are first projected to queries $( Q )$ , keys $( K )$ and values $( V )$ using projection matrices, i.e., $Q , K , V = x _ { t } W _ { Q } , x _ { t } Q _ { K } , x _ { t } Q _ { V }$ , where $W _ { Q / K / V } \in R ^ { D \times D }$ denotes the projection matrix for query, key, and value, respectively. Then, the self-attention operation is calculated as: + +$$ +A t t e n t i o n ( Q , K , V ) = s o f t m a x ( \frac { Q K ^ { T } } { \sqrt { D } } ) V . +$$ + +This SHSA module is repeated for $h$ times to formulate the MHSA module, where $h$ is the number of heads. The output features of the $h$ heads are concatenated along the channel dimension and formulate the output of the MHSA module. + +FFN FFN is placed on top of the MHSA module and applied to each token identically and separately. It consists of two linear transformations with an activation function in between. Besides, a layer normalization [2] and a shortcut are added before and aside from the MHSA and FFN, respectively. + +# 3.2 Overview architecture of ViTAE + +ViTAE aims to introduce the intrinsic IB in CNNs to vision transformers. As shown in Figure 2, ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding multi-scale context and local information into tokens, and NCs are used to further model the locality and long-range dependencies in the tokens. Taken an image $x \in R ^ { H \times W \times C }$ as input, three RCs are used to gradually downsample $x$ by $4 \times , 2 \times$ , and $2 \times$ , respectively. Thereby, the output tokens of the RCs are of size $[ H / 1 6 , W / 1 6 , D ]$ where $D$ is the token dimension (64 in our experiments). The output tokens of RCs are then flattened as $R ^ { H W / 2 5 6 \times D }$ , concatenated with the class token, and added by the sinusoid position encoding. Next, the tokens are fed into the following NCs, which keep the length of the tokens. Finally, the prediction probability is obtained using a linear classification layer on the class token from the last NC. + +# 3.3 Reduction cell + +Instead of directly splitting and flatten images into visual tokens based on a linear image patch embedding layer, we devise the reduction cell to embed multi-scale context and local information into visual tokens, which introduces the intrinsic scale-invariance and locality IBs from convolutions. Technically, RC has two parallel branches responsible for modeling locality and long-range dependency, respectively, followed by an FFN for feature transformation. We denote the input feature of the $i _ { t h } \ : \mathrm { R C }$ as $f _ { i } \in \dot { R } ^ { H _ { i } \times W _ { i } \times D _ { i } }$ . The input of the first RC is the image $x$ . In the global dependencies branch, $f _ { i }$ is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e., + +$$ +f _ { i } ^ { m s } \triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \in S _ { i } , r _ { i } \in { \mathcal { R } } ] ) , +$$ + +where $C o n v _ { i j } ( \cdot )$ indicates the $j$ th convolutional layer in the PRM $( P R M _ { i } ( \cdot ) )$ . It uses a dilation rate $s _ { i j }$ from the predefined dilation rate set $S _ { i }$ corresponding to the ith RC. Note that we use stride convolution to reduce the spatial dimension of features by a ratio $r _ { i }$ from the predefined reduction ratio set $\mathcal { R }$ . The conv features are concatenated along the channel dimension, i.e., $f _ { i } ^ { m s } \in$ $R ^ { ( W _ { i } / p ) \times ( H _ { i } / p ) \times ( | S _ { i } | D ) }$ , where $| { S _ { i } } |$ denotes the number of dilation rates in $S _ { i }$ . $f _ { i } ^ { m s }$ is then processed by an MHSA module to model long-range dependencies, i.e., + +$$ +f _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) , +$$ + +where $I m g 2 S e q ( \cdot )$ is a simple reshape operation to flatten the feature map to a 1D sequence. In this way, $f _ { i } ^ { g }$ embeds the multi-scale context in each token. In addition, we use a Parallel Convolutional Module (PCM) to embed local context within the tokens, which are fused with $f _ { i } ^ { g }$ as follows: + +$$ +f _ { i } ^ { l g } = f _ { i } ^ { g } + { \cal P } { \cal C } M _ { i } ( f _ { i } ) . +$$ + +Here, $P C M _ { i } ( \cdot )$ represents the PCM, which is composed of three stacked convolution layers and an $I m g 2 S e q ( \cdot )$ operation. It is noteworthy that the parallel convolution branch has the same spatial downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed into the following RC or NC, i.e., + +$$ +f _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) , +$$ + +where the $S e q 2 I m g ( \cdot )$ is a simple reshape operation to reshape a token sequence back to feature maps. $F F N _ { i } ( \cdot )$ represents the FFN in the ith RC. In our ViTAE, three RCs are stacked sequentially to gradually reduce the input image’s spatial dimension by $4 \times , 2 \times$ , and $2 \times$ , respectively. The feature maps generated by the last RC are of a size of $[ H / 1 6 , W / 1 6 , D ]$ , which are then flattened into visual tokens and fed into the following NCs. + +# 3.4 Normal cell + +As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell except for the absence of the PRM. Due to the relatively small $( \frac { 1 } { 1 6 } \times )$ spatial size of feature maps after RCs, it is unnecessary to use PRM in NCs. Given $f _ { 3 }$ from the third RC, we first concatenate it with the class token $t _ { c l s }$ , and then add it to the positional encodings to get the input tokens $t$ for the following NCs. Here we ignore the subscript for clarity since all NCs have an identical architecture but different learnable weights. $t _ { c l s }$ is randomly initialized at the start of training and fixed during the inference. Similar to the RC, the tokens are fed into the MHSA module, i.e., $t _ { g } = M H S A ( t )$ . Meanwhile, they are reshaped to 2D feature maps and fed into the PCM, i.e., $t _ { l } = \bar { I } m g 2 S e q ( P C M ( S e q 2 I m g ( t ) ) )$ . Note that the class token is discarded in PCM because it has no spatial connections with other visual tokens. To further reduce the parameters in NCs, we use group convolutions in PCM. The features from MHSA and PCM are then fused via element-wise sum, i.e., $t _ { l g } = t _ { g } + t _ { l }$ . Finally, $t _ { l g }$ are fed into the FFN to get the output features of NC, i.e., $t _ { n c } = F F N ( t _ { l g } ) \overline { { + } } t _ { l g }$ . Similar to ViT [19], we apply layer normalization to the class token generated by the last NC and feed it to the classification head to get the final classification result. + +# 3.5 Model details + +We use two variants of ViTAE in our experiments for a fair comparison of other models with similar model sizes. The details of them are summarized in Table 1. In the first RC, the default convolution kernel size is $7 \times 7$ with a + +Table 1: Model details of two variants of ViTAE. + +
ModelReduction Cell DilationCellsNormal Cell Heads Embed CellsParams Macs (M)(G)
ViTAE-T[1,2,3,4] √34 25674.81.5
ViTAE-S[1,2,3,4] √36384 1423.65.6
+ +stride of 4 and dilation rates of $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ . In the following two RCs, the convolution kernel size is $3 \times 3$ with a stride of 2 and dilation rates of $S _ { 2 } = [ 1 , 2 , 3 ]$ and $S _ { 3 } = [ 1 , 2 ]$ , respectively. Since the spatial dimension of tokens decreases, there is no need to use large kernels and dilation rates. PCM in both RCs and NCs comprises three convolutional layers with a kernel size of $3 \times 3$ . + +# 4 Experiments + +# 4.1 Implementation details + +We train and test the proposed ViTAE model on the standard ImageNet [38] dataset, which contains about 1.3 million images and covers 1k classes. Unless explicitly stated, the image size during training is set to $2 2 4 \times 2 2 4$ . We use the AdamW [48] optimizer with the cosine learning rate scheduler and uses the data augmentation strategy exactly the same as T2T [93] for a fair comparison, regarding the training strategies and the size of models. We use a batch size of 512 for training all our models and set the initial learning rate to be 5e-4. The results of our models can be found in Table 2, where all the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and TIMM [82]. + +# 4.2 Comparison with the state-of-the-art + +We compare our ViTAE with both CNN models and vision transformers with similar model sizes in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are reported. We categorize the methods into CNN models, vision transformers with learned IB, and vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves a $7 5 . 3 \%$ Top-1 accuracy, which is better than ResNet-18 with more parameters. The real Top-1 accuracy of the ViTAE model is $8 2 . 9 \%$ , which is comparable to ResNet-50 that has four more times of parameters than ours. Similarly, our ViTAE-S achieves $8 2 . 0 \%$ Top-1 accuracy with half of the parameters of ResNet-101 and ResNet-152, showing the superiority of learning both local and longrange features from specific structures with corresponding intrinsic IBs by design. Similar phenomena can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65], where ViTAE obtains better performance with fewer parameters. When compared with larger models which are searched according to NAS [73], our ViTAE-S achieves a similar performance when using $3 8 4 \times 3 8 4$ images as input, which further shows the potential of vision transformers with intrinsic IB. + +In addition, among the transformers with learned IB, ViT is the first pure transformer model for visual recognition. DeiT shares the same structure with ViT but uses different data augmentation and training strategies to facilitate the learning of transformers. DeiT⚗ denotes using an off-the-shelf CNN model as the teacher model to train DeiT, which introduces the intrinsic IB from CNN to transformer implicitly in a knowledge distillation manner, showing better performance than the vanilla ViT on the ImageNet dataset. It is exciting to see that our ViTAE-T with fewer parameters even outperforms the distilled model DeiT⚗, demonstrating the efficacy of introducing intrinsic IBs in transformers by design. Besides, compared with other transformers with explicit intrinsic IB, our ViTAE with fewer parameters also achieves comparable or better performance. For instance, ViTAE-T achieves comparable performance with LocalVit-T but has 1M fewer parameters, demonstrating the superiority of the proposed RCs and NCs in introducing intrinsic IBs. + +# 4.3 Ablation study + +We use T2T-ViT [93] as our baseline model in the following ablation study of our ViTAE. As shown in Table 3, we investigate the hyper-parameter settings in RCs and NCs by isolating them separately. All the models are trained for 100 epochs on ImageNet and follow the same training setting and data augmentation strategy as described in Section 4.1. + +Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set. + +
Type ModelParams (M)MACs (G)Input SizeImageNet Real
Top-1Top-5Top-1
CNNResNet-18 [26]11.73.622470.386.777.3
ResNet-50 [26]25.67.622476.793.382.5
ResNet-101 [26]44.515.222478.394.183.7
ResNet-152 [26]60.222.622478.994.484.1
EfficientNet-B0 [73]5.30.822477.193.383.5
EfficientNet-B4 [73]19.38.438082.996.488.0
MobileNetV1 [31]4.30.622472.31-
MobileNetV2(1.4) [65]6.90.622474.7--
RegNetY-600M[62]6.11.222475.5--
RegNetY-4GF[62] RegNetY-8GF[62]20.6 39.28.0 16.022480.0186.4
22481.7187.4
DeiT-T[76]5.72.622472.291.180.6
DeiT-T [76]5.72.622474.591.982.1
LocalViT-T[43]5.92.622474.892.6
LocalViT-T2T[43]4.32.422472.5-1 1
ConT-Ti [89]5.81.622474.9--
PiT-Ti [29]4.91.422473.0-1
T2T-ViT-7 [93]4.31.222471.790.979.7
ViTAE-T4.81.522475.392.782.9
ViTAE-T ↑ 3844.85.738477.293.884.4
CeiT-T [92]6.42.422476.493.483.6
ConViT-Ti[15]6.02.022473.111
Cross ViT-Ti [6]6.93.222473.411
ViTAE-6M6.52.022477.994.184.9
PVT-T[80] LocalViT-PVT [43]13.23.822475.11
PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]13.59.622494.2-
ConViT-Ti+ [15] 10.04.022478.2 76.71
2.878.11-
10.6 19.26.2224 22480.2-1
13.23.422481.0- 95.4- 86.8
22.1 DeiT-S [76] 22.19.8 9.8224 22479.9 81.295.0 95.485.7 86.8
PVT-S[80]7.622479.8-
24.5 23.55.281.31
Conformer-Ti [58] Swin-T[47]224-
CeiT-S [92]29.09.022481.3-1
CvT-13 [83]24.2 20.09.022482.0 81.695.987.3 86.7
ConViT-S[15]27.09.0 10.8224 22481.31 11
Cross ViT-S [6]26.711.222481.011
PiT-S [29]23.54.822480.9
TNT-S [23]-1
Twins-PCPVT-S[10]23.810.422481.395.6-
24.17.422481.2--
Twins-SVT-S [10]24.05.622481.7-1
T2T-ViT-14 [93]21.55.222481.595.786.8
ViTAE-S23.65.622482.095.987.0
ViTAE-S ↑ 38423.620.238483.096.287.5
+ +We use $\checkmark$ and $\times$ to denote whether or not the corresponding module is enabled during the experiments. If all columns under the RC and NC are marked $\times$ as shown in the first row, the model becomes the standard T2T-ViT model. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN after the convolutional layer or not. $\mathit { \Omega } ^ { 6 } \times 3 \mathit { \Omega } ^ { 5 }$ in the first column denotes that the dilation rate set is the same in the three RCs. “ $[ 1 , 2 , 3 , 4 ]$ $\downarrow ^ { \circ }$ denotes using lower dilation rates in deeper RCs, i.e., $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $S _ { 2 } = [ 1 , 2 , 3 ]$ , $S _ { 3 } = [ 1 , 2 ]$ . + +As can be seen, using a pre-fusion strategy and BN achieves the best $6 9 . 9 \%$ Top-1 accuracy among other settings. It is noteworthy that all the variants of NC outperform the vanilla T2T-ViT, implying the effectiveness of PCM, which introduces the intrinsic locality IB in transformers. It can also be observed that BN plays an important role in improving the model’s performance as it can help to alleviate the scale deviation between convolution’s and attention’s features. For the RC, we first investigate the impact of using different dilation rates in the PRM, as shown in the first column. As can be seen, using larger dilation rates (e.g., 4 or 5) does not deliver better performance. We suspect that larger dilation rates may lead to plain features in the deeper RCs due to the smaller resolution of feature maps. To + +Table 3: Ablation Study of RCs and NCs in our ViTAE. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN or not. “ $[ 1 , 2 , 3 , 4 ]$ $\downarrow ^ { \circ }$ denotes using smaller dilation rates in deeper RCs, i.e., $\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $\mathsf { \bar { S } } _ { 2 } = [ 1 , 2 , 3 ]$ , $ { S _ { 3 } } = [ 1 , 2 ]$ . + +
Reduction CellNormal CellTop-1
Dilation (S1 ~ S3) PCMPreBN Post
× ××× ×68.7
× ×√× ×69.1
× ××√ ×69.0
× ××√ √68.8
× ×√× √69.9
[1,2]×3× ××× 69.5
[1,2,3]×3 ××× ×69.9
[1,2,3,4] × 3 ××× ×69.2
[1,2,3,4,5] × 3 ××× ×68.9
[1,2,3,4]↓ ××× ×69.8
[1,2,3,4]↓ √×× ×71.7
[1,2,3,4]↓ √√× √72.6
+ +validate the hypothesis, we use smaller dilation rates in deeper RCs as denoted by $[ 1 , 2 , 3 , 4 ] \downarrow$ . As can be seen, it achieves comparable performance as $[ 1 , 2 , 3 ] \times$ . However, compared with $[ 1 , 2 , 3 , 4 ] \downarrow$ , $[ 1 , 2 , 3 ] \times$ increases the amount of parameters from 4.35M to $4 . 6 \mathsf { M }$ . Therefore, we select $[ 1 , 2 , 3 , 4 ] \downarrow$ as the default setting. In addition, after using PCM in the RC, it introduces the intrinsic locality IB, and the performance increases to $7 1 . 7 \%$ Top-1 accuracy. Finally, the combination of RCs and NCs achieves the best accuracy at $7 2 . 6 \%$ , demonstrating the complementarity between our RCs and NCs. + +# 4.4 Data efficiency and training efficiency + +To validate the effectiveness of the introduced intrinsic IBs in improving data efficiency and training efficiency, we compare our ViTAE with T2T-ViT at different training settings: (a) training them using $20 \%$ , $60 \%$ , and $100 \%$ ImageNet training set for equivalent 100 epochs on the full ImageNet training set, e.g., we employ 5 times epochs when using $20 \%$ data for training compared with using $100 \%$ data; and (b) training them using the full ImageNet training set for 100, 200, and 300 epochs respectively. The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For example, ViTAE using only $20 \%$ training data achieves comparable performance with T2T-ViT using all data. When $60 \%$ training data are used, ViTAE significantly outperforms T2T-ViT using all data by about an absolute $3 \%$ accuracy. It is also noteworthy that ViTAE trained for only 100 epochs has outperformed T2T-ViT trained for 300 epochs. After training ViTAE for 300 epochs, its performance is significantly boosted to $7 5 . 3 \%$ Top-1 accuracy. With the proposed RCs and NCs, the transformer layers in our ViTAE only need to focus on modeling long-range dependencies, leaving the locality and multi-scale context modeling to its convolution counterparts, i.e., PCM and PRM. Such a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to learn more efficiently with less training data and fewer training epochs. + +To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch on the smaller datasets, i.e., Cifar10 and Cifar100. The results are summarized in Table 4. It can be viewed that with only $1 / 7$ number of epochs, the ViTAE-T model achieves better classification performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms ViTAE model’s data efficiency. + +Table 4: Results of training from scratch on Cifar10/100. + +
ModelParams (M)Top-1 AccEpochsDataset
DeiT-B86.097.57000Cifar10
ViTAE-T4.897.71000Cifar10
ViTAE-T4.885.01000Cifar100
+ +# 4.5 Generalization on downstream tasks + +Table 5: Generalization of ViTAE and SOTA methods on different downstream tasks. + +
ModelParams (M)Cifar10Cifar100iNat19CarsFlowersPets
Grafit ResNet-50 [78]25.6-=75.992.598.2-
EfficientNet-B5 [73]3098.191.1--98.5-
ViT-B/16 [19]86.598.187.1-189.593.8
ViT-L/16 [19]304.397.986.4--89.793.6
DeiT-B [76]86.699.190.877.792.198.4-
T2T-ViT-14 [93]21.598.388.4---1
ViTAE-T4.897.386.073.389.597.592.6
ViTAE-S23.698.890.876.091.497.894.2
+ +We further investigate the generalization of the proposed ViTAE models on downstream tasks by finetuning them on the training sets of several fine-grained classification tasks3, including Flowers [53], Cars [36], Pets [56], and iNaturalist19. We also fine-tune the proposed ViTAE models on Cifar10 [37] and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA performance on most of the datasets using comparable or fewer parameters. These results demonstrate that the good generalization ability of our ViTAE. + +# 4.6 Visual inspection of ViTAE + +To further analyze the property of our ViTAE, we first calculate the average attention distance of each layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality, the transformer layers in the proposed NCs can better focus on modeling long-range dependencies, especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important. + +These results confirm the effectiveness of the adopted “divide-and-conquer” idea in the proposed ViTAE, i.e., introducing the intrinsic locality IB from convolutions into vision transformers makes it possible that transformer layers only need to be responsible to long-range dependencies, since locality can be well modeled by convolutions in PCM. + +![](images/0d27a138487be9062b7073116e5379b9d74b5e8122843b673cecf6fb02d9776d.jpg) +Figure 3: The average per-layer attention distance of T2T-ViT-7 and our ViTAE-T. + +Besides, we apply Grad-CAM [66] on the MHSA’s output in the last NC to qualitatively inspect ViTAE. The visualization results are provided in Figure 4. Compared with the baseline T2T-ViT, our ViTAE covers the single or multiple targets in the images more precisely and attends less to the background. Moreover, ViTAE can better handle the scale variance issue as shown in Figure 4(b). Namely, it can precisely cover the birds no matter they are in small, middle, or large size. Such observations demonstrate that introducing the intrinsic IBs of locality and scale-invariance from convolutions to transformers helps ViTAE learn more discriminate features than the pure transformers. + +![](images/c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg) +Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing multiple or single objects and the heatmaps. (b) Images containing the same class of objects at different scales and the heatmaps (Best viewed in color). + +# 5 Limitation and discussion + +In this paper, we explore two types of IBs and incorporate them into transformers through the proposed reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless, due to computational resource constraints, we have not scaled the ViTAE model and train it on largesize dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we are optimistic about its scale property from the following preliminary evidence. As illustrated in Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer layers and convolution layers owing to the skip connection and parallel structure. According to the attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting viewpoint invariance [64] can be explored in the future study. + +# 6 Conclusion + +In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE. Extensive experiments show that ViTAE outperforms representative vision transformers in various respects including classification accuracy, data efficiency, training efficiency, and generalization ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. We hope that this study will provide valuable insights to the following studies of introducing intrinsic IB into vision transformers and understanding the impact of intrinsic and learned IBs. + +Acknowledgement Dr. Jing Zhang is supported by the ARC project FL-170100117. + +# References + +processing. RCA engineer, 29(6):33–41, 1984. [2] J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. +[3] H. Bay, T. Tuytelaars, and L. Van Gool. Surf: Speeded up robust features. In European conference on computer vision, pages 404–417. Springer, 2006. [4] P. 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In International Conference on Learning Representations, 2021. \ No newline at end of file diff --git a/parse/train/_WnAQKse_uK/_WnAQKse_uK_content_list.json b/parse/train/_WnAQKse_uK/_WnAQKse_uK_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..52a99f3ce7c6b59b904f2a2c8e630d59295be0d0 --- /dev/null +++ b/parse/train/_WnAQKse_uK/_WnAQKse_uK_content_list.json @@ -0,0 +1,1010 @@ +[ + { + "type": "text", + "text": "ViTAE: Vision Transformer Advanced by Exploring Intrinsic Inductive Bias ", + "text_level": 1, + "bbox": [ + 183, + 122, + 816, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yufei Xu1∗ Qiming Zhang1∗ Jing Zhang1 Dacheng Tao2,1 ", + "bbox": [ + 225, + 220, + 772, + 237 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1The University of Sydney, Australia, 2JD Explore Academy, China ", + "bbox": [ + 398, + 253, + 647, + 284 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "{yuxu7116,qzha2506}@uni.sydney.edu.au, jing.zhang1@sydney.edu.au, dacheng.tao@gmail.com ", + "bbox": [ + 184, + 297, + 854, + 310 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 332, + 535, + 348 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformers have shown great potential in various computer vision tasks owing to their strong capability in modeling long-range dependency using the self-attention mechanism. Nevertheless, vision transformers treat an image as 1D sequence of visual tokens, lacking an intrinsic inductive bias (IB) in modeling local visual structures and dealing with scale variance. Alternatively, they require large-scale training data and longer training schedules to learn the IB implicitly. In this paper, we propose a new Vision Transformer Advanced by Exploring intrinsic IB from convolutions, i.e., ViTAE. Technically, ViTAE has several spatial pyramid reduction modules to downsample and embed the input image into tokens with rich multi-scale context by using multiple convolutions with different dilation rates. In this way, it acquires an intrinsic scale invariance IB and is able to learn robust feature representation for objects at various scales. Moreover, in each transformer layer, ViTAE has a convolution block in parallel to the multi-head selfattention module, whose features are fused and fed into the feed-forward network. Consequently, it has the intrinsic locality IB and is able to learn local features and global dependencies collaboratively. Experiments on ImageNet as well as downstream tasks prove the superiority of ViTAE over the baseline transformer and concurrent works. Source code and pretrained models will be available at code. ", + "bbox": [ + 233, + 363, + 766, + 612 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 637, + 310, + 655 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformers [79, 17, 40, 14, 46, 61] have shown a domination trend in NLP studies owing to their strong ability in modeling long-range dependencies by the self-attention mechanism [67, 81, 51]. Such success and good properties of transformers has inspired following many works that apply them in various computer vision tasks [19, 100, 97, 80, 7]. Among them, ViT [19] is the pioneering pure transformer model that embeds images into a sequence of visual tokens and models the global dependencies among them with stacked transformer blocks. Although it achieves promising performance on image classification, it requires large-scale training data and a longer training schedule. One important reason is that ViT ", + "bbox": [ + 174, + 670, + 496, + 876 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/4515266c49c6aff7e7d656459cb5d8b9e3662e222790882fee6a157e2ad56504.jpg", + "image_caption": [ + "Figure 1: Comparison of data and training efficiency of T2T-ViT-7 and ViTAE-T on ImageNet. " + ], + "image_footnote": [], + "bbox": [ + 514, + 651, + 820, + 835 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "lacks intrinsic inductive bias (IB) in modeling local visual structures (e.g., edges and corners) and dealing with objects at various scales like convolutions. Alternatively, ViT has to learn such IB implicitly from large-scale data. ", + "bbox": [ + 174, + 92, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70, 62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that computes local correlation among neighbor pixels is good at extracting local features such as edges and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94], which are then aggregated into high-level features progressively by a bulk of sequential convolutions [32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not well suited to model long-range dependencies2, which is the key advantage of transformers. An interesting question comes up: Can we improve vision transformers by leveraging the good properties of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model as the teacher and consumes extra training cost. ", + "bbox": [ + 174, + 140, + 825, + 359 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing the network structures in this paper. Current vision transformers always obtain tokens with singlescale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data. For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner. Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale neighboring contextual information and progressively structurizes the image to tokens. Motivated by the success of CNNs in dealing with scale variance, we explore a similar design in transformers, i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn scale-invariant features more efficiently from data. On the other hand, low-level local features are fundamental elements to generate high-level discriminative features. Although transformers can also learn such features at shallow layers from data, they are not skilled as convolutions by design. Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that locality is a reasonable compensation of global dependency. However, this serial structure ignores the global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the “divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and then fuse the features to account for both. In this way, we empower transformers to learn local and long-range features within each block more effectively. ", + "bbox": [ + 174, + 366, + 826, + 628 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias $( V i T A E )$ , which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell (NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale context while NCs aim to jointly model locality and global dependencies in the token sequence. Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial resolution by $1 / 1 6$ and a series of NCs to learn discriminative features from data. ViTAE outperforms representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as well as classification accuracy and generalization on downstream tasks. ", + "bbox": [ + 174, + 636, + 825, + 787 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e., scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature learning ability of transformers. Second, we design a new transformer architecture named ViTAE based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features effectively. Third, ViTAE outperforms representative vision transformers regarding classification accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves $7 5 . 3 \\%$ and $8 2 . 0 \\%$ top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively. ", + "bbox": [ + 174, + 794, + 825, + 863 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 133 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 152, + 321, + 170 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 CNNs with intrinsic IB ", + "text_level": 1, + "bbox": [ + 176, + 184, + 372, + 199 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream computer vision tasks. The convolution operations in CNNs extract local features from the neighbor pixels within the receptive field determined by the kernel size [42]. Following the intuition that local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For example, to effectively learn features of large objects, a large receptive field is needed by either using large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32, 68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid [8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image mainly encode the large objects while features from the large scale image respond more to small objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale context by using multiple convolutions with different receptive fields within a single layer, i.e., intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation rates in the reduction cells to encode multi-scale context into each visual token. ", + "bbox": [ + 174, + 209, + 825, + 458 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Vision transformers with learned IB ", + "text_level": 1, + "bbox": [ + 176, + 474, + 462, + 489 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed learns the IB from amounts of data implicitly. Following works along this direction are to simplify the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75, 22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher, introducing extra computation cost during training. Recently, some works try to introduce the intrinsic IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example, [43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise structures while [47, 33] apply attention within local windows. However, these serial structure may ignore the global context during locality modeling (and vice versa). [88] establishes connection across different scales at the cost of heavy computation. Instead, we follow the “divide-and-conquer” idea and propose to model locality and global dependencies simultaneously via a parallel structure within each transformer layer. Conformer [58], the most relevant concurrent work to us, employs a unit to explore inter-block interactions between parallel convolution and transformer blocks. In contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each other within the transformer block. In addition, Conformer is not designed to have inherent scale invariance IB. ", + "bbox": [ + 173, + 501, + 826, + 790 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methodology ", + "text_level": 1, + "bbox": [ + 174, + 809, + 313, + 827 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Revisit vision transformer ", + "text_level": 1, + "bbox": [ + 174, + 842, + 393, + 856 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We first give a brief review of vision transformer in this part. To adapt transformers to vision tasks, ViT [19] first splits an image $x \\in R ^ { H \\times W \\times C }$ into tokens with a reduction ratio of $p$ (i.e., $x _ { t } \\in R ^ { ( ( H \\times W ) / p ^ { 2 } ) \\times D } )$ , where $H , W$ and $C$ denote the height, width, and channel dimensions of the input image, $D = C p ^ { 2 }$ denotes the token dimension. Then, an extra class token is concatenated to the visual tokens before adding position embeddings in an element-wise manner. The resulting tokens are fed into the following transformer layers. Each transformer layer is composed of two parts, i.e., a multi-head self-attention module (MHSA) and a feed forward network (FFN). ", + "bbox": [ + 174, + 867, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/12ff638562a0b78bbcef09633d771c21fc798831f54a00e148f38d5e89e3596b.jpg", + "image_caption": [ + "Figure 2: The structure of the proposed ViTAE. It is constructed by stacking three RCs and several NCs. Both types of cells share a simple basic structure, i.e., an MHSA module and a parallel convolutional module followed by an FFN. In particular, RC has an extra pyramid reduction module using atrous convolutions with different dilation rates to embed multi-scale context into tokens. " + ], + "image_footnote": [], + "bbox": [ + 179, + 88, + 823, + 328 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 433, + 825, + 489 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "MHSA Multi-head self-attention extends single-head self-attention (SHSA) by using different projection matrices for each head. Specifically, the input tokens $x _ { t }$ are first projected to queries $( Q )$ , keys $( K )$ and values $( V )$ using projection matrices, i.e., $Q , K , V = x _ { t } W _ { Q } , x _ { t } Q _ { K } , x _ { t } Q _ { V }$ , where $W _ { Q / K / V } \\in R ^ { D \\times D }$ denotes the projection matrix for query, key, and value, respectively. Then, the self-attention operation is calculated as: ", + "bbox": [ + 173, + 505, + 825, + 578 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/67c1b6a153e357266ce63dcf66004997f007dc52c414762d6054a92d21ab5683.jpg", + "text": "$$\nA t t e n t i o n ( Q , K , V ) = s o f t m a x ( \\frac { Q K ^ { T } } { \\sqrt { D } } ) V .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 585, + 648, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This SHSA module is repeated for $h$ times to formulate the MHSA module, where $h$ is the number of heads. The output features of the $h$ heads are concatenated along the channel dimension and formulate the output of the MHSA module. ", + "bbox": [ + 174, + 627, + 825, + 670 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "FFN FFN is placed on top of the MHSA module and applied to each token identically and separately. It consists of two linear transformations with an activation function in between. Besides, a layer normalization [2] and a shortcut are added before and aside from the MHSA and FFN, respectively. ", + "bbox": [ + 174, + 684, + 825, + 728 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Overview architecture of ViTAE ", + "text_level": 1, + "bbox": [ + 176, + 744, + 436, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "ViTAE aims to introduce the intrinsic IB in CNNs to vision transformers. As shown in Figure 2, ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding multi-scale context and local information into tokens, and NCs are used to further model the locality and long-range dependencies in the tokens. Taken an image $x \\in R ^ { H \\times W \\times C }$ as input, three RCs are used to gradually downsample $x$ by $4 \\times , 2 \\times$ , and $2 \\times$ , respectively. Thereby, the output tokens of the RCs are of size $[ H / 1 6 , W / 1 6 , D ]$ where $D$ is the token dimension (64 in our experiments). The output tokens of RCs are then flattened as $R ^ { H W / 2 5 6 \\times D }$ , concatenated with the class token, and added by the sinusoid position encoding. Next, the tokens are fed into the following NCs, which keep the length of the tokens. Finally, the prediction probability is obtained using a linear classification layer on the class token from the last NC. ", + "bbox": [ + 173, + 770, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 Reduction cell ", + "text_level": 1, + "bbox": [ + 174, + 90, + 312, + 106 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Instead of directly splitting and flatten images into visual tokens based on a linear image patch embedding layer, we devise the reduction cell to embed multi-scale context and local information into visual tokens, which introduces the intrinsic scale-invariance and locality IBs from convolutions. Technically, RC has two parallel branches responsible for modeling locality and long-range dependency, respectively, followed by an FFN for feature transformation. We denote the input feature of the $i _ { t h } \\ : \\mathrm { R C }$ as $f _ { i } \\in \\dot { R } ^ { H _ { i } \\times W _ { i } \\times D _ { i } }$ . The input of the first RC is the image $x$ . In the global dependencies branch, $f _ { i }$ is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e., ", + "bbox": [ + 173, + 117, + 826, + 215 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/995f71351d9a0d0f5ed166388dc336a063cfcb324dbc2f6372f07fdf76e058d7.jpg", + "text": "$$\nf _ { i } ^ { m s } \\triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \\in S _ { i } , r _ { i } \\in { \\mathcal { R } } ] ) ,\n$$", + "text_format": "latex", + "bbox": [ + 279, + 223, + 715, + 242 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $C o n v _ { i j } ( \\cdot )$ indicates the $j$ th convolutional layer in the PRM $( P R M _ { i } ( \\cdot ) )$ . It uses a dilation rate $s _ { i j }$ from the predefined dilation rate set $S _ { i }$ corresponding to the ith RC. Note that we use stride convolution to reduce the spatial dimension of features by a ratio $r _ { i }$ from the predefined reduction ratio set $\\mathcal { R }$ . The conv features are concatenated along the channel dimension, i.e., $f _ { i } ^ { m s } \\in$ $R ^ { ( W _ { i } / p ) \\times ( H _ { i } / p ) \\times ( | S _ { i } | D ) }$ , where $| { S _ { i } } |$ denotes the number of dilation rates in $S _ { i }$ . $f _ { i } ^ { m s }$ is then processed by an MHSA module to model long-range dependencies, i.e., ", + "bbox": [ + 173, + 248, + 825, + 335 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/47bb46efbf400464cc18685ec76d7959be215ba831aad37913cc57da0f8d98b9.jpg", + "text": "$$\nf _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 383, + 343, + 611, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $I m g 2 S e q ( \\cdot )$ is a simple reshape operation to flatten the feature map to a 1D sequence. In this way, $f _ { i } ^ { g }$ embeds the multi-scale context in each token. In addition, we use a Parallel Convolutional Module (PCM) to embed local context within the tokens, which are fused with $f _ { i } ^ { g }$ as follows: ", + "bbox": [ + 174, + 367, + 825, + 410 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/aab14cd318e095613617d3482aa2f81617b3140ca8a4a2e53d1fa0230ff83a82.jpg", + "text": "$$\nf _ { i } ^ { l g } = f _ { i } ^ { g } + { \\cal P } { \\cal C } M _ { i } ( f _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 420, + 580, + 439 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here, $P C M _ { i } ( \\cdot )$ represents the PCM, which is composed of three stacked convolution layers and an $I m g 2 S e q ( \\cdot )$ operation. It is noteworthy that the parallel convolution branch has the same spatial downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed into the following RC or NC, i.e., ", + "bbox": [ + 173, + 446, + 825, + 530 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/716d8b1849d36a1ad308d5ce41f0df05844de9ab779da38fd00d2d340e97c83e.jpg", + "text": "$$\nf _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 539, + 625, + 559 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the $S e q 2 I m g ( \\cdot )$ is a simple reshape operation to reshape a token sequence back to feature maps. $F F N _ { i } ( \\cdot )$ represents the FFN in the ith RC. In our ViTAE, three RCs are stacked sequentially to gradually reduce the input image’s spatial dimension by $4 \\times , 2 \\times$ , and $2 \\times$ , respectively. The feature maps generated by the last RC are of a size of $[ H / 1 6 , W / 1 6 , D ]$ , which are then flattened into visual tokens and fed into the following NCs. ", + "bbox": [ + 174, + 565, + 825, + 636 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 Normal cell ", + "text_level": 1, + "bbox": [ + 174, + 654, + 294, + 667 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell except for the absence of the PRM. Due to the relatively small $( \\frac { 1 } { 1 6 } \\times )$ spatial size of feature maps after RCs, it is unnecessary to use PRM in NCs. Given $f _ { 3 }$ from the third RC, we first concatenate it with the class token $t _ { c l s }$ , and then add it to the positional encodings to get the input tokens $t$ for the following NCs. Here we ignore the subscript for clarity since all NCs have an identical architecture but different learnable weights. $t _ { c l s }$ is randomly initialized at the start of training and fixed during the inference. Similar to the RC, the tokens are fed into the MHSA module, i.e., $t _ { g } = M H S A ( t )$ . Meanwhile, they are reshaped to 2D feature maps and fed into the PCM, i.e., $t _ { l } = \\bar { I } m g 2 S e q ( P C M ( S e q 2 I m g ( t ) ) )$ . Note that the class token is discarded in PCM because it has no spatial connections with other visual tokens. To further reduce the parameters in NCs, we use group convolutions in PCM. The features from MHSA and PCM are then fused via element-wise sum, i.e., $t _ { l g } = t _ { g } + t _ { l }$ . Finally, $t _ { l g }$ are fed into the FFN to get the output features of NC, i.e., $t _ { n c } = F F N ( t _ { l g } ) \\overline { { + } } t _ { l g }$ . Similar to ViT [19], we apply layer normalization to the class token generated by the last NC and feed it to the classification head to get the final classification result. ", + "bbox": [ + 173, + 679, + 826, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 Model details ", + "text_level": 1, + "bbox": [ + 174, + 891, + 307, + 905 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use two variants of ViTAE in our experiments for a fair comparison of other models with similar model sizes. The details of them are summarized in Table 1. In the first RC, the default convolution kernel size is $7 \\times 7$ with a ", + "bbox": [ + 173, + 92, + 388, + 186 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b78024fbd1f12dcc881657db015afe79b126382b56ffdf33fae9095f44e9bcf1.jpg", + "table_caption": [ + "Table 1: Model details of two variants of ViTAE. " + ], + "table_footnote": [], + "table_body": "
ModelReduction Cell DilationCellsNormal Cell Heads Embed CellsParams Macs (M)(G)
ViTAE-T[1,2,3,4] √34 25674.81.5
ViTAE-S[1,2,3,4] √36384 1423.65.6
", + "bbox": [ + 405, + 113, + 816, + 166 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "stride of 4 and dilation rates of $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ . In the following two RCs, the convolution kernel size is $3 \\times 3$ with a stride of 2 and dilation rates of $S _ { 2 } = [ 1 , 2 , 3 ]$ and $S _ { 3 } = [ 1 , 2 ]$ , respectively. Since the spatial dimension of tokens decreases, there is no need to use large kernels and dilation rates. PCM in both RCs and NCs comprises three convolutional layers with a kernel size of $3 \\times 3$ . ", + "bbox": [ + 174, + 188, + 825, + 243 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 263, + 312, + 280 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Implementation details ", + "text_level": 1, + "bbox": [ + 174, + 295, + 374, + 310 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We train and test the proposed ViTAE model on the standard ImageNet [38] dataset, which contains about 1.3 million images and covers 1k classes. Unless explicitly stated, the image size during training is set to $2 2 4 \\times 2 2 4$ . We use the AdamW [48] optimizer with the cosine learning rate scheduler and uses the data augmentation strategy exactly the same as T2T [93] for a fair comparison, regarding the training strategies and the size of models. We use a batch size of 512 for training all our models and set the initial learning rate to be 5e-4. The results of our models can be found in Table 2, where all the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and TIMM [82]. ", + "bbox": [ + 174, + 320, + 825, + 431 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Comparison with the state-of-the-art ", + "text_level": 1, + "bbox": [ + 176, + 449, + 467, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare our ViTAE with both CNN models and vision transformers with similar model sizes in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are reported. We categorize the methods into CNN models, vision transformers with learned IB, and vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves a $7 5 . 3 \\%$ Top-1 accuracy, which is better than ResNet-18 with more parameters. The real Top-1 accuracy of the ViTAE model is $8 2 . 9 \\%$ , which is comparable to ResNet-50 that has four more times of parameters than ours. Similarly, our ViTAE-S achieves $8 2 . 0 \\%$ Top-1 accuracy with half of the parameters of ResNet-101 and ResNet-152, showing the superiority of learning both local and longrange features from specific structures with corresponding intrinsic IBs by design. Similar phenomena can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65], where ViTAE obtains better performance with fewer parameters. When compared with larger models which are searched according to NAS [73], our ViTAE-S achieves a similar performance when using $3 8 4 \\times 3 8 4$ images as input, which further shows the potential of vision transformers with intrinsic IB. ", + "bbox": [ + 174, + 474, + 825, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In addition, among the transformers with learned IB, ViT is the first pure transformer model for visual recognition. DeiT shares the same structure with ViT but uses different data augmentation and training strategies to facilitate the learning of transformers. DeiT⚗ denotes using an off-the-shelf CNN model as the teacher model to train DeiT, which introduces the intrinsic IB from CNN to transformer implicitly in a knowledge distillation manner, showing better performance than the vanilla ViT on the ImageNet dataset. It is exciting to see that our ViTAE-T with fewer parameters even outperforms the distilled model DeiT⚗, demonstrating the efficacy of introducing intrinsic IBs in transformers by design. Besides, compared with other transformers with explicit intrinsic IB, our ViTAE with fewer parameters also achieves comparable or better performance. For instance, ViTAE-T achieves comparable performance with LocalVit-T but has 1M fewer parameters, demonstrating the superiority of the proposed RCs and NCs in introducing intrinsic IBs. ", + "bbox": [ + 174, + 660, + 825, + 813 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 Ablation study ", + "text_level": 1, + "bbox": [ + 174, + 830, + 313, + 844 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use T2T-ViT [93] as our baseline model in the following ablation study of our ViTAE. As shown in Table 3, we investigate the hyper-parameter settings in RCs and NCs by isolating them separately. All the models are trained for 100 epochs on ImageNet and follow the same training setting and data augmentation strategy as described in Section 4.1. ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/e514264bc0b3536b852ece05e614d3ca959237e923c2f3d7e1d3c16e5426e296.jpg", + "table_caption": [ + "Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set. " + ], + "table_footnote": [], + "table_body": "
Type ModelParams (M)MACs (G)Input SizeImageNet Real
Top-1Top-5Top-1
CNNResNet-18 [26]11.73.622470.386.777.3
ResNet-50 [26]25.67.622476.793.382.5
ResNet-101 [26]44.515.222478.394.183.7
ResNet-152 [26]60.222.622478.994.484.1
EfficientNet-B0 [73]5.30.822477.193.383.5
EfficientNet-B4 [73]19.38.438082.996.488.0
MobileNetV1 [31]4.30.622472.31-
MobileNetV2(1.4) [65]6.90.622474.7--
RegNetY-600M[62]6.11.222475.5--
RegNetY-4GF[62] RegNetY-8GF[62]20.6 39.28.0 16.022480.0186.4
22481.7187.4
DeiT-T[76]5.72.622472.291.180.6
DeiT-T [76]5.72.622474.591.982.1
LocalViT-T[43]5.92.622474.892.6
LocalViT-T2T[43]4.32.422472.5-1 1
ConT-Ti [89]5.81.622474.9--
PiT-Ti [29]4.91.422473.0-1
T2T-ViT-7 [93]4.31.222471.790.979.7
ViTAE-T4.81.522475.392.782.9
ViTAE-T ↑ 3844.85.738477.293.884.4
CeiT-T [92]6.42.422476.493.483.6
ConViT-Ti[15]6.02.022473.111
Cross ViT-Ti [6]6.93.222473.411
ViTAE-6M6.52.022477.994.184.9
PVT-T[80] LocalViT-PVT [43]13.23.822475.11
PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]13.59.622494.2-
ConViT-Ti+ [15] 10.04.022478.2 76.71
2.878.11-
10.6 19.26.2224 22480.2-1
13.23.422481.0- 95.4- 86.8
22.1 DeiT-S [76] 22.19.8 9.8224 22479.9 81.295.0 95.485.7 86.8
PVT-S[80]7.622479.8-
24.5 23.55.281.31
Conformer-Ti [58] Swin-T[47]224-
CeiT-S [92]29.09.022481.3-1
CvT-13 [83]24.2 20.09.022482.0 81.695.987.3 86.7
ConViT-S[15]27.09.0 10.8224 22481.31 11
Cross ViT-S [6]26.711.222481.011
PiT-S [29]23.54.822480.9
TNT-S [23]-1
Twins-PCPVT-S[10]23.810.422481.395.6-
24.17.422481.2--
Twins-SVT-S [10]24.05.622481.7-1
T2T-ViT-14 [93]21.55.222481.595.786.8
ViTAE-S23.65.622482.095.987.0
ViTAE-S ↑ 38423.620.238483.096.287.5
", + "bbox": [ + 196, + 199, + 800, + 839 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use $\\checkmark$ and $\\times$ to denote whether or not the corresponding module is enabled during the experiments. If all columns under the RC and NC are marked $\\times$ as shown in the first row, the model becomes the standard T2T-ViT model. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN after the convolutional layer or not. $\\mathit { \\Omega } ^ { 6 } \\times 3 \\mathit { \\Omega } ^ { 5 }$ in the first column denotes that the dilation rate set is the same in the three RCs. “ $[ 1 , 2 , 3 , 4 ]$ $\\downarrow ^ { \\circ }$ denotes using lower dilation rates in deeper RCs, i.e., $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $S _ { 2 } = [ 1 , 2 , 3 ]$ , $S _ { 3 } = [ 1 , 2 ]$ . ", + "bbox": [ + 173, + 92, + 826, + 190 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As can be seen, using a pre-fusion strategy and BN achieves the best $6 9 . 9 \\%$ Top-1 accuracy among other settings. It is noteworthy that all the variants of NC outperform the vanilla T2T-ViT, implying the effectiveness of PCM, which introduces the intrinsic locality IB in transformers. It can also be observed that BN plays an important role in improving the model’s performance as it can help to alleviate the scale deviation between convolution’s and attention’s features. For the RC, we first investigate the impact of using different dilation rates in the PRM, as shown in the first column. As can be seen, using larger dilation rates (e.g., 4 or 5) does not deliver better performance. We suspect that larger dilation rates may lead to plain features in the deeper RCs due to the smaller resolution of feature maps. To ", + "bbox": [ + 174, + 194, + 421, + 497 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8cb6fb2c3865a2b27116e792990f87e65f07efdf975a9d033b0acebcd1d733eb.jpg", + "table_caption": [ + "Table 3: Ablation Study of RCs and NCs in our ViTAE. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN or not. “ $[ 1 , 2 , 3 , 4 ]$ $\\downarrow ^ { \\circ }$ denotes using smaller dilation rates in deeper RCs, i.e., $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $\\mathsf { \\bar { S } } _ { 2 } = [ 1 , 2 , 3 ]$ , $ { S _ { 3 } } = [ 1 , 2 ]$ . " + ], + "table_footnote": [], + "table_body": "
Reduction CellNormal CellTop-1
Dilation (S1 ~ S3) PCMPreBN Post
× ××× ×68.7
× ×√× ×69.1
× ××√ ×69.0
× ××√ √68.8
× ×√× √69.9
[1,2]×3× ××× 69.5
[1,2,3]×3 ××× ×69.9
[1,2,3,4] × 3 ××× ×69.2
[1,2,3,4,5] × 3 ××× ×68.9
[1,2,3,4]↓ ××× ×69.8
[1,2,3,4]↓ √×× ×71.7
[1,2,3,4]↓ √√× √72.6
", + "bbox": [ + 450, + 292, + 805, + 478 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "validate the hypothesis, we use smaller dilation rates in deeper RCs as denoted by $[ 1 , 2 , 3 , 4 ] \\downarrow$ . As can be seen, it achieves comparable performance as $[ 1 , 2 , 3 ] \\times$ . However, compared with $[ 1 , 2 , 3 , 4 ] \\downarrow$ , $[ 1 , 2 , 3 ] \\times$ increases the amount of parameters from 4.35M to $4 . 6 \\mathsf { M }$ . Therefore, we select $[ 1 , 2 , 3 , 4 ] \\downarrow$ as the default setting. In addition, after using PCM in the RC, it introduces the intrinsic locality IB, and the performance increases to $7 1 . 7 \\%$ Top-1 accuracy. Finally, the combination of RCs and NCs achieves the best accuracy at $7 2 . 6 \\%$ , demonstrating the complementarity between our RCs and NCs. ", + "bbox": [ + 174, + 498, + 826, + 580 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Data efficiency and training efficiency ", + "text_level": 1, + "bbox": [ + 176, + 611, + 473, + 626 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To validate the effectiveness of the introduced intrinsic IBs in improving data efficiency and training efficiency, we compare our ViTAE with T2T-ViT at different training settings: (a) training them using $20 \\%$ , $60 \\%$ , and $100 \\%$ ImageNet training set for equivalent 100 epochs on the full ImageNet training set, e.g., we employ 5 times epochs when using $20 \\%$ data for training compared with using $100 \\%$ data; and (b) training them using the full ImageNet training set for 100, 200, and 300 epochs respectively. The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For example, ViTAE using only $20 \\%$ training data achieves comparable performance with T2T-ViT using all data. When $60 \\%$ training data are used, ViTAE significantly outperforms T2T-ViT using all data by about an absolute $3 \\%$ accuracy. It is also noteworthy that ViTAE trained for only 100 epochs has outperformed T2T-ViT trained for 300 epochs. After training ViTAE for 300 epochs, its performance is significantly boosted to $7 5 . 3 \\%$ Top-1 accuracy. With the proposed RCs and NCs, the transformer layers in our ViTAE only need to focus on modeling long-range dependencies, leaving the locality and multi-scale context modeling to its convolution counterparts, i.e., PCM and PRM. Such a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to learn more efficiently with less training data and fewer training epochs. ", + "bbox": [ + 174, + 641, + 825, + 863 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch on the smaller datasets, i.e., Cifar10 and Cifar100. The results are summarized in Table 4. It can be viewed that with only $1 / 7$ number of epochs, the ViTAE-T model achieves better classification performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms ViTAE model’s data efficiency. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 90, + 825, + 119 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/63a31781efc1cf86aa1701ec0651b0baf9513eb3aeb90723d610223388fffb83.jpg", + "table_caption": [ + "Table 4: Results of training from scratch on Cifar10/100. " + ], + "table_footnote": [], + "table_body": "
ModelParams (M)Top-1 AccEpochsDataset
DeiT-B86.097.57000Cifar10
ViTAE-T4.897.71000Cifar10
ViTAE-T4.885.01000Cifar100
", + "bbox": [ + 290, + 154, + 705, + 214 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 Generalization on downstream tasks ", + "text_level": 1, + "bbox": [ + 176, + 239, + 465, + 253 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1811ce2c1001783f4b47fef070c93481a3eed9f0f8f8b509c00a30ea1da1a596.jpg", + "table_caption": [ + "Table 5: Generalization of ViTAE and SOTA methods on different downstream tasks. " + ], + "table_footnote": [], + "table_body": "
ModelParams (M)Cifar10Cifar100iNat19CarsFlowersPets
Grafit ResNet-50 [78]25.6-=75.992.598.2-
EfficientNet-B5 [73]3098.191.1--98.5-
ViT-B/16 [19]86.598.187.1-189.593.8
ViT-L/16 [19]304.397.986.4--89.793.6
DeiT-B [76]86.699.190.877.792.198.4-
T2T-ViT-14 [93]21.598.388.4---1
ViTAE-T4.897.386.073.389.597.592.6
ViTAE-S23.698.890.876.091.497.894.2
", + "bbox": [ + 217, + 287, + 779, + 406 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We further investigate the generalization of the proposed ViTAE models on downstream tasks by finetuning them on the training sets of several fine-grained classification tasks3, including Flowers [53], Cars [36], Pets [56], and iNaturalist19. We also fine-tune the proposed ViTAE models on Cifar10 [37] and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA performance on most of the datasets using comparable or fewer parameters. These results demonstrate that the good generalization ability of our ViTAE. ", + "bbox": [ + 173, + 422, + 826, + 506 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.6 Visual inspection of ViTAE ", + "text_level": 1, + "bbox": [ + 174, + 523, + 400, + 537 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To further analyze the property of our ViTAE, we first calculate the average attention distance of each layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality, the transformer layers in the proposed NCs can better focus on modeling long-range dependencies, especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important. ", + "bbox": [ + 173, + 549, + 821, + 632 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "These results confirm the effectiveness of the adopted “divide-and-conquer” idea in the proposed ViTAE, i.e., introducing the intrinsic locality IB from convolutions into vision transformers makes it possible that transformer layers only need to be responsible to long-range dependencies, since locality can be well modeled by convolutions in PCM. ", + "bbox": [ + 173, + 632, + 517, + 728 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/0d27a138487be9062b7073116e5379b9d74b5e8122843b673cecf6fb02d9776d.jpg", + "image_caption": [ + "Figure 3: The average per-layer attention distance of T2T-ViT-7 and our ViTAE-T. " + ], + "image_footnote": [], + "bbox": [ + 531, + 648, + 823, + 750 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Besides, we apply Grad-CAM [66] on the MHSA’s output in the last NC to qualitatively inspect ViTAE. The visualization results are provided in Figure 4. Compared with the baseline T2T-ViT, our ViTAE covers the single or multiple targets in the images more precisely and attends less to the background. Moreover, ViTAE can better handle the scale variance issue as shown in Figure 4(b). Namely, it can precisely cover the birds no matter they are in small, middle, or large size. Such observations demonstrate that introducing the intrinsic IBs of locality and scale-invariance from convolutions to transformers helps ViTAE learn more discriminate features than the pure transformers. ", + "bbox": [ + 174, + 734, + 519, + 804 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 805, + 825, + 873 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg", + "image_caption": [ + "Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing multiple or single objects and the heatmaps. (b) Images containing the same class of objects at different scales and the heatmaps (Best viewed in color). " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 823, + 294 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Limitation and discussion ", + "text_level": 1, + "bbox": [ + 176, + 385, + 419, + 401 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we explore two types of IBs and incorporate them into transformers through the proposed reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless, due to computational resource constraints, we have not scaled the ViTAE model and train it on largesize dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we are optimistic about its scale property from the following preliminary evidence. As illustrated in Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer layers and convolution layers owing to the skip connection and parallel structure. According to the attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting viewpoint invariance [64] can be explored in the future study. ", + "bbox": [ + 173, + 416, + 826, + 595 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 614, + 299, + 631 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE. Extensive experiments show that ViTAE outperforms representative vision transformers in various respects including classification accuracy, data efficiency, training efficiency, and generalization ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. We hope that this study will provide valuable insights to the following studies of introducing intrinsic IB into vision transformers and understanding the impact of intrinsic and learned IBs. ", + "bbox": [ + 174, + 646, + 825, + 771 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement Dr. Jing Zhang is supported by the ARC project FL-170100117. ", + "bbox": [ + 173, + 776, + 725, + 791 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 90, + 266, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "processing. RCA engineer, 29(6):33–41, 1984. [2] J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. \n[3] H. Bay, T. Tuytelaars, and L. Van Gool. Surf: Speeded up robust features. In European conference on computer vision, pages 404–417. Springer, 2006. [4] P. J. Burt and E. H. Adelson. The laplacian pyramid as a compact image code. 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Alternatively, ViT has to learn such IB", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 237, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 237, + 106 + ], + "score": 1.0, + "content": "implicitly from large-scale data.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 111, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 155 + ], + "score": 1.0, + "content": "start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "computes local correlation among neighbor pixels is good at extracting local features such as edges", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "which are then aggregated into high-level features progressively by a bulk of sequential convolutions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "[32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 233 + ], + "score": 1.0, + "content": "representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "well suited to model long-range dependencies2, which is the key advantage of transformers. An", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "interesting question comes up: Can we improve vision transformers by leveraging the good properties", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 298, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 298, + 287 + ], + "score": 1.0, + "content": "as the teacher and consumes extra training cost.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "the network structures in this paper. Current vision transformers always obtain tokens with single-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "scale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 507, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 507, + 336 + ], + "score": 1.0, + "content": "For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "neighboring contextual information and progressively structurizes the image to tokens. Motivated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "by the success of CNNs in dealing with scale variance, we explore a similar design in transformers,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "scale-invariant features more efficiently from data. On the other hand, low-level local features are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "fundamental elements to generate high-level discriminative features. Although transformers can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "also learn such features at shallow layers from data, they are not skilled as convolutions by design.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "locality is a reasonable compensation of global dependency. However, this serial structure ignores the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 479 + ], + "score": 1.0, + "content": "“divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 475, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 490 + ], + "score": 1.0, + "content": "then fuse the features to account for both. In this way, we empower transformers to learn local and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 487, + 329, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 329, + 500 + ], + "score": 1.0, + "content": "long-range features within each block more effectively.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 141, + 525 + ], + "score": 0.26, + "content": "( V i T A E )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "score": 1.0, + "content": ", which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "(NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "context while NCs aim to jointly model locality and global dependencies in the token sequence.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "score": 1.0, + "content": "context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 160, + 603 + ], + "score": 1.0, + "content": "resolution by", + "type": "text" + }, + { + "bbox": [ + 160, + 591, + 181, + 603 + ], + "score": 0.44, + "content": "1 / 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "and a series of NCs to learn discriminative features from data. ViTAE outperforms", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 613, + 394, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 394, + 625 + ], + "score": 1.0, + "content": "well as classification accuracy and generalization on downstream tasks.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e.,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "learning ability of transformers. Second, we design a new transformer architecture named ViTAE", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 700, + 313, + 713 + ], + "score": 1.0, + "content": "2Despite projection in transformer can be viewed as", + "type": "text" + }, + { + "bbox": [ + 314, + 704, + 335, + 711 + ], + "score": 0.88, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "convolution [9], the term of convolution here", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 246, + 723 + ], + "score": 1.0, + "content": "refers to those with larger kernels, e.g.,", + "type": "text" + }, + { + "bbox": [ + 246, + 712, + 268, + 721 + ], + "score": 0.86, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 711, + 505, + 723 + ], + "score": 1.0, + "content": ", which are widely used in typical CNNs to extract spatial features.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "lacks intrinsic inductive bias (IB) in modeling local visual structures (e.g., edges and corners) and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "dealing with objects at various scales like convolutions. Alternatively, ViT has to learn such IB", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 237, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 237, + 106 + ], + "score": 1.0, + "content": "implicitly from large-scale data.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 73, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 111, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 155 + ], + "score": 1.0, + "content": "start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "computes local correlation among neighbor pixels is good at extracting local features such as edges", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "which are then aggregated into high-level features progressively by a bulk of sequential convolutions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "[32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 233 + ], + "score": 1.0, + "content": "representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "well suited to model long-range dependencies2, which is the key advantage of transformers. An", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "interesting question comes up: Can we improve vision transformers by leveraging the good properties", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 298, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 298, + 287 + ], + "score": 1.0, + "content": "as the teacher and consumes extra training cost.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 110, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "the network structures in this paper. Current vision transformers always obtain tokens with single-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "scale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 507, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 507, + 336 + ], + "score": 1.0, + "content": "For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "neighboring contextual information and progressively structurizes the image to tokens. Motivated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "by the success of CNNs in dealing with scale variance, we explore a similar design in transformers,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 392 + ], + "score": 1.0, + "content": "into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "scale-invariant features more efficiently from data. On the other hand, low-level local features are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "fundamental elements to generate high-level discriminative features. Although transformers can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "also learn such features at shallow layers from data, they are not skilled as convolutions by design.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "locality is a reasonable compensation of global dependency. However, this serial structure ignores the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 506, + 467 + ], + "score": 1.0, + "content": "global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 479 + ], + "score": 1.0, + "content": "“divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 475, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 490 + ], + "score": 1.0, + "content": "then fuse the features to account for both. In this way, we empower transformers to learn local and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 487, + 329, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 329, + 500 + ], + "score": 1.0, + "content": "long-range features within each block more effectively.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 289, + 507, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 141, + 525 + ], + "score": 0.26, + "content": "( V i T A E )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "score": 1.0, + "content": ", which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "(NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "context while NCs aim to jointly model locality and global dependencies in the token sequence.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 504, + 592 + ], + "score": 1.0, + "content": "context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 160, + 603 + ], + "score": 1.0, + "content": "resolution by", + "type": "text" + }, + { + "bbox": [ + 160, + 591, + 181, + 603 + ], + "score": 0.44, + "content": "1 / 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "and a series of NCs to learn discriminative features from data. ViTAE outperforms", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 613, + 394, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 394, + 625 + ], + "score": 1.0, + "content": "well as classification accuracy and generalization on downstream tasks.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 504, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e.,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "learning ability of transformers. Second, we design a new transformer architecture named ViTAE", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "effectively. Third, ViTAE outperforms representative vision transformers regarding classification", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 486, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 134, + 105 + ], + "score": 0.87, + "content": "7 5 . 3 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 134, + 93, + 151, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 152, + 94, + 179, + 105 + ], + "score": 0.87, + "content": "8 2 . 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 180, + 93, + 486, + 108 + ], + "score": 1.0, + "content": "top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 628, + 506, + 686 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "effectively. Third, ViTAE outperforms representative vision transformers regarding classification", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 486, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 134, + 105 + ], + "score": 0.87, + "content": "7 5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 93, + 151, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 152, + 94, + 179, + 105 + ], + "score": 0.87, + "content": "8 2 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 93, + 486, + 108 + ], + "score": 1.0, + "content": "top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 121, + 197, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 198, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 198, + 136 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 108, + 146, + 228, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 229, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 229, + 158 + ], + "score": 1.0, + "content": "2.1 CNNs with intrinsic IB", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 179 + ], + "score": 1.0, + "content": "CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "computer vision tasks. The convolution operations in CNNs extract local features from the neighbor", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "pixels within the receptive field determined by the kernel size [42]. Following the intuition that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 229, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 506, + 246 + ], + "score": 1.0, + "content": "example, to effectively learn features of large objects, a large receptive field is needed by either using", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "[8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "score": 1.0, + "content": "resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "mainly encode the large objects while features from the large scale image respond more to small", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 507, + 322 + ], + "score": 1.0, + "content": "context by using multiple convolutions with different receptive fields within a single layer, i.e.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 352, + 424, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 424, + 363 + ], + "score": 1.0, + "content": "rates in the reduction cells to encode multi-scale context into each visual token.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 376, + 283, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 285, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 285, + 389 + ], + "score": 1.0, + "content": "2.2 Vision transformers with learned IB", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "learns the IB from amounts of data implicitly. Following works along this direction are to simplify", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "score": 1.0, + "content": "22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "introducing extra computation cost during training. Recently, some works try to introduce the intrinsic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "[43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "structures while [47, 33] apply attention within local windows. However, these serial structure may", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "ignore the global context during locality modeling (and vice versa). [88] establishes connection", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "across different scales at the cost of heavy computation. Instead, we follow the “divide-and-conquer”", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "idea and propose to model locality and global dependencies simultaneously via a parallel structure", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "within each transformer layer. Conformer [58], the most relevant concurrent work to us, employs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "a unit to explore inter-block interactions between parallel convolution and transformer blocks. In", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "score": 1.0, + "content": "other within the transformer block. In addition, Conformer is not designed to have inherent scale", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 614, + 165, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 165, + 626 + ], + "score": 1.0, + "content": "invariance IB.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 641, + 192, + 655 + ], + "lines": [ + { + "bbox": [ + 104, + 639, + 194, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 639, + 194, + 659 + ], + "score": 1.0, + "content": "3 Methodology", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 241, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 243, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 243, + 679 + ], + "score": 1.0, + "content": "3.1 Revisit vision transformer", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "We first give a brief review of vision transformer in this part. To adapt transformers to vision", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 695, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 257, + 712 + ], + "score": 1.0, + "content": "tasks, ViT [19] first splits an image", + "type": "text" + }, + { + "bbox": [ + 257, + 697, + 321, + 708 + ], + "score": 0.92, + "content": "x \\in R ^ { H \\times W \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 695, + 477, + 712 + ], + "score": 1.0, + "content": "into tokens with a reduction ratio of", + "type": "text" + }, + { + "bbox": [ + 477, + 700, + 484, + 709 + ], + "score": 0.74, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 695, + 507, + 712 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 707, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 201, + 722 + ], + "score": 0.91, + "content": "x _ { t } \\in R ^ { ( ( H \\times W ) / p ^ { 2 } ) \\times D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 707, + 232, + 725 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 232, + 711, + 259, + 721 + ], + "score": 0.33, + "content": "H , W", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 707, + 278, + 725 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 278, + 711, + 288, + 721 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 707, + 507, + 725 + ], + "score": 1.0, + "content": "denote the height, width, and channel dimensions of", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 303, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 72, + 506, + 108 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 121, + 197, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 198, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 198, + 136 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 108, + 146, + 228, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 229, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 229, + 158 + ], + "score": 1.0, + "content": "2.1 CNNs with intrinsic IB", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 179 + ], + "score": 1.0, + "content": "CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "computer vision tasks. The convolution operations in CNNs extract local features from the neighbor", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "pixels within the receptive field determined by the kernel size [42]. Following the intuition that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 234 + ], + "score": 1.0, + "content": "multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 229, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 506, + 246 + ], + "score": 1.0, + "content": "example, to effectively learn features of large objects, a large receptive field is needed by either using", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "[8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "score": 1.0, + "content": "resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "mainly encode the large objects while features from the large scale image respond more to small", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 507, + 322 + ], + "score": 1.0, + "content": "context by using multiple convolutions with different receptive fields within a single layer, i.e.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 352, + 424, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 424, + 363 + ], + "score": 1.0, + "content": "rates in the reduction cells to encode multi-scale context into each visual token.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 165, + 507, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 376, + 283, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 285, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 285, + 389 + ], + "score": 1.0, + "content": "2.2 Vision transformers with learned IB", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "learns the IB from amounts of data implicitly. Following works along this direction are to simplify", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "score": 1.0, + "content": "22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "introducing extra computation cost during training. Recently, some works try to introduce the intrinsic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "[43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "structures while [47, 33] apply attention within local windows. However, these serial structure may", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "ignore the global context during locality modeling (and vice versa). [88] establishes connection", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "across different scales at the cost of heavy computation. 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In", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "score": 1.0, + "content": "other within the transformer block. 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As shown in Figure 2,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "multi-scale context and local information into tokens, and NCs are used to further model the locality", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 641, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 348, + 658 + ], + "score": 1.0, + "content": "and long-range dependencies in the tokens. 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It is constructed by stacking three RCs and several", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "NCs. Both types of cells share a simple basic structure, i.e., an MHSA module and a parallel", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "convolutional module followed by an FFN. 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Then, the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 446, + 267, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 267, + 459 + ], + "score": 1.0, + "content": "self-attention operation is calculated as:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 400, + 507, + 459 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 464, + 397, + 492 + ], + "lines": [ + { + "bbox": [ + 214, + 464, + 397, + 492 + ], + "spans": [ + { + "bbox": [ + 214, + 464, + 397, + 492 + ], + "score": 0.93, + "content": "A t t e n t i o n ( Q , K , V ) = s o f t m a x ( \\frac { Q K ^ { T } } { \\sqrt { D } } ) V .", + "type": "interline_equation", + "image_path": "67c1b6a153e357266ce63dcf66004997f007dc52c414762d6054a92d21ab5683.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 214, + 464, + 397, + 478.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 214, + 478.0, + 397, + 492.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 247, + 508 + ], + "score": 1.0, + "content": "This SHSA module is repeated for", + "type": "text" + }, + { + "bbox": [ + 247, + 498, + 254, + 507 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 497, + 440, + 508 + ], + "score": 1.0, + "content": "times to formulate the MHSA module, where", + "type": "text" + }, + { + "bbox": [ + 440, + 498, + 447, + 507 + ], + "score": 0.84, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 497, + 505, + 508 + ], + "score": 1.0, + "content": "is the number", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 508, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 259, + 519 + ], + "score": 1.0, + "content": "of heads. The output features of the", + "type": "text" + }, + { + "bbox": [ + 259, + 509, + 267, + 518 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 508, + 504, + 519 + ], + "score": 1.0, + "content": "heads are concatenated along the channel dimension and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 520, + 281, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 281, + 531 + ], + "score": 1.0, + "content": "formulate the output of the MHSA module.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 497, + 505, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 507, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 507, + 556 + ], + "score": 1.0, + "content": "FFN FFN is placed on top of the MHSA module and applied to each token identically and separately.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "It consists of two linear transformations with an activation function in between. 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As shown in Figure 2,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "multi-scale context and local information into tokens, and NCs are used to further model the locality", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 641, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 641, + 348, + 658 + ], + "score": 1.0, + "content": "and long-range dependencies in the tokens. Taken an image", + "type": "text" + }, + { + "bbox": [ + 349, + 643, + 410, + 654 + ], + "score": 0.92, + "content": "x \\in R ^ { H \\times W \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 641, + 506, + 658 + ], + "score": 1.0, + "content": "as input, three RCs are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 234, + 667 + ], + "score": 1.0, + "content": "used to gradually downsample", + "type": "text" + }, + { + "bbox": [ + 234, + 657, + 241, + 665 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 655, + 254, + 667 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 255, + 655, + 288, + 666 + ], + "score": 0.31, + "content": "4 \\times , 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 655, + 309, + 667 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 309, + 655, + 323, + 666 + ], + "score": 0.88, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ", respectively. Thereby, the output tokens of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 185, + 678 + ], + "score": 1.0, + "content": "the RCs are of size", + "type": "text" + }, + { + "bbox": [ + 185, + 666, + 256, + 678 + ], + "score": 0.91, + "content": "[ H / 1 6 , W / 1 6 , D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 665, + 284, + 678 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 285, + 667, + 294, + 676 + ], + "score": 0.84, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "is the token dimension (64 in our experiments). The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 675, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 271, + 691 + ], + "score": 1.0, + "content": "output tokens of RCs are then flattened as", + "type": "text" + }, + { + "bbox": [ + 272, + 677, + 324, + 688 + ], + "score": 0.92, + "content": "R ^ { H W / 2 5 6 \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 675, + 507, + 691 + ], + "score": 1.0, + "content": ", concatenated with the class token, and added", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "by the sinusoid position encoding. 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The input of the first RC is the image", + "type": "text" + }, + { + "bbox": [ + 387, + 150, + 393, + 158 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 145, + 507, + 162 + ], + "score": 1.0, + "content": ". In the global dependencies", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 138, + 172 + ], + "score": 1.0, + "content": "branch,", + "type": "text" + }, + { + "bbox": [ + 139, + 159, + 148, + 170 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e.,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 177, + 438, + 192 + ], + "lines": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "spans": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "score": 0.89, + "content": "f _ { i } ^ { m s } \\triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \\in S _ { i } , r _ { i } \\in { \\mathcal { R } } ] ) ,", + "type": "interline_equation", + "image_path": "995f71351d9a0d0f5ed166388dc336a063cfcb324dbc2f6372f07fdf76e058d7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 134, + 210 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 198, + 176, + 210 + ], + "score": 0.92, + "content": "C o n v _ { i j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 198, + 232, + 210 + ], + "score": 1.0, + "content": "indicates the", + "type": "text" + }, + { + "bbox": [ + 233, + 199, + 238, + 210 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 198, + 381, + 210 + ], + "score": 1.0, + "content": "th convolutional layer in the PRM", + "type": "text" + }, + { + "bbox": [ + 382, + 198, + 427, + 210 + ], + "score": 0.87, + "content": "( P R M _ { i } ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 198, + 506, + 210 + ], + "score": 1.0, + "content": ". It uses a dilation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 125, + 222 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 126, + 211, + 138, + 221 + ], + "score": 0.86, + "content": "s _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 209, + 294, + 222 + ], + "score": 1.0, + "content": "from the predefined dilation rate set", + "type": "text" + }, + { + "bbox": [ + 295, + 209, + 305, + 220 + ], + "score": 0.87, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "corresponding to the ith RC. Note that we use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 408, + 232 + ], + "score": 1.0, + "content": "stride convolution to reduce the spatial dimension of features by a ratio", + "type": "text" + }, + { + "bbox": [ + 409, + 222, + 418, + 231 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "from the predefined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 228, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 181, + 246 + ], + "score": 1.0, + "content": "reduction ratio set", + "type": "text" + }, + { + "bbox": [ + 182, + 231, + 191, + 241 + ], + "score": 0.79, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 228, + 476, + 246 + ], + "score": 1.0, + "content": ". The conv features are concatenated along the channel dimension, i.e.,", + "type": "text" + }, + { + "bbox": [ + 476, + 231, + 505, + 243 + ], + "score": 0.89, + "content": "f _ { i } ^ { m s } \\in", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 239, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 107, + 241, + 203, + 253 + ], + "score": 0.9, + "content": "R ^ { ( W _ { i } / p ) \\times ( H _ { i } / p ) \\times ( | S _ { i } | D ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 239, + 233, + 258 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 233, + 243, + 249, + 255 + ], + "score": 0.91, + "content": "| { S _ { i } } |", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 239, + 402, + 258 + ], + "score": 1.0, + "content": "denotes the number of dilation rates in", + "type": "text" + }, + { + "bbox": [ + 402, + 243, + 412, + 254 + ], + "score": 0.79, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 239, + 416, + 258 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 417, + 243, + 434, + 255 + ], + "score": 0.84, + "content": "f _ { i } ^ { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 239, + 506, + 258 + ], + "score": 1.0, + "content": "is then processed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 354, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 354, + 267 + ], + "score": 1.0, + "content": "by an MHSA module to model long-range dependencies, i.e.,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 272, + 374, + 286 + ], + "lines": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "spans": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "score": 0.93, + "content": "f _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) ,", + "type": "interline_equation", + "image_path": "47bb46efbf400464cc18685ec76d7959be215ba831aad37913cc57da0f8d98b9.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 132, + 305 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 292, + 185, + 304 + ], + "score": 0.84, + "content": "I m g 2 S e q ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "is a simple reshape operation to flatten the feature map to a 1D sequence. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 127, + 317 + ], + "score": 1.0, + "content": "way,", + "type": "text" + }, + { + "bbox": [ + 127, + 303, + 139, + 315 + ], + "score": 0.89, + "content": "f _ { i } ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "embeds the multi-scale context in each token. In addition, we use a Parallel Convolutional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 482, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 423, + 327 + ], + "score": 1.0, + "content": "Module (PCM) to embed local context within the tokens, which are fused with", + "type": "text" + }, + { + "bbox": [ + 423, + 314, + 434, + 326 + ], + "score": 0.9, + "content": "f _ { i } ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 312, + 482, + 327 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 333, + 355, + 348 + ], + "lines": [ + { + "bbox": [ + 256, + 333, + 355, + 348 + ], + "spans": [ + { + "bbox": [ + 256, + 333, + 355, + 348 + ], + "score": 0.93, + "content": "f _ { i } ^ { l g } = f _ { i } ^ { g } + { \\cal P } { \\cal C } M _ { i } ( f _ { i } ) .", + "type": "interline_equation", + "image_path": "aab14cd318e095613617d3482aa2f81617b3140ca8a4a2e53d1fa0230ff83a82.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 256, + 333, + 355, + 348 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 354, + 504, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 131, + 367 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 354, + 171, + 366 + ], + "score": 0.91, + "content": "P C M _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 354, + 504, + 367 + ], + "score": 1.0, + "content": "represents the PCM, which is composed of three stacked convolution layers and an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 364, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 107, + 366, + 158, + 378 + ], + "score": 0.86, + "content": "I m g 2 S e q ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 364, + 506, + 379 + ], + "score": 1.0, + "content": "operation. It is noteworthy that the parallel convolution branch has the same spatial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 243, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 243, + 423 + ], + "score": 1.0, + "content": "into the following RC or NC, i.e.,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 427, + 383, + 443 + ], + "lines": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "spans": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "score": 0.92, + "content": "f _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) ,", + "type": "interline_equation", + "image_path": "716d8b1849d36a1ad308d5ce41f0df05844de9ab779da38fd00d2d340e97c83e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 149, + 462 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 149, + 449, + 201, + 461 + ], + "score": 0.88, + "content": "S e q 2 I m g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "is a simple reshape operation to reshape a token sequence back to feature", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 133, + 472 + ], + "score": 1.0, + "content": "maps.", + "type": "text" + }, + { + "bbox": [ + 133, + 460, + 171, + 471 + ], + "score": 0.91, + "content": "F F N _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "represents the FFN in the ith RC. In our ViTAE, three RCs are stacked sequentially", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 336, + 484 + ], + "score": 1.0, + "content": "to gradually reduce the input image’s spatial dimension by", + "type": "text" + }, + { + "bbox": [ + 337, + 471, + 368, + 482 + ], + "score": 0.52, + "content": "4 \\times , 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 470, + 388, + 484 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 388, + 471, + 402, + 482 + ], + "score": 0.87, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 470, + 505, + 484 + ], + "score": 1.0, + "content": ", respectively. The feature", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 289, + 494 + ], + "score": 1.0, + "content": "maps generated by the last RC are of a size of", + "type": "text" + }, + { + "bbox": [ + 290, + 482, + 361, + 494 + ], + "score": 0.93, + "content": "[ H / 1 6 , W / 1 6 , D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 482, + 505, + 494 + ], + "score": 1.0, + "content": ", which are then flattened into visual", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 263, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 263, + 505 + ], + "score": 1.0, + "content": "tokens and fed into the following NCs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 518, + 180, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 182, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 182, + 531 + ], + "score": 1.0, + "content": "3.4 Normal cell", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 350, + 564 + ], + "score": 1.0, + "content": "except for the absence of the PRM. Due to the relatively small", + "type": "text" + }, + { + "bbox": [ + 351, + 549, + 376, + 563 + ], + "score": 0.89, + "content": "( \\frac { 1 } { 1 6 } \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "spatial size of feature maps after", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 299, + 572 + ], + "score": 1.0, + "content": "RCs, it is unnecessary to use PRM in NCs. Given", + "type": "text" + }, + { + "bbox": [ + 300, + 561, + 310, + 572 + ], + "score": 0.89, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "from the third RC, we first concatenate it with the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 152, + 586 + ], + "score": 1.0, + "content": "class token", + "type": "text" + }, + { + "bbox": [ + 152, + 573, + 167, + 583 + ], + "score": 0.89, + "content": "t _ { c l s }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 570, + 430, + 586 + ], + "score": 1.0, + "content": ", and then add it to the positional encodings to get the input tokens", + "type": "text" + }, + { + "bbox": [ + 430, + 573, + 435, + 582 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "for the following", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "NCs. Here we ignore the subscript for clarity since all NCs have an identical architecture but different", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 182, + 606 + ], + "score": 1.0, + "content": "learnable weights.", + "type": "text" + }, + { + "bbox": [ + 182, + 594, + 198, + 604 + ], + "score": 0.9, + "content": "t _ { c l s }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 592, + 507, + 606 + ], + "score": 1.0, + "content": "is randomly initialized at the start of training and fixed during the inference.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 365, + 618 + ], + "score": 1.0, + "content": "Similar to the RC, the tokens are fed into the MHSA module, i.e.,", + "type": "text" + }, + { + "bbox": [ + 365, + 604, + 433, + 616 + ], + "score": 0.88, + "content": "t _ { g } = M H S A ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 603, + 506, + 618 + ], + "score": 1.0, + "content": ". Meanwhile, they", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 613, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 348, + 629 + ], + "score": 1.0, + "content": "are reshaped to 2D feature maps and fed into the PCM, i.e.,", + "type": "text" + }, + { + "bbox": [ + 349, + 615, + 502, + 627 + ], + "score": 0.87, + "content": "t _ { l } = \\bar { I } m g 2 S e q ( P C M ( S e q 2 I m g ( t ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 613, + 506, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "Note that the class token is discarded in PCM because it has no spatial connections with other visual", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "tokens. To further reduce the parameters in NCs, we use group convolutions in PCM. The features", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 371, + 662 + ], + "score": 1.0, + "content": "from MHSA and PCM are then fused via element-wise sum, i.e.,", + "type": "text" + }, + { + "bbox": [ + 372, + 649, + 424, + 660 + ], + "score": 0.88, + "content": "t _ { l g } = t _ { g } + t _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 648, + 461, + 662 + ], + "score": 1.0, + "content": ". Finally,", + "type": "text" + }, + { + "bbox": [ + 461, + 649, + 473, + 660 + ], + "score": 0.87, + "content": "t _ { l g }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "are fed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 311, + 672 + ], + "score": 1.0, + "content": "into the FFN to get the output features of NC, i.e.,", + "type": "text" + }, + { + "bbox": [ + 311, + 659, + 404, + 671 + ], + "score": 0.91, + "content": "t _ { n c } = F F N ( t _ { l g } ) \\overline { { + } } t _ { l g }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 658, + 506, + 672 + ], + "score": 1.0, + "content": ". Similar to ViT [19], we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 682 + ], + "score": 1.0, + "content": "apply layer normalization to the class token generated by the last NC and feed it to the classification", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 681, + 268, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 268, + 692 + ], + "score": 1.0, + "content": "head to get the final classification result.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 107, + 706, + 188, + 717 + ], + "lines": [ + { + "bbox": [ + 105, + 705, + 189, + 719 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 189, + 719 + ], + "score": 1.0, + "content": "3.5 Model details", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 72, + 191, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 192, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 192, + 85 + ], + "score": 1.0, + "content": "3.3 Reduction cell", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 93, + 506, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Instead of directly splitting and flatten images into visual tokens based on a linear image patch", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 116 + ], + "score": 1.0, + "content": "embedding layer, we devise the reduction cell to embed multi-scale context and local information", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 507, + 128 + ], + "score": 1.0, + "content": "into visual tokens, which introduces the intrinsic scale-invariance and locality IBs from convolutions.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "Technically, RC has two parallel branches responsible for modeling locality and long-range depen-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "dency, respectively, followed by an FFN for feature transformation. 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The input of the first RC is the image", + "type": "text" + }, + { + "bbox": [ + 387, + 150, + 393, + 158 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 145, + 507, + 162 + ], + "score": 1.0, + "content": ". In the global dependencies", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 138, + 172 + ], + "score": 1.0, + "content": "branch,", + "type": "text" + }, + { + "bbox": [ + 139, + 159, + 148, + 170 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e.,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 93, + 507, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 177, + 438, + 192 + ], + "lines": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "spans": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "score": 0.89, + "content": "f _ { i } ^ { m s } \\triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \\in S _ { i } , r _ { i } \\in { \\mathcal { R } } ] ) ,", + "type": "interline_equation", + "image_path": "995f71351d9a0d0f5ed166388dc336a063cfcb324dbc2f6372f07fdf76e058d7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 171, + 177, + 438, + 192 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 134, + 210 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 198, + 176, + 210 + ], + "score": 0.92, + "content": "C o n v _ { i j } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 198, + 232, + 210 + ], + "score": 1.0, + "content": "indicates the", + "type": "text" + }, + { + "bbox": [ + 233, + 199, + 238, + 210 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 198, + 381, + 210 + ], + "score": 1.0, + "content": "th convolutional layer in the PRM", + "type": "text" + }, + { + "bbox": [ + 382, + 198, + 427, + 210 + ], + "score": 0.87, + "content": "( P R M _ { i } ( \\cdot ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 198, + 506, + 210 + ], + "score": 1.0, + "content": ". It uses a dilation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 125, + 222 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 126, + 211, + 138, + 221 + ], + "score": 0.86, + "content": "s _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 209, + 294, + 222 + ], + "score": 1.0, + "content": "from the predefined dilation rate set", + "type": "text" + }, + { + "bbox": [ + 295, + 209, + 305, + 220 + ], + "score": 0.87, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "corresponding to the ith RC. Note that we use", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 408, + 232 + ], + "score": 1.0, + "content": "stride convolution to reduce the spatial dimension of features by a ratio", + "type": "text" + }, + { + "bbox": [ + 409, + 222, + 418, + 231 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "from the predefined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 228, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 181, + 246 + ], + "score": 1.0, + "content": "reduction ratio set", + "type": "text" + }, + { + "bbox": [ + 182, + 231, + 191, + 241 + ], + "score": 0.79, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 228, + 476, + 246 + ], + "score": 1.0, + "content": ". The conv features are concatenated along the channel dimension, i.e.,", + "type": "text" + }, + { + "bbox": [ + 476, + 231, + 505, + 243 + ], + "score": 0.89, + "content": "f _ { i } ^ { m s } \\in", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 239, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 107, + 241, + 203, + 253 + ], + "score": 0.9, + "content": "R ^ { ( W _ { i } / p ) \\times ( H _ { i } / p ) \\times ( | S _ { i } | D ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 239, + 233, + 258 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 233, + 243, + 249, + 255 + ], + "score": 0.91, + "content": "| { S _ { i } } |", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 239, + 402, + 258 + ], + "score": 1.0, + "content": "denotes the number of dilation rates in", + "type": "text" + }, + { + "bbox": [ + 402, + 243, + 412, + 254 + ], + "score": 0.79, + "content": "S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 239, + 416, + 258 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 417, + 243, + 434, + 255 + ], + "score": 0.84, + "content": "f _ { i } ^ { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 239, + 506, + 258 + ], + "score": 1.0, + "content": "is then processed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 354, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 354, + 267 + ], + "score": 1.0, + "content": "by an MHSA module to model long-range dependencies, i.e.,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 198, + 506, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 272, + 374, + 286 + ], + "lines": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "spans": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "score": 0.93, + "content": "f _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) ,", + "type": "interline_equation", + "image_path": "47bb46efbf400464cc18685ec76d7959be215ba831aad37913cc57da0f8d98b9.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 235, + 272, + 374, + 286 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 132, + 305 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 292, + 185, + 304 + ], + "score": 0.84, + "content": "I m g 2 S e q ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "is a simple reshape operation to flatten the feature map to a 1D sequence. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 302, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 127, + 317 + ], + "score": 1.0, + "content": "way,", + "type": "text" + }, + { + "bbox": [ + 127, + 303, + 139, + 315 + ], + "score": 0.89, + "content": "f _ { i } ^ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 302, + 506, + 317 + ], + "score": 1.0, + "content": "embeds the multi-scale context in each token. 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It is noteworthy that the parallel convolution branch has the same spatial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 243, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 243, + 423 + ], + "score": 1.0, + "content": "into the following RC or NC, i.e.,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 354, + 506, + 423 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 427, + 383, + 443 + ], + "lines": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "spans": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "score": 0.92, + "content": "f _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) ,", + "type": "interline_equation", + "image_path": "716d8b1849d36a1ad308d5ce41f0df05844de9ab779da38fd00d2d340e97c83e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 226, + 427, + 383, + 443 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 149, + 462 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 149, + 449, + 201, + 461 + ], + "score": 0.88, + "content": "S e q 2 I m g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "is a simple reshape operation to reshape a token sequence back to feature", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 133, + 472 + ], + "score": 1.0, + "content": "maps.", + "type": "text" + }, + { + "bbox": [ + 133, + 460, + 171, + 471 + ], + "score": 0.91, + "content": "F F N _ { i } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "represents the FFN in the ith RC. 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The feature", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 289, + 494 + ], + "score": 1.0, + "content": "maps generated by the last RC are of a size of", + "type": "text" + }, + { + "bbox": [ + 290, + 482, + 361, + 494 + ], + "score": 0.93, + "content": "[ H / 1 6 , W / 1 6 , D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 482, + 505, + 494 + ], + "score": 1.0, + "content": ", which are then flattened into visual", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 263, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 263, + 505 + ], + "score": 1.0, + "content": "tokens and fed into the following NCs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 448, + 506, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 518, + 180, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 182, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 182, + 531 + ], + "score": 1.0, + "content": "3.4 Normal cell", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 350, + 564 + ], + "score": 1.0, + "content": "except for the absence of the PRM. Due to the relatively small", + "type": "text" + }, + { + "bbox": [ + 351, + 549, + 376, + 563 + ], + "score": 0.89, + "content": "( \\frac { 1 } { 1 6 } \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "spatial size of feature maps after", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 299, + 572 + ], + "score": 1.0, + "content": "RCs, it is unnecessary to use PRM in NCs. Given", + "type": "text" + }, + { + "bbox": [ + 300, + 561, + 310, + 572 + ], + "score": 0.89, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "from the third RC, we first concatenate it with the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 152, + 586 + ], + "score": 1.0, + "content": "class token", + "type": "text" + }, + { + "bbox": [ + 152, + 573, + 167, + 583 + ], + "score": 0.89, + "content": "t _ { c l s }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 570, + 430, + 586 + ], + "score": 1.0, + "content": ", and then add it to the positional encodings to get the input tokens", + "type": "text" + }, + { + "bbox": [ + 430, + 573, + 435, + 582 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "for the following", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "NCs. 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ModelReduction Cell DilationCellsNormal Cell Heads Embed CellsParams Macs (M)(G)
ViTAE-T[1,2,3,4] √34 25674.81.5
ViTAE-S[1,2,3,4] √36384 1423.65.6
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Since", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 171, + 507, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 507, + 182 + ], + "score": 1.0, + "content": "the spatial dimension of tokens decreases, there is no need to use large kernels and dilation rates.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 181, + 476, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 448, + 195 + ], + "score": 1.0, + "content": "PCM in both RCs and NCs comprises three convolutional layers with a kernel size of", + "type": "text" + }, + { + "bbox": [ + 449, + 182, + 472, + 192 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 181, + 476, + 195 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 209, + 191, + 222 + ], + "lines": [ + { + "bbox": [ + 104, + 207, + 193, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 207, + 193, + 226 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 234, + 229, + 246 + ], + "lines": [ + { + "bbox": [ + 105, + 233, + 230, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 230, + 248 + ], + "score": 1.0, + "content": "4.1 Implementation details", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 504, + 267 + ], + "score": 1.0, + "content": "We train and test the proposed ViTAE model on the standard ImageNet [38] dataset, which contains", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "about 1.3 million images and covers 1k classes. Unless explicitly stated, the image size during training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 140, + 288 + ], + "score": 1.0, + "content": "is set to", + "type": "text" + }, + { + "bbox": [ + 140, + 276, + 183, + 287 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 276, + 505, + 288 + ], + "score": 1.0, + "content": ". We use the AdamW [48] optimizer with the cosine learning rate scheduler and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "uses the data augmentation strategy exactly the same as T2T [93] for a fair comparison, regarding the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "training strategies and the size of models. We use a batch size of 512 for training all our models and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "set the initial learning rate to be 5e-4. The results of our models can be found in Table 2, where all", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 330, + 158, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 158, + 343 + ], + "score": 1.0, + "content": "TIMM [82].", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 286, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 289, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 289, + 369 + ], + "score": 1.0, + "content": "4.2 Comparison with the state-of-the-art", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "We compare our ViTAE with both CNN models and vision transformers with similar model sizes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "reported. We categorize the methods into CNN models, vision transformers with learned IB, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 114, + 433 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 420, + 142, + 431 + ], + "score": 0.85, + "content": "7 5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Top-1 accuracy, which is better than ResNet-18 with more parameters. 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Similar phenomena", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65],", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "where ViTAE obtains better performance with fewer parameters. 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ModelReduction Cell DilationCellsNormal Cell Heads Embed CellsParams Macs (M)(G)
ViTAE-T[1,2,3,4] √34 25674.81.5
ViTAE-S[1,2,3,4] √36384 1423.65.6
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The results of our models can be found in Table 2, where all", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 330, + 158, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 158, + 343 + ], + "score": 1.0, + "content": "TIMM [82].", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 255, + 505, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 286, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 289, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 289, + 369 + ], + "score": 1.0, + "content": "4.2 Comparison with the state-of-the-art", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "We compare our ViTAE with both CNN models and vision transformers with similar model sizes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "reported. We categorize the methods into CNN models, vision transformers with learned IB, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 114, + 433 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 420, + 142, + 431 + ], + "score": 0.85, + "content": "7 5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Top-1 accuracy, which is better than ResNet-18 with more parameters. The real Top-1", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 236, + 443 + ], + "score": 1.0, + "content": "accuracy of the ViTAE model is", + "type": "text" + }, + { + "bbox": [ + 236, + 431, + 263, + 441 + ], + "score": 0.9, + "content": "8 2 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 430, + 505, + 443 + ], + "score": 1.0, + "content": ", which is comparable to ResNet-50 that has four more times", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 345, + 455 + ], + "score": 1.0, + "content": "of parameters than ours. Similarly, our ViTAE-S achieves", + "type": "text" + }, + { + "bbox": [ + 345, + 442, + 373, + 452 + ], + "score": 0.86, + "content": "8 2 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "Top-1 accuracy with half of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 451, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 467 + ], + "score": 1.0, + "content": "parameters of ResNet-101 and ResNet-152, showing the superiority of learning both local and long-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "range features from specific structures with corresponding intrinsic IBs by design. Similar phenomena", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65],", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "where ViTAE obtains better performance with fewer parameters. When compared with larger models", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "score": 1.0, + "content": "which are searched according to NAS [73], our ViTAE-S achieves a similar performance when using", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 150, + 518 + ], + "score": 0.89, + "content": "3 8 4 \\times 3 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "images as input, which further shows the potential of vision transformers with intrinsic IB.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 376, + 506, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "In addition, among the transformers with learned IB, ViT is the first pure transformer model for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "visual recognition. DeiT shares the same structure with ViT but uses different data augmentation and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "training strategies to facilitate the learning of transformers. DeiT⚗ denotes using an off-the-shelf CNN", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "model as the teacher model to train DeiT, which introduces the intrinsic IB from CNN to transformer", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "score": 1.0, + "content": "implicitly in a knowledge distillation manner, showing better performance than the vanilla ViT on the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "ImageNet dataset. It is exciting to see that our ViTAE-T with fewer parameters even outperforms", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "the distilled model DeiT⚗, demonstrating the efficacy of introducing intrinsic IBs in transformers", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "by design. Besides, compared with other transformers with explicit intrinsic IB, our ViTAE with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "fewer parameters also achieves comparable or better performance. For instance, ViTAE-T achieves", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 622, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 504, + 634 + ], + "score": 1.0, + "content": "comparable performance with LocalVit-T but has 1M fewer parameters, demonstrating the superiority", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 632, + 341, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 341, + 645 + ], + "score": 1.0, + "content": "of the proposed RCs and NCs in introducing intrinsic IBs.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 524, + 506, + 645 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 192, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "score": 1.0, + "content": "4.3 Ablation study", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "We use T2T-ViT [93] as our baseline model in the following ablation study of our ViTAE. As shown", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "in Table 3, we investigate the hyper-parameter settings in RCs and NCs by isolating them separately.", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "All the models are trained for 100 epochs on ImageNet and follow the same training setting and data", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 309, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 309, + 722 + ], + "score": 1.0, + "content": "augmentation strategy as described in Section 4.1.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 677, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 120, + 158, + 490, + 665 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 137, + 146, + 471, + 159 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 145, + 473, + 160 + ], + "spans": [ + { + "bbox": [ + 137, + 145, + 473, + 160 + ], + "score": 1.0, + "content": "Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 120, + 158, + 490, + 665 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 158, + 490, + 665 + ], + "spans": [ + { + "bbox": [ + 120, + 158, + 490, + 665 + ], + "score": 0.981, + "html": "
Type ModelParams (M)MACs (G)Input SizeImageNet Real
Top-1Top-5Top-1
CNNResNet-18 [26]11.73.622470.386.777.3
ResNet-50 [26]25.67.622476.793.382.5
ResNet-101 [26]44.515.222478.394.183.7
ResNet-152 [26]60.222.622478.994.484.1
EfficientNet-B0 [73]5.30.822477.193.383.5
EfficientNet-B4 [73]19.38.438082.996.488.0
MobileNetV1 [31]4.30.622472.31-
MobileNetV2(1.4) [65]6.90.622474.7--
RegNetY-600M[62]6.11.222475.5--
RegNetY-4GF[62] RegNetY-8GF[62]20.6 39.28.0 16.022480.0186.4
22481.7187.4
DeiT-T[76]5.72.622472.291.180.6
DeiT-T [76]5.72.622474.591.982.1
LocalViT-T[43]5.92.622474.892.6
LocalViT-T2T[43]4.32.422472.5-1 1
ConT-Ti [89]5.81.622474.9--
PiT-Ti [29]4.91.422473.0-1
T2T-ViT-7 [93]4.31.222471.790.979.7
ViTAE-T4.81.522475.392.782.9
ViTAE-T ↑ 3844.85.738477.293.884.4
CeiT-T [92]6.42.422476.493.483.6
ConViT-Ti[15]6.02.022473.111
Cross ViT-Ti [6]6.93.222473.411
ViTAE-6M6.52.022477.994.184.9
PVT-T[80] LocalViT-PVT [43]13.23.822475.11
PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]13.59.622494.2-
ConViT-Ti+ [15] 10.04.022478.2 76.71
2.878.11-
10.6 19.26.2224 22480.2-1
13.23.422481.0- 95.4- 86.8
22.1 DeiT-S [76] 22.19.8 9.8224 22479.9 81.295.0 95.485.7 86.8
PVT-S[80]7.622479.8-
24.5 23.55.281.31
Conformer-Ti [58] Swin-T[47]224-
CeiT-S [92]29.09.022481.3-1
CvT-13 [83]24.2 20.09.022482.0 81.695.987.3 86.7
ConViT-S[15]27.09.0 10.8224 22481.31 11
Cross ViT-S [6]26.711.222481.011
PiT-S [29]23.54.822480.9
TNT-S [23]-1
Twins-PCPVT-S[10]23.810.422481.395.6-
24.17.422481.2--
Twins-SVT-S [10]24.05.622481.7-1
T2T-ViT-14 [93]21.55.222481.595.786.8
ViTAE-S23.65.622482.095.987.0
ViTAE-S ↑ 38423.620.238483.096.287.5
", + "type": "table", + "image_path": "e514264bc0b3536b852ece05e614d3ca959237e923c2f3d7e1d3c16e5426e296.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 120, + 158, + 490, + 327.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 327.0, + 490, + 496.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 120, + 496.0, + 490, + 665.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 120, + 158, + 490, + 665 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 137, + 146, + 471, + 159 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 145, + 473, + 160 + ], + "spans": [ + { + "bbox": [ + 137, + 145, + 473, + 160 + ], + "score": 1.0, + "content": "Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 120, + 158, + 490, + 665 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 158, + 490, + 665 + ], + "spans": [ + { + "bbox": [ + 120, + 158, + 490, + 665 + ], + "score": 0.981, + "html": "
Type ModelParams (M)MACs (G)Input SizeImageNet Real
Top-1Top-5Top-1
CNNResNet-18 [26]11.73.622470.386.777.3
ResNet-50 [26]25.67.622476.793.382.5
ResNet-101 [26]44.515.222478.394.183.7
ResNet-152 [26]60.222.622478.994.484.1
EfficientNet-B0 [73]5.30.822477.193.383.5
EfficientNet-B4 [73]19.38.438082.996.488.0
MobileNetV1 [31]4.30.622472.31-
MobileNetV2(1.4) [65]6.90.622474.7--
RegNetY-600M[62]6.11.222475.5--
RegNetY-4GF[62] RegNetY-8GF[62]20.6 39.28.0 16.022480.0186.4
22481.7187.4
DeiT-T[76]5.72.622472.291.180.6
DeiT-T [76]5.72.622474.591.982.1
LocalViT-T[43]5.92.622474.892.6
LocalViT-T2T[43]4.32.422472.5-1 1
ConT-Ti [89]5.81.622474.9--
PiT-Ti [29]4.91.422473.0-1
T2T-ViT-7 [93]4.31.222471.790.979.7
ViTAE-T4.81.522475.392.782.9
ViTAE-T ↑ 3844.85.738477.293.884.4
CeiT-T [92]6.42.422476.493.483.6
ConViT-Ti[15]6.02.022473.111
Cross ViT-Ti [6]6.93.222473.411
ViTAE-6M6.52.022477.994.184.9
PVT-T[80] LocalViT-PVT [43]13.23.822475.11
PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]13.59.622494.2-
ConViT-Ti+ [15] 10.04.022478.2 76.71
2.878.11-
10.6 19.26.2224 22480.2-1
13.23.422481.0- 95.4- 86.8
22.1 DeiT-S [76] 22.19.8 9.8224 22479.9 81.295.0 95.485.7 86.8
PVT-S[80]7.622479.8-
24.5 23.55.281.31
Conformer-Ti [58] Swin-T[47]224-
CeiT-S [92]29.09.022481.3-1
CvT-13 [83]24.2 20.09.022482.0 81.695.987.3 86.7
ConViT-S[15]27.09.0 10.8224 22481.31 11
Cross ViT-S [6]26.711.222481.011
PiT-S [29]23.54.822480.9
TNT-S [23]-1
Twins-PCPVT-S[10]23.810.422481.395.6-
24.17.422481.2--
Twins-SVT-S [10]24.05.622481.7-1
T2T-ViT-14 [93]21.55.222481.595.786.8
ViTAE-S23.65.622482.095.987.0
ViTAE-S ↑ 38423.620.238483.096.287.5
", + "type": "table", + "image_path": "e514264bc0b3536b852ece05e614d3ca959237e923c2f3d7e1d3c16e5426e296.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 120, + 158, + 490, + 327.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 327.0, + 490, + 496.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 120, + 496.0, + 490, + 665.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 506, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 136, + 85 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 136, + 73, + 145, + 83 + ], + "score": 0.7, + "content": "\\checkmark", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 73, + 162, + 85 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 74, + 172, + 83 + ], + 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It is noteworthy that all the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 257, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 257, + 209 + ], + "score": 1.0, + "content": "variants of NC outperform the vanilla", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 258, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 258, + 222 + ], + "score": 1.0, + "content": "T2T-ViT, implying the effectiveness", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 219, + 258, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 258, + 232 + ], + "score": 1.0, + "content": "of PCM, which introduces the intrin-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 258, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 258, + 243 + ], + "score": 1.0, + "content": "sic locality IB in transformers. It can", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 241, + 258, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 258, + 253 + ], + "score": 1.0, + "content": "also be observed that BN plays an im-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 258, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 258, + 265 + ], + "score": 1.0, + "content": "portant role in improving the model’s", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 264, + 259, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 259, + 275 + ], + "score": 1.0, + "content": "performance as it can help to allevi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 259, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 259, + 285 + ], + "score": 1.0, + "content": "ate the scale deviation between convo-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 285, + 258, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 258, + 297 + ], + "score": 1.0, + "content": "lution’s and attention’s features. For", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 296, + 258, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 258, + 308 + ], + "score": 1.0, + "content": "the RC, we first investigate the impact", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 258, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 258, + 318 + ], + "score": 1.0, + "content": "of using different dilation rates in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 318, + 259, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 259, + 330 + ], + "score": 1.0, + "content": "PRM, as shown in the first column.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 258, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 258, + 341 + ], + "score": 1.0, + "content": "As can be seen, using larger dilation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 340, + 258, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 258, + 352 + ], + "score": 1.0, + "content": "rates (e.g., 4 or 5) does not deliver", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 351, + 258, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 258, + 362 + ], + "score": 1.0, + "content": "better performance. We suspect that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 362, + 257, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 257, + 374 + ], + "score": 1.0, + "content": "larger dilation rates may lead to plain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 372, + 257, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 257, + 384 + ], + "score": 1.0, + "content": "features in the deeper RCs due to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 384, + 257, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 257, + 395 + ], + "score": 1.0, + "content": "smaller resolution of feature maps. 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Reduction CellNormal CellTop-1
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[1,2,3,4] × 3 ××× ×69.2
[1,2,3,4,5] × 3 ××× ×68.9
[1,2,3,4]↓ ××× ×69.8
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However, compared with", + "type": "text" + }, + { + "bbox": [ + 456, + 406, + 503, + 417 + ], + "score": 0.82, + "content": "[ 1 , 2 , 3 , 4 ] \\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 405, + 507, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 415, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 107, + 416, + 145, + 428 + ], + "score": 0.86, + "content": "[ 1 , 2 , 3 ] \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 415, + 347, + 429 + ], + "score": 1.0, + "content": "increases the amount of parameters from 4.35M to", + "type": "text" + }, + { + "bbox": [ + 348, + 416, + 370, + 427 + ], + "score": 0.31, + "content": "4 . 6 \\mathsf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 415, + 456, + 429 + ], + "score": 1.0, + "content": ". 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Finally, the combination of RCs and NCs", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 449, + 507, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 222, + 461 + ], + "score": 1.0, + "content": "achieves the best accuracy at", + "type": "text" + }, + { + "bbox": [ + 223, + 449, + 249, + 460 + ], + "score": 0.87, + "content": "7 2 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 449, + 507, + 461 + ], + "score": 1.0, + "content": ", demonstrating the complementarity between our RCs and NCs.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49.5 + }, + { + "type": "title", + "bbox": [ + 108, + 484, + 290, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 292, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 292, + 500 + ], + "score": 1.0, + "content": "4.4 Data efficiency and training efficiency", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 508, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 104, + 506, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 506, + 523 + ], + "score": 1.0, + "content": "To validate the effectiveness of the introduced intrinsic IBs in improving data efficiency and training", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "efficiency, we compare our ViTAE with T2T-ViT at different training settings: (a) training them", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 131, + 543 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 531, + 151, + 542 + ], + "score": 0.85, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 530, + 154, + 543 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 154, + 531, + 174, + 542 + ], + "score": 0.83, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 530, + 195, + 543 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 196, + 531, + 220, + 541 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "ImageNet training set for equivalent 100 epochs on the full ImageNet", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 104, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 331, + 556 + ], + "score": 1.0, + "content": "training set, e.g., we employ 5 times epochs when using", + "type": "text" + }, + { + "bbox": [ + 331, + 542, + 351, + 552 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "data for training compared with using", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 131, + 563 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "data; and (b) training them using the full ImageNet training set for 100, 200, and 300 epochs", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "respectively. The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 225, + 597 + ], + "score": 1.0, + "content": "example, ViTAE using only", + "type": "text" + }, + { + "bbox": [ + 225, + 585, + 245, + 596 + ], + "score": 0.89, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "training data achieves comparable performance with T2T-ViT", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 192, + 610 + ], + "score": 1.0, + "content": "using all data. When", + "type": "text" + }, + { + "bbox": [ + 193, + 596, + 213, + 607 + ], + "score": 0.86, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 595, + 506, + 610 + ], + "score": 1.0, + "content": "training data are used, ViTAE significantly outperforms T2T-ViT using", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 226, + 619 + ], + "score": 1.0, + "content": "all data by about an absolute", + "type": "text" + }, + { + "bbox": [ + 226, + 608, + 241, + 617 + ], + "score": 0.85, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "accuracy. It is also noteworthy that ViTAE trained for only 100", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "epochs has outperformed T2T-ViT trained for 300 epochs. After training ViTAE for 300 epochs, its", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 262, + 641 + ], + "score": 1.0, + "content": "performance is significantly boosted to", + "type": "text" + }, + { + "bbox": [ + 263, + 629, + 290, + 640 + ], + "score": 0.87, + "content": "7 5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "Top-1 accuracy. With the proposed RCs and NCs, the", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "transformer layers in our ViTAE only need to focus on modeling long-range dependencies, leaving the", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "locality and multi-scale context modeling to its convolution counterparts, i.e., PCM and PRM. Such", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 106, + 673, + 393, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 393, + 685 + ], + "score": 1.0, + "content": "learn more efficiently with less training data and fewer training epochs.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 61.5 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "on the smaller datasets, i.e., Cifar10 and Cifar100. The results are summarized in Table 4. It can", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 209, + 723 + ], + "score": 1.0, + "content": "be viewed that with only", + "type": "text" + }, + { + "bbox": [ + 209, + 711, + 225, + 723 + ], + "score": 0.68, + "content": "1 / 7", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "number of epochs, the ViTAE-T model achieves better classification", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 71 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + 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layer or not.", + "type": "text" + }, + { + "bbox": [ + 274, + 117, + 294, + 127 + ], + "score": 0.84, + "content": "\\mathit { \\Omega } ^ { 6 } \\times 3 \\mathit { \\Omega } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "in the first column denotes that the dilation rate set", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 231, + 140 + ], + "score": 1.0, + "content": "is the same in the three RCs. “", + "type": "text" + }, + { + "bbox": [ + 232, + 127, + 271, + 138 + ], + "score": 0.33, + "content": "[ 1 , 2 , 3 , 4 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 128, + 285, + 138 + ], + "score": 0.72, + "content": "\\downarrow ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "denotes using lower dilation rates in deeper RCs, i.e.,", 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It is noteworthy that all the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 257, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 257, + 209 + ], + "score": 1.0, + "content": "variants of NC outperform the vanilla", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 258, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 258, + 222 + ], + "score": 1.0, + "content": "T2T-ViT, implying the effectiveness", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 219, + 258, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 258, + 232 + ], + "score": 1.0, + "content": "of PCM, which introduces the intrin-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 258, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 258, + 243 + ], + "score": 1.0, + "content": "sic locality IB in transformers. It can", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 241, + 258, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 258, + 253 + ], + "score": 1.0, + "content": "also be observed that BN plays an im-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 258, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 258, + 265 + ], + "score": 1.0, + "content": "portant role in improving the model’s", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 264, + 259, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 259, + 275 + ], + "score": 1.0, + "content": "performance as it can help to allevi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 259, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 259, + 285 + ], + "score": 1.0, + "content": "ate the scale deviation between convo-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 285, + 258, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 258, + 297 + ], + "score": 1.0, + "content": "lution’s and attention’s features. For", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 296, + 258, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 258, + 308 + ], + "score": 1.0, + "content": "the RC, we first investigate the impact", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 258, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 258, + 318 + ], + "score": 1.0, + "content": "of using different dilation rates in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 318, + 259, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 259, + 330 + ], + "score": 1.0, + "content": "PRM, as shown in the first column.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 258, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 258, + 341 + ], + "score": 1.0, + "content": "As can be seen, using larger dilation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 340, + 258, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 258, + 352 + ], + "score": 1.0, + "content": "rates (e.g., 4 or 5) does not deliver", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 351, + 258, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 258, + 362 + ], + "score": 1.0, + "content": "better performance. We suspect that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 362, + 257, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 257, + 374 + ], + "score": 1.0, + "content": "larger dilation rates may lead to plain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 372, + 257, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 257, + 384 + ], + "score": 1.0, + "content": "features in the deeper RCs due to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 384, + 257, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 257, + 395 + ], + "score": 1.0, + "content": "smaller resolution of feature maps. 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Reduction CellNormal CellTop-1
Dilation (S1 ~ S3) PCMPreBN Post
× ××× ×68.7
× ×√× ×69.1
× ××√ ×69.0
× ××√ √68.8
× ×√× √69.9
[1,2]×3× ××× 69.5
[1,2,3]×3 ××× ×69.9
[1,2,3,4] × 3 ××× ×69.2
[1,2,3,4,5] × 3 ××× ×68.9
[1,2,3,4]↓ ××× ×69.8
[1,2,3,4]↓ √×× ×71.7
[1,2,3,4]↓ √√× √72.6
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However, compared with", + "type": "text" + }, + { + "bbox": [ + 456, + 406, + 503, + 417 + ], + "score": 0.82, + "content": "[ 1 , 2 , 3 , 4 ] \\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 405, + 507, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 415, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 107, + 416, + 145, + 428 + ], + "score": 0.86, + "content": "[ 1 , 2 , 3 ] \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 415, + 347, + 429 + ], + "score": 1.0, + "content": "increases the amount of parameters from 4.35M to", + "type": "text" + }, + { + "bbox": [ + 348, + 416, + 370, + 427 + ], + "score": 0.31, + "content": "4 . 6 \\mathsf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 415, + 456, + 429 + ], + "score": 1.0, + "content": ". Therefore, we select", + "type": "text" + }, + { + "bbox": [ + 456, + 417, + 504, + 428 + ], + "score": 0.67, + "content": "[ 1 , 2 , 3 , 4 ] \\downarrow", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 427, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 507, + 440 + ], + "score": 1.0, + "content": "as the default setting. In addition, after using PCM in the RC, it introduces the intrinsic locality IB,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 241, + 451 + ], + "score": 1.0, + "content": "and the performance increases to", + "type": "text" + }, + { + "bbox": [ + 241, + 438, + 269, + 449 + ], + "score": 0.87, + "content": "7 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "Top-1 accuracy. 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The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 225, + 597 + ], + "score": 1.0, + "content": "example, ViTAE using only", + "type": "text" + }, + { + "bbox": [ + 225, + 585, + 245, + 596 + ], + "score": 0.89, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "training data achieves comparable performance with T2T-ViT", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 192, + 610 + ], + "score": 1.0, + "content": "using all data. 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Such", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 106, + 673, + 393, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 393, + 685 + ], + "score": 1.0, + "content": "learn more efficiently with less training data and fewer training epochs.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 61.5, + "bbox_fs": [ + 104, + 506, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "on the smaller datasets, i.e., Cifar10 and Cifar100. 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It can", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 209, + 723 + ], + "score": 1.0, + "content": "be viewed that with only", + "type": "text" + }, + { + "bbox": [ + 209, + 711, + 225, + 723 + ], + "score": 0.68, + "content": "1 / 7", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "number of epochs, the ViTAE-T model achieves better classification", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 233, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 233, + 97 + ], + "score": 1.0, + "content": "ViTAE model’s data efficiency.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 71, + "bbox_fs": [ + 105, + 688, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 233, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 233, + 97 + ], + "score": 1.0, + "content": "ViTAE model’s data efficiency.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table", + "bbox": [ + 178, + 122, + 432, + 170 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 190, + 105, + 420, + 117 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 191, + 104, + 420, + 117 + ], + "spans": [ + { + "bbox": [ + 191, + 104, + 420, + 117 + ], + "score": 1.0, + "content": "Table 4: Results of training from scratch on Cifar10/100.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 178, + 122, + 432, + 170 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 178, + 122, + 432, + 170 + ], + "spans": [ + { + "bbox": [ + 178, + 122, + 432, + 170 + ], + "score": 0.974, + "html": "
ModelParams (M)Top-1 AccEpochsDataset
DeiT-B86.097.57000Cifar10
ViTAE-T4.897.71000Cifar10
ViTAE-T4.885.01000Cifar100
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ModelParams (M)Cifar10Cifar100iNat19CarsFlowersPets
Grafit ResNet-50 [78]25.6-=75.992.598.2-
EfficientNet-B5 [73]3098.191.1--98.5-
ViT-B/16 [19]86.598.187.1-189.593.8
ViT-L/16 [19]304.397.986.4--89.793.6
DeiT-B [76]86.699.190.877.792.198.4-
T2T-ViT-14 [93]21.598.388.4---1
ViTAE-T4.897.386.073.389.597.592.6
ViTAE-S23.698.890.876.091.497.894.2
", + "type": "table", + "image_path": "1811ce2c1001783f4b47fef070c93481a3eed9f0f8f8b509c00a30ea1da1a596.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 133, + 228, + 477, + 259.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 133, + 259.3333333333333, + 477, + 290.66666666666663 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 133, + 290.66666666666663, + 477, + 321.99999999999994 + ], + "spans": [], + "index": 10 + } + ] + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 347 + ], + "score": 1.0, + "content": "We further investigate the generalization of the proposed ViTAE models on downstream tasks by fine-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 345, + 507, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 507, + 360 + ], + "score": 1.0, + "content": "tuning them on the training sets of several fine-grained classification tasks3, including Flowers [53],", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "Cars [36], Pets [56], and iNaturalist19. We also fine-tune the proposed ViTAE models on Cifar10 [37]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "performance on most of the datasets using comparable or fewer parameters. These results demonstrate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 390, + 307, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 307, + 402 + ], + "score": 1.0, + "content": "that the good generalization ability of our ViTAE.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 415, + 245, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 246, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 246, + 428 + ], + "score": 1.0, + "content": "4.6 Visual inspection of ViTAE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 503, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "To further analyze the property of our ViTAE, we first calculate the average attention distance of each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 471 + ], + "score": 1.0, + "content": "shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "the transformer layers in the proposed NCs can better focus on modeling long-range dependencies,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 317, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 317, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 317, + 513 + ], + "score": 1.0, + "content": "These results confirm the effectiveness of the adopted", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 318, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 318, + 524 + ], + "score": 1.0, + "content": "“divide-and-conquer” idea in the proposed ViTAE,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 523, + 318, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 318, + 534 + ], + "score": 1.0, + "content": "i.e., introducing the intrinsic locality IB from con-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 533, + 317, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 317, + 546 + ], + "score": 1.0, + "content": "volutions into vision transformers makes it possible", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 544, + 317, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 317, + 556 + ], + "score": 1.0, + "content": "that transformer layers only need to be responsible to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 317, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 317, + 568 + ], + "score": 1.0, + "content": "long-range dependencies, since locality can be well", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 567, + 245, + 578 + ], + "spans": [ + { + 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ModelParams (M)Top-1 AccEpochsDataset
DeiT-B86.097.57000Cifar10
ViTAE-T4.897.71000Cifar10
ViTAE-T4.885.01000Cifar100
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ModelParams (M)Cifar10Cifar100iNat19CarsFlowersPets
Grafit ResNet-50 [78]25.6-=75.992.598.2-
EfficientNet-B5 [73]3098.191.1--98.5-
ViT-B/16 [19]86.598.187.1-189.593.8
ViT-L/16 [19]304.397.986.4--89.793.6
DeiT-B [76]86.699.190.877.792.198.4-
T2T-ViT-14 [93]21.598.388.4---1
ViTAE-T4.897.386.073.389.597.592.6
ViTAE-S23.698.890.876.091.497.894.2
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We also fine-tune the proposed ViTAE models on Cifar10 [37]", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "performance on most of the datasets using comparable or fewer parameters. These results demonstrate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 390, + 307, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 307, + 402 + ], + "score": 1.0, + "content": "that the good generalization ability of our ViTAE.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 336, + 507, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 415, + 245, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 246, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 246, + 428 + ], + "score": 1.0, + "content": "4.6 Visual inspection of ViTAE", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 503, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "To further analyze the property of our ViTAE, we first calculate the average attention distance of each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 471 + ], + "score": 1.0, + "content": "shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "the transformer layers in the proposed NCs can better focus on modeling long-range dependencies,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 434, + 505, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 317, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 317, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 317, + 513 + ], + "score": 1.0, + "content": "These results confirm the effectiveness of the adopted", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 318, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 318, + 524 + ], + "score": 1.0, + "content": "“divide-and-conquer” idea in the proposed ViTAE,", + "type": "text" + } + ], 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+ }, + { + "type": "text", + "bbox": [ + 107, + 582, + 318, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 317, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 317, + 594 + ], + "score": 1.0, + "content": "Besides, we apply Grad-CAM [66] on the MHSA’s", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 593, + 318, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 318, + 605 + ], + "score": 1.0, + "content": "output in the last NC to qualitatively inspect ViTAE.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 318, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 318, + 616 + ], + "score": 1.0, + "content": "The visualization results are provided in Figure 4.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 615, + 317, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 317, + 627 + ], + "score": 1.0, + "content": "Compared with the baseline T2T-ViT, our ViTAE", + "type": "text" + 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Moreover, ViTAE can better handle the scale", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "variance issue as shown in Figure 4(b). Namely, it can precisely cover the birds no matter they are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "in small, middle, or large size. Such observations demonstrate that introducing the intrinsic IBs of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "locality and scale-invariance from convolutions to transformers helps ViTAE learn more discriminate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 250, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 250, + 693 + ], + "score": 1.0, + "content": "features than the pure transformers.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 582, + 318, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 692 + ], + "lines": [], + "index": 46, + "bbox_fs": [ + 105, + 637, + 505, + 693 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 72, + 504, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 72, + 504, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 72, + 504, + 233 + ], + "spans": [ + { + "bbox": [ + 108, + 72, + 504, + 233 + ], + "score": 0.974, + "type": "image", + "image_path": "c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 72, + 504, + 125.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 125.66666666666666, + 504, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 179.33333333333331, + 504, + 232.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 252, + 506, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "multiple or single objects and the heatmaps. (b) Images containing the same class of objects at", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 274, + 335, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 335, + 287 + ], + "score": 1.0, + "content": "different scales and the heatmaps (Best viewed in color).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 257, + 318 + ], + "lines": [ + { + "bbox": [ + 104, + 303, + 259, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 259, + 320 + ], + "score": 1.0, + "content": "5 Limitation and discussion", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "In this paper, we explore two types of IBs and incorporate them into transformers through the proposed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "due to computational resource constraints, we have not scaled the ViTAE model and train it on large-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "size dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "are optimistic about its scale property from the following preliminary evidence. As illustrated in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "layers and convolution layers owing to the skip connection and parallel structure. According to the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 464 + ], + "score": 1.0, + "content": "data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 461, + 353, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 353, + 474 + ], + "score": 1.0, + "content": "viewpoint invariance [64] can be explored in the future study.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 487, + 183, + 500 + ], + "lines": [ + { + "bbox": [ + 104, + 485, + 185, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 185, + 503 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "Extensive experiments show that ViTAE outperforms representative vision transformers in various", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "respects including classification accuracy, data efficiency, training efficiency, and generalization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "hope that this study will provide valuable insights to the following studies of introducing intrinsic IB", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 600, + 437, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 437, + 612 + ], + "score": 1.0, + "content": "into vision transformers and understanding the impact of intrinsic and learned IBs.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 444, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 446, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 446, + 629 + ], + "score": 1.0, + "content": "Acknowledgement Dr. Jing Zhang is supported by the ARC project FL-170100117.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 72, + 504, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 72, + 504, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 72, + 504, + 233 + ], + "spans": [ + { + "bbox": [ + 108, + 72, + 504, + 233 + ], + "score": 0.974, + "type": "image", + "image_path": "c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 72, + 504, + 125.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 125.66666666666666, + 504, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 179.33333333333331, + 504, + 232.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 252, + 506, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "multiple or single objects and the heatmaps. (b) Images containing the same class of objects at", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 274, + 335, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 335, + 287 + ], + "score": 1.0, + "content": "different scales and the heatmaps (Best viewed in color).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 257, + 318 + ], + "lines": [ + { + "bbox": [ + 104, + 303, + 259, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 259, + 320 + ], + "score": 1.0, + "content": "5 Limitation and discussion", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "In this paper, we explore two types of IBs and incorporate them into transformers through the proposed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "due to computational resource constraints, we have not scaled the ViTAE model and train it on large-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "size dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "are optimistic about its scale property from the following preliminary evidence. As illustrated in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "layers and convolution layers owing to the skip connection and parallel structure. According to the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 464 + ], + "score": 1.0, + "content": "data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 461, + 353, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 353, + 474 + ], + "score": 1.0, + "content": "viewpoint invariance [64] can be explored in the future study.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 330, + 506, + 474 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 487, + 183, + 500 + ], + "lines": [ + { + "bbox": [ + 104, + 485, + 185, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 185, + 503 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "Extensive experiments show that ViTAE outperforms representative vision transformers in various", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "respects including classification accuracy, data efficiency, training efficiency, and generalization", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. 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Type ModelParams (M)MACs (G)Input SizeImageNet Real
Top-1Top-5Top-1
CNNResNet-18 [26]11.73.622470.386.777.3
ResNet-50 [26]25.67.622476.793.382.5
ResNet-101 [26]44.515.222478.394.183.7
ResNet-152 [26]60.222.622478.994.484.1
EfficientNet-B0 [73]5.30.822477.193.383.5
EfficientNet-B4 [73]19.38.438082.996.488.0
MobileNetV1 [31]4.30.622472.31-
MobileNetV2(1.4) [65]6.90.622474.7--
RegNetY-600M[62]6.11.222475.5--
RegNetY-4GF[62] RegNetY-8GF[62]20.6 39.28.0 16.022480.0186.4
22481.7187.4
DeiT-T[76]5.72.622472.291.180.6
DeiT-T [76]5.72.622474.591.982.1
LocalViT-T[43]5.92.622474.892.6
LocalViT-T2T[43]4.32.422472.5-1 1
ConT-Ti [89]5.81.622474.9--
PiT-Ti [29]4.91.422473.0-1
T2T-ViT-7 [93]4.31.222471.790.979.7
ViTAE-T4.81.522475.392.782.9
ViTAE-T ↑ 3844.85.738477.293.884.4
CeiT-T [92]6.42.422476.493.483.6
ConViT-Ti[15]6.02.022473.111
Cross ViT-Ti [6]6.93.222473.411
ViTAE-6M6.52.022477.994.184.9
PVT-T[80] LocalViT-PVT [43]13.23.822475.11
PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]13.59.622494.2-
ConViT-Ti+ [15] 10.04.022478.2 76.71
2.878.11-
10.6 19.26.2224 22480.2-1
13.23.422481.0- 95.4- 86.8
22.1 DeiT-S [76] 22.19.8 9.8224 22479.9 81.295.0 95.485.7 86.8
PVT-S[80]7.622479.8-
24.5 23.55.281.31
Conformer-Ti [58] Swin-T[47]224-
CeiT-S [92]29.09.022481.3-1
CvT-13 [83]24.2 20.09.022482.0 81.695.987.3 86.7
ConViT-S[15]27.09.0 10.8224 22481.31 11
Cross ViT-S [6]26.711.222481.011
PiT-S [29]23.54.822480.9
TNT-S [23]-1
Twins-PCPVT-S[10]23.810.422481.395.6-
24.17.422481.2--
Twins-SVT-S [10]24.05.622481.7-1
T2T-ViT-14 [93]21.55.222481.595.786.8
ViTAE-S23.65.622482.095.987.0
ViTAE-S ↑ 38423.620.238483.096.287.5
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Reduction CellNormal CellTop-1
Dilation (S1 ~ S3) PCMPreBN Post
× ××× ×68.7
× ×√× ×69.1
× ××√ ×69.0
× ××√ √68.8
× ×√× √69.9
[1,2]×3× ××× 69.5
[1,2,3]×3 ××× ×69.9
[1,2,3,4] × 3 ××× ×69.2
[1,2,3,4,5] × 3 ××× ×68.9
[1,2,3,4]↓ ××× ×69.8
[1,2,3,4]↓ √×× ×71.7
[1,2,3,4]↓ √√× √72.6
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"width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/_mQp5cr_iNy/_mQp5cr_iNy.md b/parse/train/_mQp5cr_iNy/_mQp5cr_iNy.md new file mode 100644 index 0000000000000000000000000000000000000000..cf509faa54a003fc0a340cf94bee303c12ab2806 --- /dev/null +++ b/parse/train/_mQp5cr_iNy/_mQp5cr_iNy.md @@ -0,0 +1,423 @@ +# ADVERSARIALLY GUIDED ACTOR-CRITIC + +# Johan Ferret∗ + +Yannis Flet-Berliac∗ +Inria, Scool team +Univ. Lille, CRIStAL, CNRS +yannis.flet-berliac@inria.fr + +Google Research, Brain team Inria, Scool team Univ. Lille, CRIStAL, CNRS + +# Olivier Pietquin + +Google Research, Brain team + +Philippe Preux +Inria, Scool team +Univ. Lille, CRIStAL, CNRS + +Matthieu Geist Google Research, Brain team + +# ABSTRACT + +Despite definite success in deep reinforcement learning problems, actor-critic algorithms are still confronted with sample inefficiency in complex environments, particularly in tasks where efficient exploration is a bottleneck. These methods consider a policy (the actor) and a value function (the critic) whose respective losses are built using different motivations and approaches. This paper introduces a third protagonist: the adversary. While the adversary mimics the actor by minimizing the KL-divergence between their respective action distributions, the actor, in addition to learning to solve the task, tries to differentiate itself from the adversary predictions. This novel objective stimulates the actor to follow strategies that could not have been correctly predicted from previous trajectories, making its behavior innovative in tasks where the reward is extremely rare. Our experimental analysis shows that the resulting Adversarially Guided Actor-Critic (AGAC) algorithm leads to more exhaustive exploration. Notably, AGAC outperforms current state-of-the-art methods on a set of various hard-exploration and procedurally-generated tasks. + +# 1 INTRODUCTION + +Research in deep reinforcement learning (RL) has proven to be successful across a wide range of problems (Silver et al., 2014; Schulman et al., 2016; Lillicrap et al., 2016; Mnih et al., 2016). Nevertheless, generalization and exploration in RL still represent key challenges that leave most current methods ineffective. First, a battery of recent studies (Farebrother et al., 2018; Zhang et al., 2018a; Song et al., 2020; Cobbe et al., 2020) indicates that current RL methods fail to generalize correctly even when agents have been trained in a diverse set of environments. Second, exploration has been extensively studied in RL; however, most hard-exploration problems use the same environment for training and evaluation. Hence, since a well-designed exploration strategy should maximize the information received from a trajectory about an environment, the exploration capabilities may not be appropriately assessed if that information is memorized. In this line of research, we choose to study the exploration capabilities of our method and its ability to generalize to new scenarios. Our evaluation domains will, therefore, be tasks with sparse reward in procedurally-generated environments. + +In this work, we propose Adversarially Guided Actor-Critic (AGAC), which reconsiders the actor-critic framework by introducing a third protagonist: the adversary. Its role is to predict the actor’s actions correctly. Meanwhile, the actor must not only find the optimal actions to maximize the sum of expected returns, but also counteract the predictions of the adversary. This formulation is lightly inspired by adversarial methods, specifically generative adversarial networks (GANs) (Goodfellow et al., 2014). Such a link between GANs and actor-critic methods has been formalized by Pfau & Vinyals (2016); however, in the context of a third protagonist, we draw a different analogy. The adversary can be interpreted as playing the role of a discriminator that must predict the actions of the actor, and the actor can be considered as playing the role of a generator that behaves to deceive the predictions of the adversary. This approach has the advantage, as with GANs, that the optimization procedure generates a diversity of meaningful data, corresponding to sequences of actions in AGAC. + +This paper analyses and explores how AGAC explicitly drives diversity in the behaviors of the agent while remaining reward-focused, and to which extent this approach allows to adapt to the evolving state space of procedurally-generated environments where the map is constructed differently with each new episode. Moreover, because stability is a legitimate concern since specific instances of adversarial networks were shown to be prone to hyperparameter sensitivity issues (Arjovsky & Bottou, 2017), we also examine this aspect in our experiments. + +The contributions of this work are as follow: (i) we propose a novel actor-critic formulation inspired from adversarial learning (AGAC), (ii) we analyse empirically AGAC on key reinforcement learning aspects such as diversity, exploration and stability, (iii) we demonstrate significant gains in performance on several sparse-reward hard-exploration tasks including procedurally-generated tasks. + +# 2 RELATED WORK + +Actor-critic methods (Barto et al., 1983; Sutton, 1984) have been extended to the deep learning setting by Mnih et al. (2016), who combined deep neural networks and multiple distributed actors with an actor-critic setting, with strong results on Atari. Since then, many additions have been proposed, be it architectural improvements (Vinyals et al., 2019), better advantage or value estimation (Schulman et al., 2016; Flet-Berliac et al., 2021), or the incorporation of off-policy elements (Wang et al., 2017; Oh et al., 2018; Flet-Berliac & Preux, 2020). Regularization was shown to improve actor-critic methods, either by enforcing trust regions (Schulman et al., 2015; 2017; Wu et al., 2017), or by correcting for off-policiness (Munos et al., 2016; Gruslys et al., 2018); and recent works analyzed its impact from a theoretical standpoint (Geist et al., 2019; Ahmed et al., 2019; Vieillard et al., 2020a;b). Related to our work, Han & Sung (2020) use the entropy of the mixture between the policy induced from a replay buffer and the current policy as a regularizer. To the best of our knowledge, none of these methods explored the use of an adversarial objective to drive exploration. + +While introduced in supervised learning, adversarial learning (Goodfellow et al., 2015; Miyato et al., 2016; Kurakin et al., 2017) was leveraged in several RL works. Ho & Ermon (2016) propose an imitation learning method that uses a discriminator whose task is to distinguish between expert trajectories and those of the agent while the agent tries to match expert behavior to fool the discriminator. Bahdanau et al. (2019) use a discriminator to distinguish goal states from non-goal states based on a textual instruction, and use the resulting model as a reward function. Florensa et al. (2018) use a GAN to produce sub-goals at the right level of difficulty for the current agent, inducing a form of curriculum. Additionally, Pfau & Vinyals (2016) provide a parallel between GANs and the actor-critic framework. + +While exploration is driven in part by the core RL algorithms (Fortunato et al., 2018; Han & Sung, 2020; Ferret et al., 2021), it is often necessary to resort to exploration-specific techniques. For instance, intrinsic motivation encourages exploratory behavior from the agent. Some works use state-visitation counts or pseudo-counts to promote exhaustive exploration (Bellemare et al., 2016a), while others use curiosity rewards, expressed in the magnitude of prediction error from the agent, to push it towards unfamiliar areas of the state space (Burda et al., 2018). Ecoffet et al. (2019) propose a technique akin to tree traversal to explore while learning to come back to promising areas. Eysenbach et al. (2018) show that encouraging diversity helps with exploration, even in the absence of reward. + +Last but not least, generalization is a key challenge in RL. Zhang et al. (2018b) showed that, even when the environment is not deterministic, agents can overfit to their training distribution and that it is difficult to distinguish agents likely to generalize to new environments from those that will not. In the same vein, recent work has advocated using procedurally-generated environments, in which a new instance of the environment is sampled when a new episode starts, to assess generalization capabilities better (Justesen et al., 2018; Cobbe et al., 2020). Finally, methods based on network randomization (Igl et al., 2019), noise injection (Lee et al., 2020), and credit assignment (Ferret et al., 2020) have been proposed to reduce the generalization gap for RL agents. + +# 3 BACKGROUND AND NOTATIONS + +We place ourselves in the Markov Decision Processes (Puterman, 1994) framework. A Markov Decision Process (MDP) is a tuple $M = \{ { \mathcal { S } } , { \mathcal { A } } , { \mathcal { P } } , R , \gamma \}$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $\mathcal { P }$ is the transition kernel, $\mathcal { R }$ is the bounded reward function and $\gamma \in [ 0 , 1 )$ is the discount factor. Let $\pi$ denote a stochastic policy mapping states to distributions over actions. We place ourselves in the infinite-horizon setting, i.e., we seek a policy that optimizes $\begin{array} { r } { J ( \pi ) = \mathbb { E } _ { \pi } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r \left( s _ { t } , a _ { t } \right) ] } \end{array}$ . The value of a state is the quantity $\begin{array} { r } { V ^ { \pi } ( s ) = \mathbb { E } _ { \pi } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r \left( s _ { t } , a _ { t } \right) | s _ { 0 } = s ] } \end{array}$ t=0 and the value of a state-action pair of performing action $a$ in state $s$ and then following policy is defined as: $\begin{array} { r } { Q ^ { \pi } ( s , a ) = \mathbb { E } _ { \pi } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r \left( s _ { t } , a _ { t } \right) \vert s _ { 0 } = s , a _ { 0 } = a \right] . } \end{array}$ . The advantage function, which quantifies how an action $a$ is better than the average action in state $s$ , is $A ^ { \pi } ( s , a ) \bar { = } Q ^ { \pi } ( s , a ) - V ^ { \pi } ( \bar { s } )$ . Finally, the entropy $\mathcal { H } ^ { \pi }$ of a policy is calculated as: $\mathcal { H } ^ { \pi } ( s ) = \mathbb { E } _ { \pi ( \cdot | s ) } \left[ - \log \pi ( \cdot | s ) \right]$ . + +Actor-Critic and Deep Policy Gradients. An actor-critic algorithm is composed of two main components: a policy and a value predictor. In deep RL, both the policy and the value function are obtained via parametric estimators; we denote $\theta$ and $\phi$ their respective parameters. The policy is updated via policy gradient, while the value is usually updated via temporal difference or Monte Carlo rollouts. In practice, for a sequence of transitions $\left\{ s _ { t } , a _ { t } , r _ { t } , s _ { t + 1 } \right\} _ { t \in [ 0 , N ] }$ , we use the following policy gradient loss (including the commonly used entropic penalty): + +$$ +\mathcal { L } _ { P G } = - \frac { 1 } { N } \sum _ { t ^ { \prime } = t } ^ { t + N } ( A _ { t ^ { \prime } } \log \pi \left( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } , \theta \right) + \alpha \mathcal { H } ^ { \pi } ( s _ { t ^ { \prime } } , \theta ) ) , +$$ + +where 2016) $\alpha$ is tfin t: $A _ { t }$ timato, with (Schulman et al.,a fixed hyperpa$\begin{array} { r } { A _ { t } = \sum _ { t ^ { \prime } = t } ^ { t + N } ( \gamma \lambda ) ^ { t ^ { \prime } - t } ( r _ { t ^ { \prime } } + \gamma V _ { \phi _ { \mathrm { o l d } } } ( s _ { t ^ { \prime } + 1 } ) - V _ { \phi _ { \mathrm { o l d } } } ( s _ { t ^ { \prime } } ) ) } \end{array}$ $\lambda$ $V _ { \phi _ { \mathrm { o l d } } }$ +value function, we solve the non-linear regression problem minimizeφ $\begin{array} { r } { \sum _ { t ^ { \prime } = t } ^ { t + N } ( V _ { \phi } ( s _ { t ^ { \prime } } ) - \hat { V } _ { t ^ { \prime } } ) ^ { 2 } } \end{array}$ where +$\hat { V } _ { t } = A _ { t } + V _ { \phi _ { \mathrm { o l d } } } ( s _ { t ^ { \prime } } )$ . + +# 4 ADVERSARIALLY GUIDED ACTOR-CRITIC + +To foster diversified behavior in its trajectories, AGAC introduces a third protagonist to the actor-critic framework: the adversary. The role of the adversary is to accurately predict the actor’s actions, by minimizing the discrepancy between its action distribution $\pi _ { \mathrm { a d v } }$ and the distribution induced by the policy $\pi$ . Meanwhile, in addition to finding the optimal actions to maximize the sum of expected returns, the actor must also counteract the adversary’s predictions by maximizing the discrepancy between $\pi$ and $\pi _ { \mathrm { a d v } }$ (see Appendix B for an illustration). This discrepancy, used as a form of exploration bonus, is defined as the difference of action log-probabilities (see Eq. (1)), whose expectation is the Kullback–Leibler divergence: + +$$ +D _ { \mathrm { K L } } ( \pi ( \cdot | s ) \| \pi _ { \mathrm { a d v } } ( \cdot | s ) ) = \mathbb { E } _ { \pi ( \cdot | s ) } \left[ \log \pi ( \cdot | s ) - \log \pi _ { \mathrm { a d v } } ( \cdot | s ) \right] . +$$ + +Formally, for each state-action pair $( s _ { t } , a _ { t } )$ in a trajectory, an action-dependent bonus $\log \pi ( a _ { t } | s _ { t } ) -$ $\log \pi _ { \mathrm { a d v } } \mathbf { \dot { ( } } a _ { t } | s _ { t } )$ is added to the advantage. In addition, the value target of the critic is modified to include the action-independent equivalent, which is the KL-divergence $D _ { \mathrm { K L } } ( \pi ( \cdot | s _ { t } ) \| \pi _ { \mathrm { a d v } } ( \cdot | s _ { t } ) )$ . We discuss the role of these mirrored terms below, and the implications of AGAC’s modified objective from a more theoretical standpoint in the next section. In addition to the parameters $\theta$ (resp. $\theta _ { \mathrm { o l d } }$ the parameter of the policy at the previous iteration) and $\phi$ defined above (resp. $\phi _ { \mathrm { o l d } }$ that of the critic), we denote $\psi$ (resp. $\psi _ { \mathrm { o l d . } }$ ) that of the adversary. + +AGAC minimizes the following loss: + +$$ +\mathcal { L } _ { \mathrm { A G A C } } = \mathcal { L } _ { \mathrm { P G } } + \beta _ { V } \mathcal { L } _ { V } + \beta _ { \mathrm { a d v } } \mathcal { L } _ { \mathrm { a d v } } . +$$ + +In the new objective $\begin{array} { r } { \mathcal { L } _ { P G } = - \frac { 1 } { N } \sum _ { t = 0 } ^ { N } ( A _ { t } ^ { \tt A G A C } \log \pi \left( a _ { t } | s _ { t } , \theta \right) + \alpha \mathcal { H } ^ { \pi } ( s _ { t } , \theta ) ) , } \end{array}$ AGAC modifies $A _ { t }$ as: + +$$ +A _ { t } ^ { \mathtt { A G A C } } = A _ { t } + c \left( \log \pi ( a _ { t } | s _ { t } , \theta _ { \mathrm { o l d } } ) - \log \pi _ { \mathrm { a d v } } ( a _ { t } | s _ { t } , \psi _ { \mathrm { o l d } } ) \right) , +$$ + +with $c$ a varying hyperparameter that controls the dependence on the action log-probability difference. To encourage exploration without preventing asymptotic stability, $c$ is linearly annealed during the course of training. ${ \mathcal { L } } _ { V }$ is the objective function of the critic defined as: + +$$ +\mathcal { L } _ { V } = \frac { 1 } { N } \sum _ { t = 0 } ^ { N } \left( V _ { \phi } ( s _ { t } ) - \left( \hat { V } _ { t } + c D _ { \mathrm { K L } } ( \pi ( \cdot | s _ { t } , \theta _ { \mathrm { o l d } } ) | | \pi _ { \mathrm { a d v } } ( \cdot | s _ { t } , \psi _ { \mathrm { o l d } } ) ) \right) \right) ^ { 2 } . +$$ + +Finally, ${ \mathcal { L } } _ { \mathrm { a d v } }$ is the objective function of the adversary: + +$$ +\mathcal { L } _ { \mathrm { a d v } } = \frac { 1 } { N } \sum _ { t = 0 } ^ { N } D _ { \mathrm { K L } } \big ( \pi \big ( \cdot | s _ { t } , \theta _ { \mathrm { o l d } } \big ) \big | \big | \pi _ { \mathrm { a d v } } \big ( \cdot | s _ { t } , \psi \big ) \big ) . +$$ + +Eqs. (1), (2) and (3) are the three equations that our method modifies (we color in blue the specific parts) in the traditional actor-critic framework. The terms $\beta _ { V }$ and $\beta _ { \mathrm { a d v } }$ are fixed hyperparameters. + +Under the proposed actor-critic formulation, the probability of sampling an action is increased if the modified advantage is positive, i.e. (i) the corresponding return is larger than the predicted value and/or (ii) the action log-probability difference is large. More precisely, our method favors transitions whose actions were less accurately predicted than the average action, i.e. $\log \pi ( a | s ) -$ $\begin{array} { r } { \log \pi _ { \mathrm { a d v } } ( a | s ) \geq D _ { \mathrm { K L } } ( \pi ( \cdot | s ) \| \pi _ { \mathrm { a d v } } ( \cdot | s ) ) } \end{array}$ . This is particularly visible for $\lambda 1$ , in which case the generalized advantage is $A _ { t } = G _ { t } - V _ { \phi _ { \mathrm { o l d } } } ( s _ { t } )$ , resulting in the appearance of both aforementioned mirrored terms in the modified advantage: + +$$ +A _ { t } ^ { \mathtt { A G A C } } = G _ { t } - \hat { V } _ { t } ^ { \phi _ { \mathrm { o d d } } } + c \left( \log \pi ( a _ { t } | s _ { t } ) - \log \pi _ { \mathrm { a d v } } ( a _ { t } | s _ { t } ) - \hat { D } _ { \mathrm { K L } } ^ { \phi _ { \mathrm { o d d } } } ( \pi ( \cdot | s _ { t } ) | | \pi _ { \mathrm { a d v } } ( \cdot | s _ { t } ) ) \right) , +$$ + +with KL- $G _ { t }$ the observed return, rgence (estimated co $\hat { V } _ { t } ^ { \phi _ { \mathrm { o l d } } }$ the estents of eturn and from Eq. $\hat { D } _ { \mathrm { K L } } ^ { \phi _ { \mathrm { o l d } } } \big ( \pi \big ( \cdot | s _ { t } \big ) \big | \big | \pi _ { \mathrm { a d v } } \big ( \cdot | s _ { t } \big ) \big )$ the estimated $V _ { \phi _ { \mathrm { o l d } } } ( s _ { t } )$ + +To avoid instability, in practice the adversary is a separate estimator, updated with a smaller learning rate than the actor. This way, it represents a delayed and more steady version of the actor’s policy, which prevents the agent from having to constantly adapt or focus solely on fooling the adversary. + +# 4.1 BUILDING MOTIVATION + +In the following, we provide an interpretation of AGAC by studying the dynamics of attraction and repulsion between the actor and the adversary. To simplify, we study the equivalent of AGAC in a policy iteration (PI) scheme. PI being the dynamic programming scheme underlying the standard actor-critic, we have reasons to think that some of our findings translate to the original AGAC algorithm. In PI, the quantity of interest is the action-value, which AGAC would modify as: + +$$ +Q _ { \pi _ { k } } ^ { \mathtt { A G A C } } = Q _ { \pi _ { k } } + c ( \log \pi _ { k } - \log \pi _ { \mathrm { a d v } } ) , +$$ + +with $\pi _ { k }$ the policy at iteration $k$ . Incorporating the entropic penalty, the new policy $\pi _ { k + 1 }$ verifies: + +$$ +\pi _ { k + 1 } = \underset { \pi } { \arg \operatorname* { m a x } } \ : \mathcal { I } _ { \mathrm { P I } } ( \pi ) = \underset { \pi } { \arg \operatorname* { m a x } } \ : \mathbb { E } _ { s } \mathbb { E } _ { a \sim \pi ( \cdot \vert s ) } [ Q _ { \pi _ { k } } ^ { \mathrm { { A G A C } } } ( s , a ) - \alpha \log \pi ( a \vert s ) ] . +$$ + +We can rewrite this objective: + +$$ +\begin{array} { r l } & { \mathcal { T } _ { \mathrm { { p I } } } ( \pi ) = \mathbb { E } _ { s } \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ Q _ { \pi _ { k } } ^ { \mathrm { { A G } A C } } ( s , a ) - \alpha \log \pi ( a | s ) ] } \\ & { \qquad = \mathbb { E } _ { s } \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ Q _ { \pi _ { k } } ( s , a ) + c ( \log \pi _ { k } ( a | s ) - \log \pi _ { \mathrm { a d v } } ( a | s ) ) - \alpha \log \pi ( a | s ) ] } \\ & { \qquad = \mathbb { E } _ { s } \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ Q _ { \pi _ { k } } ( s , a ) + c ( \log \pi _ { k } ( a | s ) - \log \pi ( a | s ) + \log \pi ( a | s ) - \log \pi _ { \mathrm { a d v } } ( a | s ) ) - \alpha \log \mathrm { { g } } } \\ & { \qquad = \mathbb { E } _ { s } \Big [ \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ Q _ { \pi _ { k } } ( s , a ) ] \underbrace { - c D _ { \mathrm { K L } } ( \pi ( \cdot | s ) | | \pi _ { k } ( \cdot | s ) ) } _ { \pi _ { k } \mathrm { ~ i s ~ a t u r a c i t i v e } } \underbrace { + c D _ { \mathrm { K L } } ( \pi ( \cdot | s ) | | \pi _ { \mathrm { a d v } } ( \cdot | s ) ) } _ { \pi _ { \mathrm { a d v } } \mathrm { ~ i s ~ r e p u l s i v e } } \underbrace { + \alpha \mathcal { H } ( \pi ( \cdot | s ) | | \pi _ { \mathrm { a d v } } ( \cdot | s ) ) } _ { \mathrm { c n ~ t e r ~ i s r e p u l s i n c } } . } \end{array} +$$ + +Thus, in the PI scheme, AGAC finds a policy that maximizes $Q$ -values, while at the same time remaining close to the current policy and far from a mixture of the previous policies (i.e., $\pi _ { k - 1 }$ , $\pi _ { k - 2 }$ , $\pi _ { k - 3 } , \ldots )$ . Note that we experimentally observe (see Section 5.3) that our method performs better with a smaller learning rate for the adversarial network than that of the other networks, which could imply that a stable repulsive term is beneficial. + +This optimization problem is strongly concave in $\pi$ (thanks to the entropy term), and is state-wise a Legendre-Fenchel transform. Its solution is given by (see Appendix E for the full derivation): + +$$ +\pi _ { k + 1 } \propto \left( \frac { \pi _ { k } } { \pi _ { \mathrm { a d v } } } \right) ^ { \frac { c } { \alpha } } \exp \frac { Q _ { \pi _ { k } } } { \alpha } . +$$ + +This result gives us some insight into the behavior of the objective function. Notably, in our example, if $\pi _ { \mathrm { a d v } }$ is fixed and $c = \alpha$ , we recover a KL-regularized PI scheme (Geist et al., 2019) with the modified reward $r - c \log \pi _ { \mathrm { a d v } }$ . + +# 4.2 IMPLEMENTATION + +In all of the experiments, we use PPO (Schulman et al., 2017) as the base algorithm and build on it to incorporate our method. Hence, + +$$ +\mathcal { L } _ { P G } = - \frac { 1 } { N } \sum _ { t ^ { \prime } = t } ^ { t + N } \operatorname* { m i n } \left( \frac { \pi ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } , \theta ) } { \pi ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } , \theta _ { \mathrm { o l d } } ) } A _ { t ^ { \prime } } ^ { \mathrm { R G K } } , \mathrm { c l i p } \left( \frac { \pi ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } , \theta ) } { \pi ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } , \theta _ { \mathrm { o l d } } ) } , 1 - \epsilon , 1 + \epsilon \right) A _ { t ^ { \prime } } ^ { \mathrm { R G K } } \right) , +$$ + +with $A _ { t ^ { \prime } } ^ { \tt A G A C }$ given in Eq. (1), $N$ the temporal length considered for one update of parameters and $\epsilon$ the clipping parameter. Similar to RIDE (Raileanu & Rocktäschel, 2019), we also discount PPO by episodic state visitation counts, except for VizDoom (cf. Section 5.1). The actor, critic and adversary use the convolutional architecture of the Nature paper of DQN (Mnih et al., 2015) with different hidden sizes (see Appendix D for architecture details). The three neural networks are optimized using Adam (Kingma & Ba, 2015). Our method does not use RNNs in its architecture; instead, in all our experiments, we use frame stacking. Indeed, Hausknecht & Stone (2015) interestingly demonstrate that although recurrence is a reliable method for processing state observation, it does not confer any systematic advantage over stacking observations in the input layer of a CNN. Note that the parameters are not shared between the policy, the critic and the adversary and that we did not observe any noticeable difference in computational complexity when using AGAC compared to PPO. We direct the reader to Appendix C for a list of hyperparameters. In particular, the $c$ coefficient of the adversarial bonus is linearly annealed. + +At each training step, we perform a stochastic optimization step to minimize $\mathcal { L } _ { \mathtt { A G A C } }$ using stop-gradient: + +$$ +\begin{array} { r l r l } & { \boldsymbol { \vartheta } \mathrm { A d a m } ( \theta , \nabla _ { \theta } \mathcal { L } _ { P G } , \eta _ { 1 } ) , } & & { \boldsymbol { \phi } \mathrm { A d a m } ( \phi , \nabla _ { \phi } \mathcal { L } _ { V } , \eta _ { 1 } ) , } & & { \boldsymbol { \psi } \mathrm { A d a m } ( \psi , \nabla _ { \psi } \mathcal { L } _ { \sf a d v } , \eta _ { 2 } ) . } \end{array} +$$ + +# 5 EXPERIMENTS + +In this section, we describe our experimental study in which we investigate: (i) whether the adversarial bonus alone (e.g. without episodic state visitation count) is sufficient to outperform other methods in VizDoom, a sparse-reward task with high-dimensional observations, (ii) whether AGAC succeeds in partially-observable and procedurally-generated environments with high sparsity in the rewards, compared to other methods, (iii) how well AGAC is capable of exploring in environments without extrinsic reward, (iv) the training stability of our method. In all of the experiments, lines are average performances and shaded areas represent one standard deviation. The code for our method is released at github.com/yfletberliac/adversarially-guided-actor-critic. + +![](images/263aa3eeaa25d94147cc96fb59396bbc06f2d865b71d0e66161e08136afe2782.jpg) +Figure 1: (a,b) Frames from the 3-D navigation task VizdoomMyWayHome. (c) MiniGridKeyCorridorS6R3. (d) MiniGrid-ObstructedMazeFull. + +Environments. To carefully evaluate the performance of our method, its ability to develop robust exploration strategies and its generalization to unseen states, we choose tasks that have been used in prior work, which are tasks with high-dimensional observations, sparse reward and procedurallygenerated environments. In VizDoom (Kempka et al., 2016), the agent must learn to move along corridors and through rooms without any reward feedback from the 3-D environment. The MiniGrid environments (Chevalier-Boisvert et al., 2018) are a set of challenging partially-observable and sparse-reward gridworlds. In this type of procedurally-generated environments, memorization is impossible due to the huge size of the state space, so the agent must learn to generalize across the different layouts of the environment. Each gridworld has different characteristics: in the MultiRoom tasks, the agent is placed in the first room and should reach a goal placed in the most distant room. In the KeyCorridor tasks, the agent must navigate to pick up an object placed in a room locked by a door whose key is in another room. Finally, in the ObstructedMaze tasks, the agent must pick up a box that is placed in a corner of a 3x3 maze in which the doors are also locked, the keys are hidden in boxes and balls obstruct the doors. All considered environments (see Fig. 1 for some examples) are available as part of OpenAI Gym (Brockman et al., 2016). + +Baselines. For a fair assessment of our method, we compare to some of the most prominent methods specialized in hard-exploration tasks: RIDE (Raileanu & Rocktäschel, 2019), based on an intrinsic reward associated with the magnitude of change between two consecutive state representations and state visitation, Count as Count-Based Exploration (Bellemare et al., 2016b), which we couple with IMPALA (Espeholt et al., 2018), RND (Burda et al., 2018) in which an exploration bonus is positively correlated to the error of predicting features from the observations and ICM (Pathak et al., 2017), where a module only predicts the changes in the environment that are produced by the actions of the agent. Finally, we compare to most the recent and best performing method at the time of writing in procedurally-generated environments: AMIGo (Campero et al., 2021) in which a goal-generating teacher provides count-based intrinsic goals. + +5.1 ADVERSARIALLY-BASED EXPLORATION (NO EPISODIC COUNT) + +Table 1: Average return in VizDoom at different timesteps. + +
Nb.of Timesteps2M4M6M8M10M
AGAC0.74± 0.050.96 ± 0.0010.96 ± 0.0010.97 ± 0.0010.97 ± 0.001
RIDE0.0.0.95 ± 0.0010.97 ± 0.0010.97 ± 0.001
ICM0.0.0.95 ±0.0010.97 ± 0.0010.97 ± 0.001
AMIGo0.0.0.0.0.
RND0.0.0.0.0.
Count0.0.0.0.0.
+ +In this section, we assess the benefits of using an adversarially-based exploration bonus and examine how AGAC performs without the help of count-based exploration. In order to provide a comparison to state-of-the-art methods, we choose VizDoom, a hard-exploration problem used in prior work. In this game, the map consists of 9 rooms connected by corridors where 270 steps separate the initial position of the agent and the goal under an optimal policy. Episodes are terminated either when the agent finds the goal or if the episode exceeds 2100 timesteps. Importantly, while other algorithms (Raileanu & Rocktäschel, 2019; Campero et al., 2021) benefit from count-based exploration, this study has been conducted with our method not benefiting from episodic count whatsoever. Results in Table 1 indicate that AGAC clearly outperforms other methods in sample-efficiency. Only the methods ICM and RIDE succeed in matching the score of AGAC, and with about twice as much transitions $( \sim 3 \mathbf { M }$ vs. 6M). Interestingly, AMIGo performs similarly to Count and RND. We find this result surprising because AMIGo has proven to perform well in the MiniGrid environments. Nevertheless, it appears that concurrent works to ours experienced similar issues with the accompanying implementation1. The results of AGAC support the capabilities of the adversarial bonus and show that it can, on its own, achieve significant gains in performance. However, the VizDoom task is not procedurally-generated; hence we have not evaluated the generalization to new states yet. In the following section, we use MiniGrid to investigate this. + +# 5.2 HARD-EXPLORATION TASKS WITH PARTIALLY-OBSERVABLE ENVIRONMENTS + +We now evaluate our method on multiple hard-exploration procedurally-generated tasks from MiniGrid. Details about MiniGrid can be found in Appendix C.1. Fig. 2 indicates that AGAC significantly outperforms other methods on these tasks in sample-efficiency and performance. AGAC also outperforms the current state-of-the-art method, AMIGo, despite the fact that it uses the fully-observable version of MiniGrid. Note that we find the same poor performance results when training AMIGo in MiniGrid, similar to Vizdoom results. For completeness, we also report in Table 2 of Appendix A.1 the performance results with the scores reported in the original papers Raileanu & Rocktäschel (2019) and Campero et al. (2021). We draw similar conclusions: AGAC clearly outperforms the state-of-the-art RIDE, AMIGo, Count, RND and ICM. + +![](images/c694a663b7b25ee719de7eb448621e3cb5826c21e451c62ee01f34f8d0c5792f.jpg) +Figure 2: Performance evaluation of AGAC. + +In all the considered tasks, the agent must learn to generalize across a very large state space because the layouts are generated procedurally. We consider three main arguments to explain why our method is successful: (i) our method makes use of partial observations: in this context, the adversary has a harder time predicting the actor’s actions; nevertheless, the mistakes of the former benefit the latter in the form of an exploration bonus, which pushes the agent to explore further in order to deceive the adversary, (ii) the exploration bonus (i.e. intrinsic reward) does not dissipate compared to most other methods, as observed in Fig. 9 in Appendix A.4, (iii) our method does not make assumptions about the environment dynamics (e.g., changes in the environment produced by an action as in Raileanu & Rocktäschel (2019)) since this can hinder learning when the space of state changes induced by an action is too large (such as the action of moving a block in ObstructedMaze). + +In Appendix A.3, we also include experiments in two environments with extremely sparse reward signals: KeyCorridorS8R3 and ObstructedMazeFull. Despite the challenge, AGAC still manages to find rewards and can perform well by taking advantage of the diversified behaviour induced by our method. To the best of our knowledge, no other method ever succeeded to perform well $\mathit { \Theta } > 0$ average return) in those tasks. We think that given more computing time, AGAC’s score could go higher. + +# 5.3 TRAINING STABILITY + +![](images/df1a9efb68034f4a5f52a20d9d5ef22a9514eee7144535790fe25105fc795095.jpg) +Figure 3: Sensitivity analysis of AGAC in KeyCorridorS4R3. + +Here we want to analyse the stability of the method when changing hyperparameters. The most important parameters in AGAC are $c$ , the coefficient for the adversarial bonus, and the learning rates ratio $\begin{array} { r } { \nu = \frac { \eta _ { 2 } } { \eta _ { 1 } } } \end{array}$ . We choose KeyCorridorS4R3 as the evaluation task because among all the tasks considered, its difficulty is at a medium level. Fig. 3 shows the learning curves. For readability, we plot the average return only; the standard deviation is sensibly the same for all curves. We observe that deviating from the hyperparameter values found using grid search results in a slower training. Moreover, although reasonable, $c$ appears to have more sensitivity than $\nu$ . + +![](images/052df8089e69d4ed8859a825df3b5e90f5f628f6ab3239fa565170ac0833dc68.jpg) +Figure 5: State visitation heatmaps for RND, Count, a random uniform policy, RIDE, and AGAC trained in a singleton environment (top row) and procedurally-generated environments (bottom row) without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task. + +# 5.4 EXPLORATION IN REWARD-FREE ENVIRONMENT + +To better understand the effectiveness of our method and inspect how the agent collects rewards that would not otherwise be achievable by simple exploration heuristics or other methods, we analyze the performance of AGAC in another (procedurally-generated) challenging environment, MultiRoomN10S6, when there is no reward signal, i.e. no extrinsic reward. Beyond the good performance of our method when extrinsic rewards are given to the agent, Fig. 4 indicates that the exploration induced by our method makes the agent succeed in a significant proportion of the episodes: in the configuration “NoExtrinsicReward” the reward signal is not given (the goal is invisible to the agent) and the performance of AGAC stabilizes around an average return of $\sim 0 . 1 5$ . Since the return of an episode is either 0 or 1 (depending on whether the agent reached the goal state or not), and because this value is aggregated across several episodes, the results indicate that reward-free AGAC succeeds in $\sim 1 5 \%$ of the tasks. Comparatively, random agents have a zero average return. This poor performance is in accordance with the results in Raileanu & Rocktäschel (2019) and reflects the complexity of the task: in order to go from one room to another, an agent must perform a specific action to open a door and cross it within the time limit of 200 timesteps. In the following, we visually investigate how different methods explore the environments. + +![](images/2b9e22a230d11ca43128f658217d5ed34a847b70794472a70049ba6a59245c82.jpg) +Figure 4: Average return on N10S6 with and without extrinsic reward. + +# 5.5 VISUALIZING COVERAGE AND DIVERSITY + +In this section, we first investigate how different methods explore environments without being guided by extrinsic rewards (the green goal is invisible to the agent) on both procedurally-generated and singleton environments. In singleton environments, an agent has to solve the same task in the same environment/maze in every episode. Fig. 5 shows the state visitation heatmaps (darker areas correspond to more visits) after a training of 10M timesteps. We observe that most of the methods explore inefficiently in a singleton environment and that only RIDE succeeds in reaching the fifth room while AGAC reaches the last (tenth) room. After training the agents in procedurally-generated environments, the methods explore even less efficiently while AGAC succeeds in exploring all rooms. + +![](images/79b4e4dfaa987603232a227680bcdf155166caf72f8802e24284152dfcbbd4e8.jpg) +Figure 6: State visitation heatmaps of the last ten episodes of an agent trained in procedurallygenerated environments without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task. The agent is continuously engaging in new strategies. + +We now qualitatively study the diversity of an agent’s behavior when trained with AGAC. Fig. 6 presents the state visitation heatmaps of the last ten episodes for an agent trained in procedurallygenerated environments in the MultiRoomN10S6 task without extrinsic reward. The heatmaps correspond to the behavior of the resulting policy, which is still learning from the AGAC objective. Looking at the figure, we can see that the strategies vary at each update with, for example, backand-forth and back-to-start behaviors. Although there are no extrinsic reward, the strategies seem to diversify from one update to the next. Finally, Fig. 7 in Appendix A.2 shows the state visitation heatmaps in a different configuration: when the agent has been trained on a singleton environment in the MultiRoomN10S6 task without extrinsic reward. Same as previously, the agent is updated between each episode. Looking at the figure, we can make essentially the same observations as previously, with a noteworthy behavior in the fourth heatmap of the bottom row where it appears the agent went to the fourth room to remain inside it. Those episodes indicate that, although the agent sees the same environment repeatedly, the successive adversarial updates force it to continuously adapt its behavior and try new strategies. + +# 6 DISCUSSION + +This paper introduced AGAC, a modification to the traditional actor-critic framework: an adversary network is added as a third protagonist. The mechanics of AGAC have been discussed from a policy iteration point of view, and we provided theoretical insight into the inner workings of the proposed algorithm: the adversary forces the agent to remain close to the current policy while moving away from the previous ones. In a nutshell, the influence of the adversary makes the actor conservatively diversified. + +In the experimental study, we have evaluated the adversarially-based bonus in VizDoom and empirically demonstrated its effectiveness and superiority compared to other relevant methods (some benefiting from count-based exploration). Then, we have conducted several performance experiments using AGAC and have shown a significant performance improvement over some of the most popular exploration methods (RIDE, AMIGo, Count, RND and ICM) on a set of various challenging tasks from MiniGrid. These procedurally-generated environments have served another purpose which is to validate the capacity of our method to generalize to unseen scenarios. In addition, the training stability of our method has been studied, showing a greater but acceptable sensitivity for $c$ , the adversarial bonus coefficient. 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Human-level control through deep reinforcement learning. nature, 518(7540):529–533, 2015. + +Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, pp. 1928–1937, 2016. + +Rémi Munos, Tom Stepleton, Anna Harutyunyan, and Marc Bellemare. Safe and efficient off-policy reinforcement learning. In Advances in Neural Information Processing Systems, pp. 1054–1062, 2016. + +Junhyuk Oh, Yijie Guo, Satinder Singh, and Honglak Lee. Self-imitation learning. In International Conference on Machine Learning, 2018. + +Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell. Curiosity-driven exploration by self-supervised prediction. In International Conference on Machine Learning, pp. 2778–2787, 2017. + +David Pfau and Oriol Vinyals. Connecting generative adversarial networks and actor-critic methods. arXiv preprint arXiv:1610.01945, 2016. + +Martin Puterman. Markov Decision Processes. Wiley, 1994. ISBN 978-0471727828. + +Roberta Raileanu and Tim Rocktäschel. Ride: Rewarding impact-driven exploration for procedurallygenerated environments. In International Conference on Learning Representations, 2019. + +John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, pp. 1889–1897, 2015. + +John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. High-dimensional continuous control using generalized advantage estimation. In International Conference on Learning Representations, 2016. + +John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. + +David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International Conference on Machine Learning, 2014. + +Xingyou Song, Yiding Jiang, Yilun Du, and Behnam Neyshabur. Observational overfitting in reinforcement learning. In International Conference on Learning Representations, 2020. + +Richard Stuart Sutton. Temporal Credit Assignment in Reinforcement Learning. PhD thesis, University of Massachusetts Amherst, 1984. + +Nino Vieillard, Tadashi Kozuno, Bruno Scherrer, Olivier Pietquin, Rémi Munos, and Matthieu Geist. Leverage the average: an analysis of regularization in rl. In Advances in Neural Information Processing Systems, 2020a. + +Nino Vieillard, Olivier Pietquin, and Matthieu Geist. Munchausen reinforcement learning. In Advances in Neural Information Processing Systems, 2020b. + +Oriol Vinyals, Igor Babuschkin, Wojciech Czarnecki, Michaël Mathieu, Andrew Dudzik, Junyoung Chung, David Choi, Richard Powell, Timo Ewalds, Petko Georgiev, Junhyuk Oh, Dan Horgan, Manuel Kroiss, Ivo Danihelka, Aja Huang, Laurent Sifre, Trevor Cai, John Agapiou, Max Jaderberg, and David Silver. Grandmaster level in StarCraft II using multi-agent reinforcement learning. Nature, 575, 11 2019. + +Ziyu Wang, Victor Bapst, Nicolas Heess, Volodymyr Mnih, Remi Munos, Koray Kavukcuoglu, and Nando de Freitas. Sample efficient actor-critic with experience replay. In International Conference on Learning Representations, 2017. + +Yuhuai Wu, Elman Mansimov, Roger B Grosse, Shun Liao, and Jimmy Ba. Scalable trust-region method for deep reinforcement learning using kronecker-factored approximation. In Advances in Neural Information Processing Systems, pp. 5279–5288, 2017. + +Amy Zhang, Nicolas Ballas, and Joelle Pineau. A dissection of overfitting and generalization in continuous reinforcement learning. arXiv preprint arXiv:1806.07937, 2018a. + +Chiyuan Zhang, Oriol Vinyals, Remi Munos, and Samy Bengio. A study on overfitting in deep reinforcement learning. arXiv preprint arXiv:1804.06893, 2018b. + +# A ADDITIONAL EXPERIMENTS + +# A.1 MINIGRID PERFORMANCE + +In this section, we report the final performance of all methods considered in the MiniGrid experiments of Fig. 2 with the scores reported in Raileanu & Rocktäschel (2019) and Campero et al. (2021). All methods have a budget of 200M frames. + +Table 2: Final average performance of all methods on several MiniGrid environments. + +
TaskKC-S4R3KC-S5R3MR-N10S10OM-2DlhbOM-1QOM-2Q
AGAC0.950.930.520.640.780.63
RIDE0.190.0.400.0.0.
AMIGo0.540.0.0.200.0.
RND0.0.0.0.030.0.
Count0.0.0.0.0.0.
ICM0.0.0.0.0.0.
+ +A.2 STATE VISITATION HEATMAPS IN SINGLETON ENVIRONMENT WITH NO EXTRINSIC REWARD + +In this section, we provide additional state visitation heatmaps. The agent has been trained on a singleton environment from the MultiRoomN10S6 task without extrinsic reward. The last ten episodes of the training suggest that although the agent experiences the same maze over and over again, the updates force it to change behavior and try new strategies. + +![](images/bdafbf5ffde8d9779028392dcc80b27fab8981f669d6dfdeaf60ea0f5e559087.jpg) +Figure 7: State visitation heatmaps of the last ten episodes of an agent trained in a singleton environment with no extrinsic reward 10M timesteps in the MultiRoomN10S6 task. The agent is continuously engaging into new strategies. + +# A.3 (EXTREMELY) HARD-EXPLORATION TASKS WITH PARTIALLY-OBSERVABLE ENVIRONMENTS + +In this section, we include additional experiments on one of the hardest tasks available in MiniGrid. The first is KeyCorridorS8R3, where the size of the rooms has been increased. In it, the agent has to pick up an object which is behind a locked door: the key is hidden in another room and the agent has to explore the environment to find it. The second, ObstructedMazeFull, is similar to ObstructedMaze4Q, where the agent has to pick up a box which is placed in one of the four corners of a 3x3 maze: the doors are locked, the keys are hidden in boxes and the doors are obstructed by balls. In those difficult tasks, only our method succeeds in exploring well enough to find rewards. + +![](images/2c2ebfe32501ea6424be77611fdb79a4abd159715b1b940e6c5d877763053c79.jpg) +Figure 8: Performance evaluation of AGAC compared to RIDE, AMIGo, Count, RND and ICM on extremely hard-exploration problems. + +# A.4 MEAN INTRINSIC REWARD + +In this section, we report the mean intrinsic reward computed for an agent trained in MultiRoomN12S10 to conveniently compare our results with that of Raileanu & Rocktäschel (2019). We observe in Fig. 9 that the intrinsic reward is consistently larger for our method and that, contrary to other methods, does not converge to low values. Please note that, in all considered experiments, the adversarial bonus coefficient $c$ in Eq. 2 and 3 is linearly annealed throughout the training since it is mainly useful at the beginning of learning when the rewards have not yet been met. In the long run, this coefficient may prevent the agent from solving the task by forcing it to always favour exploration over exploitation. + +![](images/28473b81bbecae89bf11a7987beadae0deed242b6ba51bdf0240df3613591c4e.jpg) +Figure 9: Average intrinsic reward for different methods trained in MultiRoomN12S10. + +![](images/3ce38e04ad09fb53cf941a1929d15591fb412b83302fde934dd965c50f3e5f5f.jpg) +Figure 10: A simple schematic illustration of AGAC. Left: the adversary minimizes the KL-divergence with respect to the action probability distribution of the actor. Right: the actor receives a bonus when counteracting the predictions of the adversary. + +# C EXPERIMENTAL DETAILS AND HYPERPARAMETERS + +# C.1 MINIGRID SETUP + +Here, we describe in more details the experimental setup we used in our MiniGrid experiments. + +There are several different MiniGrid scenarios that we consider in this paper. MultiRoom corresponds to a set of navigation tasks, where the goal is to go from a starting state to a goal state. The notation MultiRoom-N2S4 means that there are 2 rooms in total, and that each room has a maximal side of 4. In order to go from one room to another, the agent must perform a specific action to open a door. Episodes are terminated with zero reward after a maximum of $2 0 \times N$ steps with $N$ the number of rooms. In KeyCorridor, the agent also has to pick up a key, since the goal state is behind a door that only lets it in with the key. The notation KeyCorridor-S3R4 means that there are 4 side corridors, leading to rooms that have a maximal side of 3. The maximum number of steps is 270. In ObstructedMaze, keys are hidden in boxes, and doors are obstructed by balls the agent has to get out of its way. The notation ObstructedMaze-1Dl means that there are two connected rooms of maximal side 6 and 1 door (versus a 3x3 matrix and 2 doors if the leading characters are $2 D$ ), adding $h$ as a suffix places keys in boxes, and adding $b$ as a suffix adds balls in front of doors. Using $Q$ as a suffix is equivalent to using lhb (that is, both hiding keys and placing balls to be moved). The maximum number of steps is 576. ObstructedMazeFull is the hardest configuration for this scenario, since it has the maximal number of keys, balls to move, and doors possible. + +In each scenario, the agent has access to a partial view of the environment, a $7 \mathrm { x } 7 $ square that includes itself and points in the direction of its previous movement. + +# C.2 HYPERPARAMETERS + +In all experiments, we train six different instances of our algorithm with different random seeds. In Table 3, we report the list of hyperparameters. + +Table 3: Hyperparameters used in AGAC. + +
ParameterValue
Horizon T2048
Nb. epochs4
Nb.minibatches8
Nb.frames stacked4
NonlinearityELU (Clevert et al., 2016)
Discount γ0.99
GAE parameter 入0.95
PPO clipping parameter e0.2
βv0.5
C c anneal schedule4·10-4 (4·10-5 in VizDoom)
linear
βadv4·10-5
Adam stepsize m13·10-4
Adam stepsize m29.10-5 = 0.3· m1
+ +# D IMPLEMENTATION DETAILS + +In Fig. 11 is depicted the architecture of our method. + +![](images/166400757cb98035c5bef5beb612bef9f83aa90345f67690ff8af762d98632a2.jpg) +Figure 11: Artificial neural architecture of the critic, the actor and the adversary. + +# E PROOF OF SECTION 4.1 RESULTS + +In this section, we provide a short proof for the result of the optimization problem in Section 4.1. We recall the result here: + +$$ +\pi _ { k + 1 } = \mathop { \arg \operatorname* { m a x } } _ { \pi } \mathcal { I } _ { \mathrm { P I } } ( \pi ) \propto \left( \frac { \pi _ { k } } { \pi _ { \mathrm { a d v } } } \right) ^ { \frac { c } { \alpha } } \exp \frac { Q _ { \pi _ { k } } } { \alpha } , +$$ + +with the objective function: + +$$ +\begin{array} { r } { \mathcal { I } _ { \mathrm { P I } } ( \pi ) = \mathbb { E } _ { s } \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ Q _ { \pi _ { k } } ( s , a ) + c ( \log \pi _ { k } ( a | s ) - \log \pi _ { \mathrm { a d v } } ( a | s ) ) - \alpha \log \pi ( a | s ) ] . } \end{array} +$$ + +Proof. We first consider a simpler optimization problem: a $\because \operatorname { g m a x } _ { \pi } \langle \pi , Q _ { \pi _ { k } } \rangle + \alpha \mathcal { H } ( \pi )$ , whose solution is known (Vieillard et al., 2020a, Appendix A). The expression for the maximizer is the $\alpha$ -scaled softmax: + +$$ +\pi ^ { * } = \frac { \exp ( \frac { Q _ { \pi _ { k } } } { \alpha } ) } { \langle 1 , \exp ( \frac { Q _ { \pi _ { k } } } { \alpha } ) \rangle } . +$$ + +We now turn towards the optimization problem of interest, which we can rewrite as: + +$$ +\underset { \pi } { \arg \operatorname* { m a x } } \langle \pi , Q _ { \pi _ { k } } + c \left( \log \pi _ { k } - \log \pi _ { \mathrm { a d v } } \right) \rangle + \alpha \mathcal { H } ( \pi ) . +$$ + +By the simple change of variable $\tilde { Q } _ { \pi _ { k } } = Q _ { \pi _ { k } } + c ( \log \pi _ { k } - \log \pi _ { \mathrm { a d v } } )$ , we can reuse the previous solution (replacing $Q _ { \pi _ { k } }$ by $\tilde { Q } _ { \pi _ { k } }$ ). With the simplification: + +$$ +\exp \frac { Q _ { \pi _ { k } } + c \left( \log \pi _ { k } - \log \pi _ { \mathrm { a d v } } \right) } { \alpha } = \left( \frac { \pi _ { k } } { \pi _ { \mathrm { a d v } } } \right) ^ { \frac { c } { \alpha } } \exp \frac { Q _ { \pi _ { k } } } { \alpha } , +$$ + +we obtain the result and conclude the proof. \ No newline at end of file diff --git a/parse/train/_mQp5cr_iNy/_mQp5cr_iNy_content_list.json b/parse/train/_mQp5cr_iNy/_mQp5cr_iNy_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..041617dfde76af1c8ac21f89c9db71cb21878476 --- /dev/null +++ b/parse/train/_mQp5cr_iNy/_mQp5cr_iNy_content_list.json @@ -0,0 +1,2169 @@ +[ + { + "type": "text", + "text": "ADVERSARIALLY GUIDED ACTOR-CRITIC ", + "text_level": 1, + "bbox": [ + 176, + 98, + 676, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Johan Ferret∗ ", + "text_level": 1, + "bbox": [ + 619, + 146, + 717, + 159 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yannis Flet-Berliac∗ \nInria, Scool team \nUniv. Lille, CRIStAL, CNRS \nyannis.flet-berliac@inria.fr ", + "bbox": [ + 184, + 145, + 459, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Google Research, Brain team Inria, Scool team Univ. Lille, CRIStAL, CNRS ", + "bbox": [ + 619, + 160, + 813, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Olivier Pietquin ", + "text_level": 1, + "bbox": [ + 184, + 222, + 297, + 236 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Google Research, Brain team ", + "bbox": [ + 184, + 237, + 377, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Philippe Preux \nInria, Scool team \nUniv. Lille, CRIStAL, CNRS ", + "bbox": [ + 398, + 222, + 593, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Matthieu Geist Google Research, Brain team ", + "bbox": [ + 614, + 222, + 810, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 301, + 544, + 315 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite definite success in deep reinforcement learning problems, actor-critic algorithms are still confronted with sample inefficiency in complex environments, particularly in tasks where efficient exploration is a bottleneck. These methods consider a policy (the actor) and a value function (the critic) whose respective losses are built using different motivations and approaches. This paper introduces a third protagonist: the adversary. While the adversary mimics the actor by minimizing the KL-divergence between their respective action distributions, the actor, in addition to learning to solve the task, tries to differentiate itself from the adversary predictions. This novel objective stimulates the actor to follow strategies that could not have been correctly predicted from previous trajectories, making its behavior innovative in tasks where the reward is extremely rare. Our experimental analysis shows that the resulting Adversarially Guided Actor-Critic (AGAC) algorithm leads to more exhaustive exploration. Notably, AGAC outperforms current state-of-the-art methods on a set of various hard-exploration and procedurally-generated tasks. ", + "bbox": [ + 233, + 330, + 766, + 523 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 546, + 336, + 563 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Research in deep reinforcement learning (RL) has proven to be successful across a wide range of problems (Silver et al., 2014; Schulman et al., 2016; Lillicrap et al., 2016; Mnih et al., 2016). Nevertheless, generalization and exploration in RL still represent key challenges that leave most current methods ineffective. First, a battery of recent studies (Farebrother et al., 2018; Zhang et al., 2018a; Song et al., 2020; Cobbe et al., 2020) indicates that current RL methods fail to generalize correctly even when agents have been trained in a diverse set of environments. Second, exploration has been extensively studied in RL; however, most hard-exploration problems use the same environment for training and evaluation. Hence, since a well-designed exploration strategy should maximize the information received from a trajectory about an environment, the exploration capabilities may not be appropriately assessed if that information is memorized. In this line of research, we choose to study the exploration capabilities of our method and its ability to generalize to new scenarios. Our evaluation domains will, therefore, be tasks with sparse reward in procedurally-generated environments. ", + "bbox": [ + 174, + 577, + 825, + 743 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we propose Adversarially Guided Actor-Critic (AGAC), which reconsiders the actor-critic framework by introducing a third protagonist: the adversary. Its role is to predict the actor’s actions correctly. Meanwhile, the actor must not only find the optimal actions to maximize the sum of expected returns, but also counteract the predictions of the adversary. This formulation is lightly inspired by adversarial methods, specifically generative adversarial networks (GANs) (Goodfellow et al., 2014). Such a link between GANs and actor-critic methods has been formalized by Pfau & Vinyals (2016); however, in the context of a third protagonist, we draw a different analogy. The adversary can be interpreted as playing the role of a discriminator that must predict the actions of the actor, and the actor can be considered as playing the role of a generator that behaves to deceive the predictions of the adversary. This approach has the advantage, as with GANs, that the optimization procedure generates a diversity of meaningful data, corresponding to sequences of actions in AGAC. ", + "bbox": [ + 174, + 751, + 825, + 904 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper analyses and explores how AGAC explicitly drives diversity in the behaviors of the agent while remaining reward-focused, and to which extent this approach allows to adapt to the evolving state space of procedurally-generated environments where the map is constructed differently with each new episode. Moreover, because stability is a legitimate concern since specific instances of adversarial networks were shown to be prone to hyperparameter sensitivity issues (Arjovsky & Bottou, 2017), we also examine this aspect in our experiments. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions of this work are as follow: (i) we propose a novel actor-critic formulation inspired from adversarial learning (AGAC), (ii) we analyse empirically AGAC on key reinforcement learning aspects such as diversity, exploration and stability, (iii) we demonstrate significant gains in performance on several sparse-reward hard-exploration tasks including procedurally-generated tasks. ", + "bbox": [ + 174, + 194, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 272, + 344, + 289 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Actor-critic methods (Barto et al., 1983; Sutton, 1984) have been extended to the deep learning setting by Mnih et al. (2016), who combined deep neural networks and multiple distributed actors with an actor-critic setting, with strong results on Atari. Since then, many additions have been proposed, be it architectural improvements (Vinyals et al., 2019), better advantage or value estimation (Schulman et al., 2016; Flet-Berliac et al., 2021), or the incorporation of off-policy elements (Wang et al., 2017; Oh et al., 2018; Flet-Berliac & Preux, 2020). Regularization was shown to improve actor-critic methods, either by enforcing trust regions (Schulman et al., 2015; 2017; Wu et al., 2017), or by correcting for off-policiness (Munos et al., 2016; Gruslys et al., 2018); and recent works analyzed its impact from a theoretical standpoint (Geist et al., 2019; Ahmed et al., 2019; Vieillard et al., 2020a;b). Related to our work, Han & Sung (2020) use the entropy of the mixture between the policy induced from a replay buffer and the current policy as a regularizer. To the best of our knowledge, none of these methods explored the use of an adversarial objective to drive exploration. ", + "bbox": [ + 174, + 305, + 825, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While introduced in supervised learning, adversarial learning (Goodfellow et al., 2015; Miyato et al., 2016; Kurakin et al., 2017) was leveraged in several RL works. Ho & Ermon (2016) propose an imitation learning method that uses a discriminator whose task is to distinguish between expert trajectories and those of the agent while the agent tries to match expert behavior to fool the discriminator. Bahdanau et al. (2019) use a discriminator to distinguish goal states from non-goal states based on a textual instruction, and use the resulting model as a reward function. Florensa et al. (2018) use a GAN to produce sub-goals at the right level of difficulty for the current agent, inducing a form of curriculum. Additionally, Pfau & Vinyals (2016) provide a parallel between GANs and the actor-critic framework. ", + "bbox": [ + 174, + 478, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While exploration is driven in part by the core RL algorithms (Fortunato et al., 2018; Han & Sung, 2020; Ferret et al., 2021), it is often necessary to resort to exploration-specific techniques. For instance, intrinsic motivation encourages exploratory behavior from the agent. Some works use state-visitation counts or pseudo-counts to promote exhaustive exploration (Bellemare et al., 2016a), while others use curiosity rewards, expressed in the magnitude of prediction error from the agent, to push it towards unfamiliar areas of the state space (Burda et al., 2018). Ecoffet et al. (2019) propose a technique akin to tree traversal to explore while learning to come back to promising areas. Eysenbach et al. (2018) show that encouraging diversity helps with exploration, even in the absence of reward. ", + "bbox": [ + 174, + 611, + 825, + 722 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Last but not least, generalization is a key challenge in RL. Zhang et al. (2018b) showed that, even when the environment is not deterministic, agents can overfit to their training distribution and that it is difficult to distinguish agents likely to generalize to new environments from those that will not. In the same vein, recent work has advocated using procedurally-generated environments, in which a new instance of the environment is sampled when a new episode starts, to assess generalization capabilities better (Justesen et al., 2018; Cobbe et al., 2020). Finally, methods based on network randomization (Igl et al., 2019), noise injection (Lee et al., 2020), and credit assignment (Ferret et al., 2020) have been proposed to reduce the generalization gap for RL agents. ", + "bbox": [ + 174, + 728, + 825, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 BACKGROUND AND NOTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 863, + 468, + 878 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We place ourselves in the Markov Decision Processes (Puterman, 1994) framework. A Markov Decision Process (MDP) is a tuple $M = \\{ { \\mathcal { S } } , { \\mathcal { A } } , { \\mathcal { P } } , R , \\gamma \\}$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $\\mathcal { P }$ is the transition kernel, $\\mathcal { R }$ is the bounded reward function and $\\gamma \\in [ 0 , 1 )$ is the discount factor. Let $\\pi$ denote a stochastic policy mapping states to distributions over actions. We place ourselves in the infinite-horizon setting, i.e., we seek a policy that optimizes $\\begin{array} { r } { J ( \\pi ) = \\mathbb { E } _ { \\pi } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r \\left( s _ { t } , a _ { t } \\right) ] } \\end{array}$ . The value of a state is the quantity $\\begin{array} { r } { V ^ { \\pi } ( s ) = \\mathbb { E } _ { \\pi } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r \\left( s _ { t } , a _ { t } \\right) | s _ { 0 } = s ] } \\end{array}$ t=0 and the value of a state-action pair of performing action $a$ in state $s$ and then following policy is defined as: $\\begin{array} { r } { Q ^ { \\pi } ( s , a ) = \\mathbb { E } _ { \\pi } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r \\left( s _ { t } , a _ { t } \\right) \\vert s _ { 0 } = s , a _ { 0 } = a \\right] . } \\end{array}$ . The advantage function, which quantifies how an action $a$ is better than the average action in state $s$ , is $A ^ { \\pi } ( s , a ) \\bar { = } Q ^ { \\pi } ( s , a ) - V ^ { \\pi } ( \\bar { s } )$ . Finally, the entropy $\\mathcal { H } ^ { \\pi }$ of a policy is calculated as: $\\mathcal { H } ^ { \\pi } ( s ) = \\mathbb { E } _ { \\pi ( \\cdot | s ) } \\left[ - \\log \\pi ( \\cdot | s ) \\right]$ . ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 826, + 218 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Actor-Critic and Deep Policy Gradients. An actor-critic algorithm is composed of two main components: a policy and a value predictor. In deep RL, both the policy and the value function are obtained via parametric estimators; we denote $\\theta$ and $\\phi$ their respective parameters. The policy is updated via policy gradient, while the value is usually updated via temporal difference or Monte Carlo rollouts. In practice, for a sequence of transitions $\\left\\{ s _ { t } , a _ { t } , r _ { t } , s _ { t + 1 } \\right\\} _ { t \\in [ 0 , N ] }$ , we use the following policy gradient loss (including the commonly used entropic penalty): ", + "bbox": [ + 173, + 229, + 825, + 315 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d72fd2d9ccdcbbd3bfe5d82996770513e2a3d0869438dd67a00990f233121939.jpg", + "text": "$$\n\\mathcal { L } _ { P G } = - \\frac { 1 } { N } \\sum _ { t ^ { \\prime } = t } ^ { t + N } ( A _ { t ^ { \\prime } } \\log \\pi \\left( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta \\right) + \\alpha \\mathcal { H } ^ { \\pi } ( s _ { t ^ { \\prime } } , \\theta ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 315, + 318, + 679, + 362 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where 2016) $\\alpha$ is tfin t: $A _ { t }$ timato, with (Schulman et al.,a fixed hyperpa$\\begin{array} { r } { A _ { t } = \\sum _ { t ^ { \\prime } = t } ^ { t + N } ( \\gamma \\lambda ) ^ { t ^ { \\prime } - t } ( r _ { t ^ { \\prime } } + \\gamma V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } + 1 } ) - V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } } ) ) } \\end{array}$ $\\lambda$ $V _ { \\phi _ { \\mathrm { o l d } } }$ \nvalue function, we solve the non-linear regression problem minimizeφ $\\begin{array} { r } { \\sum _ { t ^ { \\prime } = t } ^ { t + N } ( V _ { \\phi } ( s _ { t ^ { \\prime } } ) - \\hat { V } _ { t ^ { \\prime } } ) ^ { 2 } } \\end{array}$ where \n$\\hat { V } _ { t } = A _ { t } + V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } } )$ . ", + "bbox": [ + 173, + 363, + 826, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 ADVERSARIALLY GUIDED ACTOR-CRITIC ", + "text_level": 1, + "bbox": [ + 173, + 462, + 557, + 479 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To foster diversified behavior in its trajectories, AGAC introduces a third protagonist to the actor-critic framework: the adversary. The role of the adversary is to accurately predict the actor’s actions, by minimizing the discrepancy between its action distribution $\\pi _ { \\mathrm { a d v } }$ and the distribution induced by the policy $\\pi$ . Meanwhile, in addition to finding the optimal actions to maximize the sum of expected returns, the actor must also counteract the adversary’s predictions by maximizing the discrepancy between $\\pi$ and $\\pi _ { \\mathrm { a d v } }$ (see Appendix B for an illustration). This discrepancy, used as a form of exploration bonus, is defined as the difference of action log-probabilities (see Eq. (1)), whose expectation is the Kullback–Leibler divergence: ", + "bbox": [ + 173, + 492, + 826, + 604 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/313661e94c3f88cf48c3210e2d6b7d59e59485b696566ea7ed195b46b5b4b5e7.jpg", + "text": "$$\nD _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) = \\mathbb { E } _ { \\pi ( \\cdot | s ) } \\left[ \\log \\pi ( \\cdot | s ) - \\log \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 299, + 608, + 699, + 626 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Formally, for each state-action pair $( s _ { t } , a _ { t } )$ in a trajectory, an action-dependent bonus $\\log \\pi ( a _ { t } | s _ { t } ) -$ $\\log \\pi _ { \\mathrm { a d v } } \\mathbf { \\dot { ( } } a _ { t } | s _ { t } )$ is added to the advantage. In addition, the value target of the critic is modified to include the action-independent equivalent, which is the KL-divergence $D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s _ { t } ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s _ { t } ) )$ . We discuss the role of these mirrored terms below, and the implications of AGAC’s modified objective from a more theoretical standpoint in the next section. In addition to the parameters $\\theta$ (resp. $\\theta _ { \\mathrm { o l d } }$ the parameter of the policy at the previous iteration) and $\\phi$ defined above (resp. $\\phi _ { \\mathrm { o l d } }$ that of the critic), we denote $\\psi$ (resp. $\\psi _ { \\mathrm { o l d . } }$ ) that of the adversary. ", + "bbox": [ + 173, + 636, + 825, + 736 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "AGAC minimizes the following loss: ", + "bbox": [ + 174, + 742, + 406, + 756 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d64a10d7d6388599c70759bcc998dbb883f1cd7c8f82a1c5d7eff8959002b7a9.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { A G A C } } = \\mathcal { L } _ { \\mathrm { P G } } + \\beta _ { V } \\mathcal { L } _ { V } + \\beta _ { \\mathrm { a d v } } \\mathcal { L } _ { \\mathrm { a d v } } .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 760, + 612, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the new objective $\\begin{array} { r } { \\mathcal { L } _ { P G } = - \\frac { 1 } { N } \\sum _ { t = 0 } ^ { N } ( A _ { t } ^ { \\tt A G A C } \\log \\pi \\left( a _ { t } | s _ { t } , \\theta \\right) + \\alpha \\mathcal { H } ^ { \\pi } ( s _ { t } , \\theta ) ) , } \\end{array}$ AGAC modifies $A _ { t }$ as: ", + "bbox": [ + 174, + 786, + 825, + 806 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0bdf94573c674dfa3f9a56a558b636c141ab3375b5ef5c6dc358b9bb36bfaeca.jpg", + "text": "$$\nA _ { t } ^ { \\mathtt { A G A C } } = A _ { t } + c \\left( \\log \\pi ( a _ { t } | s _ { t } , \\theta _ { \\mathrm { o l d } } ) - \\log \\pi _ { \\mathrm { a d v } } ( a _ { t } | s _ { t } , \\psi _ { \\mathrm { o l d } } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 810, + 691, + 835 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "with $c$ a varying hyperparameter that controls the dependence on the action log-probability difference. To encourage exploration without preventing asymptotic stability, $c$ is linearly annealed during the course of training. ${ \\mathcal { L } } _ { V }$ is the objective function of the critic defined as: ", + "bbox": [ + 174, + 838, + 826, + 881 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/56a67326212937240f6febf93b590f11f5a0b77762e4ed719e94f3e048bd068b.jpg", + "text": "$$\n\\mathcal { L } _ { V } = \\frac { 1 } { N } \\sum _ { t = 0 } ^ { N } \\left( V _ { \\phi } ( s _ { t } ) - \\left( \\hat { V } _ { t } + c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s _ { t } , \\theta _ { \\mathrm { o l d } } ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s _ { t } , \\psi _ { \\mathrm { o l d } } ) ) \\right) \\right) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 885, + 741, + 929 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Finally, ${ \\mathcal { L } } _ { \\mathrm { a d v } }$ is the objective function of the adversary: ", + "bbox": [ + 173, + 103, + 532, + 118 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9342fc161baad4a1f78266251ba83b09583ca4b3ea287f19105d8a4faedf6f1e.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { a d v } } = \\frac { 1 } { N } \\sum _ { t = 0 } ^ { N } D _ { \\mathrm { K L } } \\big ( \\pi \\big ( \\cdot | s _ { t } , \\theta _ { \\mathrm { o l d } } \\big ) \\big | \\big | \\pi _ { \\mathrm { a d v } } \\big ( \\cdot | s _ { t } , \\psi \\big ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 122, + 656, + 165 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Eqs. (1), (2) and (3) are the three equations that our method modifies (we color in blue the specific parts) in the traditional actor-critic framework. The terms $\\beta _ { V }$ and $\\beta _ { \\mathrm { a d v } }$ are fixed hyperparameters. ", + "bbox": [ + 174, + 169, + 823, + 198 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Under the proposed actor-critic formulation, the probability of sampling an action is increased if the modified advantage is positive, i.e. (i) the corresponding return is larger than the predicted value and/or (ii) the action log-probability difference is large. More precisely, our method favors transitions whose actions were less accurately predicted than the average action, i.e. $\\log \\pi ( a | s ) -$ $\\begin{array} { r } { \\log \\pi _ { \\mathrm { a d v } } ( a | s ) \\geq D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } \\end{array}$ . This is particularly visible for $\\lambda 1$ , in which case the generalized advantage is $A _ { t } = G _ { t } - V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t } )$ , resulting in the appearance of both aforementioned mirrored terms in the modified advantage: ", + "bbox": [ + 173, + 203, + 826, + 303 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/640255cc1738d9bd160976e173be1c2fe4381afdd9afec2513ecf5eff1dd157f.jpg", + "text": "$$\nA _ { t } ^ { \\mathtt { A G A C } } = G _ { t } - \\hat { V } _ { t } ^ { \\phi _ { \\mathrm { o d d } } } + c \\left( \\log \\pi ( a _ { t } | s _ { t } ) - \\log \\pi _ { \\mathrm { a d v } } ( a _ { t } | s _ { t } ) - \\hat { D } _ { \\mathrm { K L } } ^ { \\phi _ { \\mathrm { o d d } } } ( \\pi ( \\cdot | s _ { t } ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s _ { t } ) ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 214, + 306, + 781, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "with KL- $G _ { t }$ the observed return, rgence (estimated co $\\hat { V } _ { t } ^ { \\phi _ { \\mathrm { o l d } } }$ the estents of eturn and from Eq. $\\hat { D } _ { \\mathrm { K L } } ^ { \\phi _ { \\mathrm { o l d } } } \\big ( \\pi \\big ( \\cdot | s _ { t } \\big ) \\big | \\big | \\pi _ { \\mathrm { a d v } } \\big ( \\cdot | s _ { t } \\big ) \\big )$ the estimated $V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t } )$ ", + "bbox": [ + 171, + 338, + 823, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To avoid instability, in practice the adversary is a separate estimator, updated with a smaller learning rate than the actor. This way, it represents a delayed and more steady version of the actor’s policy, which prevents the agent from having to constantly adapt or focus solely on fooling the adversary. ", + "bbox": [ + 173, + 375, + 826, + 417 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 BUILDING MOTIVATION ", + "text_level": 1, + "bbox": [ + 174, + 434, + 382, + 448 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the following, we provide an interpretation of AGAC by studying the dynamics of attraction and repulsion between the actor and the adversary. To simplify, we study the equivalent of AGAC in a policy iteration (PI) scheme. PI being the dynamic programming scheme underlying the standard actor-critic, we have reasons to think that some of our findings translate to the original AGAC algorithm. In PI, the quantity of interest is the action-value, which AGAC would modify as: ", + "bbox": [ + 173, + 459, + 826, + 530 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7d6b08d78f936da16bf5b153c99ede87e47398c7e59016e9231adc066ebb1aa4.jpg", + "text": "$$\nQ _ { \\pi _ { k } } ^ { \\mathtt { A G A C } } = Q _ { \\pi _ { k } } + c ( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 532, + 622, + 553 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "with $\\pi _ { k }$ the policy at iteration $k$ . Incorporating the entropic penalty, the new policy $\\pi _ { k + 1 }$ verifies: ", + "bbox": [ + 176, + 555, + 808, + 570 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1f8c6698cba2a1b0d9b9ea9b2b1f56e50845c825d13115375ba0a7eeba66a55f.jpg", + "text": "$$\n\\pi _ { k + 1 } = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot \\vert s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G A C } } } ( s , a ) - \\alpha \\log \\pi ( a \\vert s ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 574, + 751, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can rewrite this objective: ", + "bbox": [ + 173, + 603, + 369, + 617 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c8e3a798f08d00f76eae423e4f37f046cdf18b7ecf46223fc27cf82bcda6e619.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { T } _ { \\mathrm { { p I } } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G } A C } } ( s , a ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi ( a | s ) + \\log \\pi ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\mathrm { { g } } } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\Big [ \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) ] \\underbrace { - c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { k } ( \\cdot | s ) ) } _ { \\pi _ { k } \\mathrm { ~ i s ~ a t u r a c i t i v e } } \\underbrace { + c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\pi _ { \\mathrm { a d v } } \\mathrm { ~ i s ~ r e p u l s i v e } } \\underbrace { + \\alpha \\mathcal { H } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\mathrm { c n ~ t e r ~ i s r e p u l s i n c } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 621, + 839, + 717 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Thus, in the PI scheme, AGAC finds a policy that maximizes $Q$ -values, while at the same time remaining close to the current policy and far from a mixture of the previous policies (i.e., $\\pi _ { k - 1 }$ , $\\pi _ { k - 2 }$ , $\\pi _ { k - 3 } , \\ldots )$ . Note that we experimentally observe (see Section 5.3) that our method performs better with a smaller learning rate for the adversarial network than that of the other networks, which could imply that a stable repulsive term is beneficial. ", + "bbox": [ + 173, + 723, + 826, + 794 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This optimization problem is strongly concave in $\\pi$ (thanks to the entropy term), and is state-wise a Legendre-Fenchel transform. Its solution is given by (see Appendix E for the full derivation): ", + "bbox": [ + 173, + 800, + 825, + 829 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/159b54c1a7faae5b408523e6b6ce2055023d3883ef1dede8bef99403f72c8612.jpg", + "text": "$$\n\\pi _ { k + 1 } \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 833, + 596, + 871 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This result gives us some insight into the behavior of the objective function. Notably, in our example, if $\\pi _ { \\mathrm { a d v } }$ is fixed and $c = \\alpha$ , we recover a KL-regularized PI scheme (Geist et al., 2019) with the modified reward $r - c \\log \\pi _ { \\mathrm { a d v } }$ . ", + "bbox": [ + 173, + 881, + 825, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 103, + 341, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In all of the experiments, we use PPO (Schulman et al., 2017) as the base algorithm and build on it to incorporate our method. Hence, ", + "bbox": [ + 173, + 128, + 825, + 159 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/512051f4cddcfb03938253cc36c2f458c14301b0bf117b1b82988a099e86e84d.jpg", + "text": "$$\n\\mathcal { L } _ { P G } = - \\frac { 1 } { N } \\sum _ { t ^ { \\prime } = t } ^ { t + N } \\operatorname* { m i n } \\left( \\frac { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta ) } { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta _ { \\mathrm { o l d } } ) } A _ { t ^ { \\prime } } ^ { \\mathrm { R G K } } , \\mathrm { c l i p } \\left( \\frac { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta ) } { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta _ { \\mathrm { o l d } } ) } , 1 - \\epsilon , 1 + \\epsilon \\right) A _ { t ^ { \\prime } } ^ { \\mathrm { R G K } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 204, + 165, + 789, + 208 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "with $A _ { t ^ { \\prime } } ^ { \\tt A G A C }$ given in Eq. (1), $N$ the temporal length considered for one update of parameters and $\\epsilon$ the clipping parameter. Similar to RIDE (Raileanu & Rocktäschel, 2019), we also discount PPO by episodic state visitation counts, except for VizDoom (cf. Section 5.1). The actor, critic and adversary use the convolutional architecture of the Nature paper of DQN (Mnih et al., 2015) with different hidden sizes (see Appendix D for architecture details). The three neural networks are optimized using Adam (Kingma & Ba, 2015). Our method does not use RNNs in its architecture; instead, in all our experiments, we use frame stacking. Indeed, Hausknecht & Stone (2015) interestingly demonstrate that although recurrence is a reliable method for processing state observation, it does not confer any systematic advantage over stacking observations in the input layer of a CNN. Note that the parameters are not shared between the policy, the critic and the adversary and that we did not observe any noticeable difference in computational complexity when using AGAC compared to PPO. We direct the reader to Appendix C for a list of hyperparameters. In particular, the $c$ coefficient of the adversarial bonus is linearly annealed. ", + "bbox": [ + 173, + 215, + 826, + 397 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "At each training step, we perform a stochastic optimization step to minimize $\\mathcal { L } _ { \\mathtt { A G A C } }$ using stop-gradient: ", + "bbox": [ + 176, + 404, + 823, + 419 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d09977d95a2403edd3cc15d250c112feabe4fe824a2f41929c104a3d1312eb65.jpg", + "text": "$$\n\\begin{array} { r l r l } & { \\boldsymbol { \\vartheta } \\mathrm { A d a m } ( \\theta , \\nabla _ { \\theta } \\mathcal { L } _ { P G } , \\eta _ { 1 } ) , } & & { \\boldsymbol { \\phi } \\mathrm { A d a m } ( \\phi , \\nabla _ { \\phi } \\mathcal { L } _ { V } , \\eta _ { 1 } ) , } & & { \\boldsymbol { \\psi } \\mathrm { A d a m } ( \\psi , \\nabla _ { \\psi } \\mathcal { L } _ { \\sf a d v } , \\eta _ { 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 425, + 820, + 443 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 462, + 326, + 478 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we describe our experimental study in which we investigate: (i) whether the adversarial bonus alone (e.g. without episodic state visitation count) is sufficient to outperform other methods in VizDoom, a sparse-reward task with high-dimensional observations, (ii) whether AGAC succeeds in partially-observable and procedurally-generated environments with high sparsity in the rewards, compared to other methods, (iii) how well AGAC is capable of exploring in environments without extrinsic reward, (iv) the training stability of our method. In all of the experiments, lines are average performances and shaded areas represent one standard deviation. The code for our method is released at github.com/yfletberliac/adversarially-guided-actor-critic. ", + "bbox": [ + 173, + 492, + 826, + 604 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/263aa3eeaa25d94147cc96fb59396bbc06f2d865b71d0e66161e08136afe2782.jpg", + "image_caption": [ + "Figure 1: (a,b) Frames from the 3-D navigation task VizdoomMyWayHome. (c) MiniGridKeyCorridorS6R3. (d) MiniGrid-ObstructedMazeFull. " + ], + "image_footnote": [], + "bbox": [ + 199, + 630, + 799, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Environments. To carefully evaluate the performance of our method, its ability to develop robust exploration strategies and its generalization to unseen states, we choose tasks that have been used in prior work, which are tasks with high-dimensional observations, sparse reward and procedurallygenerated environments. In VizDoom (Kempka et al., 2016), the agent must learn to move along corridors and through rooms without any reward feedback from the 3-D environment. The MiniGrid environments (Chevalier-Boisvert et al., 2018) are a set of challenging partially-observable and sparse-reward gridworlds. In this type of procedurally-generated environments, memorization is impossible due to the huge size of the state space, so the agent must learn to generalize across the different layouts of the environment. Each gridworld has different characteristics: in the MultiRoom tasks, the agent is placed in the first room and should reach a goal placed in the most distant room. In the KeyCorridor tasks, the agent must navigate to pick up an object placed in a room locked by a door whose key is in another room. Finally, in the ObstructedMaze tasks, the agent must pick up a box that is placed in a corner of a 3x3 maze in which the doors are also locked, the keys are hidden in boxes and balls obstruct the doors. All considered environments (see Fig. 1 for some examples) are available as part of OpenAI Gym (Brockman et al., 2016). ", + "bbox": [ + 173, + 797, + 826, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. For a fair assessment of our method, we compare to some of the most prominent methods specialized in hard-exploration tasks: RIDE (Raileanu & Rocktäschel, 2019), based on an intrinsic reward associated with the magnitude of change between two consecutive state representations and state visitation, Count as Count-Based Exploration (Bellemare et al., 2016b), which we couple with IMPALA (Espeholt et al., 2018), RND (Burda et al., 2018) in which an exploration bonus is positively correlated to the error of predicting features from the observations and ICM (Pathak et al., 2017), where a module only predicts the changes in the environment that are produced by the actions of the agent. Finally, we compare to most the recent and best performing method at the time of writing in procedurally-generated environments: AMIGo (Campero et al., 2021) in which a goal-generating teacher provides count-based intrinsic goals. ", + "bbox": [ + 173, + 203, + 825, + 340 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 ADVERSARIALLY-BASED EXPLORATION (NO EPISODIC COUNT) ", + "bbox": [ + 176, + 358, + 655, + 372 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/bca7e5d026f86a26f3f3d9f80dde3d7915315fca1cfcd36176ea13f171e0abe1.jpg", + "table_caption": [ + "Table 1: Average return in VizDoom at different timesteps. " + ], + "table_footnote": [], + "table_body": "
Nb.of Timesteps2M4M6M8M10M
AGAC0.74± 0.050.96 ± 0.0010.96 ± 0.0010.97 ± 0.0010.97 ± 0.001
RIDE0.0.0.95 ± 0.0010.97 ± 0.0010.97 ± 0.001
ICM0.0.0.95 ±0.0010.97 ± 0.0010.97 ± 0.001
AMIGo0.0.0.0.0.
RND0.0.0.0.0.
Count0.0.0.0.0.
", + "bbox": [ + 183, + 415, + 812, + 511 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we assess the benefits of using an adversarially-based exploration bonus and examine how AGAC performs without the help of count-based exploration. In order to provide a comparison to state-of-the-art methods, we choose VizDoom, a hard-exploration problem used in prior work. In this game, the map consists of 9 rooms connected by corridors where 270 steps separate the initial position of the agent and the goal under an optimal policy. Episodes are terminated either when the agent finds the goal or if the episode exceeds 2100 timesteps. Importantly, while other algorithms (Raileanu & Rocktäschel, 2019; Campero et al., 2021) benefit from count-based exploration, this study has been conducted with our method not benefiting from episodic count whatsoever. Results in Table 1 indicate that AGAC clearly outperforms other methods in sample-efficiency. Only the methods ICM and RIDE succeed in matching the score of AGAC, and with about twice as much transitions $( \\sim 3 \\mathbf { M }$ vs. 6M). Interestingly, AMIGo performs similarly to Count and RND. We find this result surprising because AMIGo has proven to perform well in the MiniGrid environments. Nevertheless, it appears that concurrent works to ours experienced similar issues with the accompanying implementation1. The results of AGAC support the capabilities of the adversarial bonus and show that it can, on its own, achieve significant gains in performance. However, the VizDoom task is not procedurally-generated; hence we have not evaluated the generalization to new states yet. In the following section, we use MiniGrid to investigate this. ", + "bbox": [ + 173, + 523, + 825, + 760 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 HARD-EXPLORATION TASKS WITH PARTIALLY-OBSERVABLE ENVIRONMENTS ", + "text_level": 1, + "bbox": [ + 173, + 776, + 751, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now evaluate our method on multiple hard-exploration procedurally-generated tasks from MiniGrid. Details about MiniGrid can be found in Appendix C.1. Fig. 2 indicates that AGAC significantly outperforms other methods on these tasks in sample-efficiency and performance. AGAC also outperforms the current state-of-the-art method, AMIGo, despite the fact that it uses the fully-observable version of MiniGrid. Note that we find the same poor performance results when training AMIGo in MiniGrid, similar to Vizdoom results. For completeness, we also report in Table 2 of Appendix A.1 the performance results with the scores reported in the original papers Raileanu & Rocktäschel (2019) and Campero et al. (2021). We draw similar conclusions: AGAC clearly outperforms the state-of-the-art RIDE, AMIGo, Count, RND and ICM. ", + "bbox": [ + 174, + 803, + 825, + 900 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/c694a663b7b25ee719de7eb448621e3cb5826c21e451c62ee01f34f8d0c5792f.jpg", + "image_caption": [ + "Figure 2: Performance evaluation of AGAC. " + ], + "image_footnote": [], + "bbox": [ + 176, + 98, + 820, + 313 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 363, + 825, + 392 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In all the considered tasks, the agent must learn to generalize across a very large state space because the layouts are generated procedurally. We consider three main arguments to explain why our method is successful: (i) our method makes use of partial observations: in this context, the adversary has a harder time predicting the actor’s actions; nevertheless, the mistakes of the former benefit the latter in the form of an exploration bonus, which pushes the agent to explore further in order to deceive the adversary, (ii) the exploration bonus (i.e. intrinsic reward) does not dissipate compared to most other methods, as observed in Fig. 9 in Appendix A.4, (iii) our method does not make assumptions about the environment dynamics (e.g., changes in the environment produced by an action as in Raileanu & Rocktäschel (2019)) since this can hinder learning when the space of state changes induced by an action is too large (such as the action of moving a block in ObstructedMaze). ", + "bbox": [ + 174, + 398, + 825, + 537 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Appendix A.3, we also include experiments in two environments with extremely sparse reward signals: KeyCorridorS8R3 and ObstructedMazeFull. Despite the challenge, AGAC still manages to find rewards and can perform well by taking advantage of the diversified behaviour induced by our method. To the best of our knowledge, no other method ever succeeded to perform well $\\mathit { \\Theta } > 0$ average return) in those tasks. We think that given more computing time, AGAC’s score could go higher. ", + "bbox": [ + 174, + 545, + 825, + 614 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 TRAINING STABILITY ", + "text_level": 1, + "bbox": [ + 174, + 632, + 364, + 645 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/df1a9efb68034f4a5f52a20d9d5ef22a9514eee7144535790fe25105fc795095.jpg", + "image_caption": [ + "Figure 3: Sensitivity analysis of AGAC in KeyCorridorS4R3. " + ], + "image_footnote": [], + "bbox": [ + 238, + 660, + 758, + 780 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here we want to analyse the stability of the method when changing hyperparameters. The most important parameters in AGAC are $c$ , the coefficient for the adversarial bonus, and the learning rates ratio $\\begin{array} { r } { \\nu = \\frac { \\eta _ { 2 } } { \\eta _ { 1 } } } \\end{array}$ . We choose KeyCorridorS4R3 as the evaluation task because among all the tasks considered, its difficulty is at a medium level. Fig. 3 shows the learning curves. For readability, we plot the average return only; the standard deviation is sensibly the same for all curves. We observe that deviating from the hyperparameter values found using grid search results in a slower training. Moreover, although reasonable, $c$ appears to have more sensitivity than $\\nu$ . ", + "bbox": [ + 173, + 824, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/052df8089e69d4ed8859a825df3b5e90f5f628f6ab3239fa565170ac0833dc68.jpg", + "image_caption": [ + "Figure 5: State visitation heatmaps for RND, Count, a random uniform policy, RIDE, and AGAC trained in a singleton environment (top row) and procedurally-generated environments (bottom row) without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task. " + ], + "image_footnote": [], + "bbox": [ + 215, + 99, + 782, + 337 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.4 EXPLORATION IN REWARD-FREE ENVIRONMENT ", + "text_level": 1, + "bbox": [ + 174, + 430, + 552, + 444 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To better understand the effectiveness of our method and inspect how the agent collects rewards that would not otherwise be achievable by simple exploration heuristics or other methods, we analyze the performance of AGAC in another (procedurally-generated) challenging environment, MultiRoomN10S6, when there is no reward signal, i.e. no extrinsic reward. Beyond the good performance of our method when extrinsic rewards are given to the agent, Fig. 4 indicates that the exploration induced by our method makes the agent succeed in a significant proportion of the episodes: in the configuration “NoExtrinsicReward” the reward signal is not given (the goal is invisible to the agent) and the performance of AGAC stabilizes around an average return of $\\sim 0 . 1 5$ . Since the return of an episode is either 0 or 1 (depending on whether the agent reached the goal state or not), and because this value is aggregated across several episodes, the results indicate that reward-free AGAC succeeds in $\\sim 1 5 \\%$ of the tasks. Comparatively, random agents have a zero average return. This poor performance is in accordance with the results in Raileanu & Rocktäschel (2019) and reflects the complexity of the task: in order to go from one room to another, an agent must perform a specific action to open a door and cross it within the time limit of 200 timesteps. In the following, we visually investigate how different methods explore the environments. ", + "bbox": [ + 174, + 462, + 517, + 642 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/2b9e22a230d11ca43128f658217d5ed34a847b70794472a70049ba6a59245c82.jpg", + "image_caption": [ + "Figure 4: Average return on N10S6 with and without extrinsic reward. " + ], + "image_footnote": [], + "bbox": [ + 531, + 462, + 820, + 594 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 642, + 825, + 752 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.5 VISUALIZING COVERAGE AND DIVERSITY ", + "text_level": 1, + "bbox": [ + 174, + 781, + 508, + 796 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we first investigate how different methods explore environments without being guided by extrinsic rewards (the green goal is invisible to the agent) on both procedurally-generated and singleton environments. In singleton environments, an agent has to solve the same task in the same environment/maze in every episode. Fig. 5 shows the state visitation heatmaps (darker areas correspond to more visits) after a training of 10M timesteps. We observe that most of the methods explore inefficiently in a singleton environment and that only RIDE succeeds in reaching the fifth room while AGAC reaches the last (tenth) room. After training the agents in procedurally-generated environments, the methods explore even less efficiently while AGAC succeeds in exploring all rooms. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/79b4e4dfaa987603232a227680bcdf155166caf72f8802e24284152dfcbbd4e8.jpg", + "image_caption": [ + "Figure 6: State visitation heatmaps of the last ten episodes of an agent trained in procedurallygenerated environments without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task. The agent is continuously engaging in new strategies. " + ], + "image_footnote": [], + "bbox": [ + 225, + 113, + 751, + 292 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We now qualitatively study the diversity of an agent’s behavior when trained with AGAC. Fig. 6 presents the state visitation heatmaps of the last ten episodes for an agent trained in procedurallygenerated environments in the MultiRoomN10S6 task without extrinsic reward. The heatmaps correspond to the behavior of the resulting policy, which is still learning from the AGAC objective. Looking at the figure, we can see that the strategies vary at each update with, for example, backand-forth and back-to-start behaviors. Although there are no extrinsic reward, the strategies seem to diversify from one update to the next. Finally, Fig. 7 in Appendix A.2 shows the state visitation heatmaps in a different configuration: when the agent has been trained on a singleton environment in the MultiRoomN10S6 task without extrinsic reward. Same as previously, the agent is updated between each episode. Looking at the figure, we can make essentially the same observations as previously, with a noteworthy behavior in the fourth heatmap of the bottom row where it appears the agent went to the fourth room to remain inside it. Those episodes indicate that, although the agent sees the same environment repeatedly, the successive adversarial updates force it to continuously adapt its behavior and try new strategies. ", + "bbox": [ + 173, + 397, + 826, + 592 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 635, + 310, + 651 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This paper introduced AGAC, a modification to the traditional actor-critic framework: an adversary network is added as a third protagonist. The mechanics of AGAC have been discussed from a policy iteration point of view, and we provided theoretical insight into the inner workings of the proposed algorithm: the adversary forces the agent to remain close to the current policy while moving away from the previous ones. In a nutshell, the influence of the adversary makes the actor conservatively diversified. ", + "bbox": [ + 174, + 680, + 825, + 763 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the experimental study, we have evaluated the adversarially-based bonus in VizDoom and empirically demonstrated its effectiveness and superiority compared to other relevant methods (some benefiting from count-based exploration). Then, we have conducted several performance experiments using AGAC and have shown a significant performance improvement over some of the most popular exploration methods (RIDE, AMIGo, Count, RND and ICM) on a set of various challenging tasks from MiniGrid. 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", + "bbox": [ + 174, + 244, + 823, + 272 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A ADDITIONAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 178, + 102, + 444, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 MINIGRID PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 132, + 403, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we report the final performance of all methods considered in the MiniGrid experiments of Fig. 2 with the scores reported in Raileanu & Rocktäschel (2019) and Campero et al. (2021). All methods have a budget of 200M frames. ", + "bbox": [ + 174, + 159, + 826, + 202 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/303beb3dda10298731dd62b047e584dd4fffff328ba187daad4cd3ea0fd3f265.jpg", + "table_caption": [ + "Table 2: Final average performance of all methods on several MiniGrid environments. " + ], + "table_footnote": [], + "table_body": "
TaskKC-S4R3KC-S5R3MR-N10S10OM-2DlhbOM-1QOM-2Q
AGAC0.950.930.520.640.780.63
RIDE0.190.0.400.0.0.
AMIGo0.540.0.0.200.0.
RND0.0.0.0.030.0.
Count0.0.0.0.0.0.
ICM0.0.0.0.0.0.
", + "bbox": [ + 228, + 238, + 766, + 334 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 STATE VISITATION HEATMAPS IN SINGLETON ENVIRONMENT WITH NO EXTRINSIC REWARD ", + "bbox": [ + 176, + 357, + 794, + 385 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we provide additional state visitation heatmaps. The agent has been trained on a singleton environment from the MultiRoomN10S6 task without extrinsic reward. The last ten episodes of the training suggest that although the agent experiences the same maze over and over again, the updates force it to change behavior and try new strategies. ", + "bbox": [ + 174, + 396, + 826, + 454 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/bdafbf5ffde8d9779028392dcc80b27fab8981f669d6dfdeaf60ea0f5e559087.jpg", + "image_caption": [ + "Figure 7: State visitation heatmaps of the last ten episodes of an agent trained in a singleton environment with no extrinsic reward 10M timesteps in the MultiRoomN10S6 task. The agent is continuously engaging into new strategies. " + ], + "image_footnote": [], + "bbox": [ + 225, + 479, + 751, + 657 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 (EXTREMELY) HARD-EXPLORATION TASKS WITH PARTIALLY-OBSERVABLE ENVIRONMENTS ", + "text_level": 1, + "bbox": [ + 174, + 746, + 740, + 773 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we include additional experiments on one of the hardest tasks available in MiniGrid. The first is KeyCorridorS8R3, where the size of the rooms has been increased. In it, the agent has to pick up an object which is behind a locked door: the key is hidden in another room and the agent has to explore the environment to find it. The second, ObstructedMazeFull, is similar to ObstructedMaze4Q, where the agent has to pick up a box which is placed in one of the four corners of a 3x3 maze: the doors are locked, the keys are hidden in boxes and the doors are obstructed by balls. In those difficult tasks, only our method succeeds in exploring well enough to find rewards. ", + "bbox": [ + 173, + 785, + 826, + 883 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/2c2ebfe32501ea6424be77611fdb79a4abd159715b1b940e6c5d877763053c79.jpg", + "image_caption": [ + "Figure 8: Performance evaluation of AGAC compared to RIDE, AMIGo, Count, RND and ICM on extremely hard-exploration problems. " + ], + "image_footnote": [], + "bbox": [ + 205, + 99, + 789, + 236 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.4 MEAN INTRINSIC REWARD ", + "text_level": 1, + "bbox": [ + 176, + 306, + 405, + 320 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we report the mean intrinsic reward computed for an agent trained in MultiRoomN12S10 to conveniently compare our results with that of Raileanu & Rocktäschel (2019). We observe in Fig. 9 that the intrinsic reward is consistently larger for our method and that, contrary to other methods, does not converge to low values. Please note that, in all considered experiments, the adversarial bonus coefficient $c$ in Eq. 2 and 3 is linearly annealed throughout the training since it is mainly useful at the beginning of learning when the rewards have not yet been met. In the long run, this coefficient may prevent the agent from solving the task by forcing it to always favour exploration over exploitation. ", + "bbox": [ + 173, + 335, + 826, + 446 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/28473b81bbecae89bf11a7987beadae0deed242b6ba51bdf0240df3613591c4e.jpg", + "image_caption": [ + "Figure 9: Average intrinsic reward for different methods trained in MultiRoomN12S10. " + ], + "image_footnote": [], + "bbox": [ + 352, + 465, + 640, + 623 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/3ce38e04ad09fb53cf941a1929d15591fb412b83302fde934dd965c50f3e5f5f.jpg", + "image_caption": [ + "Figure 10: A simple schematic illustration of AGAC. Left: the adversary minimizes the KL-divergence with respect to the action probability distribution of the actor. Right: the actor receives a bonus when counteracting the predictions of the adversary. " + ], + "image_footnote": [], + "bbox": [ + 173, + 686, + 825, + 852 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C EXPERIMENTAL DETAILS AND HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 635, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.1 MINIGRID SETUP ", + "text_level": 1, + "bbox": [ + 176, + 133, + 339, + 148 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here, we describe in more details the experimental setup we used in our MiniGrid experiments. ", + "bbox": [ + 174, + 159, + 797, + 174 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "There are several different MiniGrid scenarios that we consider in this paper. MultiRoom corresponds to a set of navigation tasks, where the goal is to go from a starting state to a goal state. The notation MultiRoom-N2S4 means that there are 2 rooms in total, and that each room has a maximal side of 4. In order to go from one room to another, the agent must perform a specific action to open a door. Episodes are terminated with zero reward after a maximum of $2 0 \\times N$ steps with $N$ the number of rooms. In KeyCorridor, the agent also has to pick up a key, since the goal state is behind a door that only lets it in with the key. The notation KeyCorridor-S3R4 means that there are 4 side corridors, leading to rooms that have a maximal side of 3. The maximum number of steps is 270. In ObstructedMaze, keys are hidden in boxes, and doors are obstructed by balls the agent has to get out of its way. The notation ObstructedMaze-1Dl means that there are two connected rooms of maximal side 6 and 1 door (versus a 3x3 matrix and 2 doors if the leading characters are $2 D$ ), adding $h$ as a suffix places keys in boxes, and adding $b$ as a suffix adds balls in front of doors. Using $Q$ as a suffix is equivalent to using lhb (that is, both hiding keys and placing balls to be moved). The maximum number of steps is 576. ObstructedMazeFull is the hardest configuration for this scenario, since it has the maximal number of keys, balls to move, and doors possible. ", + "bbox": [ + 173, + 181, + 825, + 388 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In each scenario, the agent has access to a partial view of the environment, a $7 \\mathrm { x } 7 $ square that includes itself and points in the direction of its previous movement. ", + "bbox": [ + 173, + 396, + 821, + 424 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.2 HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 440, + 359, + 455 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In all experiments, we train six different instances of our algorithm with different random seeds. In Table 3, we report the list of hyperparameters. ", + "bbox": [ + 169, + 467, + 823, + 496 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/30708c1a7a5a9e626d9adae09cffff61d0aa525bb348f66624e07ff872fd4b7e.jpg", + "table_caption": [ + "Table 3: Hyperparameters used in AGAC. " + ], + "table_footnote": [], + "table_body": "
ParameterValue
Horizon T2048
Nb. epochs4
Nb.minibatches8
Nb.frames stacked4
NonlinearityELU (Clevert et al., 2016)
Discount γ0.99
GAE parameter 入0.95
PPO clipping parameter e0.2
βv0.5
C c anneal schedule4·10-4 (4·10-5 in VizDoom)
linear
βadv4·10-5
Adam stepsize m13·10-4
Adam stepsize m29.10-5 = 0.3· m1
", + "bbox": [ + 281, + 534, + 707, + 751 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 441, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In Fig. 11 is depicted the architecture of our method. ", + "bbox": [ + 173, + 135, + 519, + 150 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/166400757cb98035c5bef5beb612bef9f83aa90345f67690ff8af762d98632a2.jpg", + "image_caption": [ + "Figure 11: Artificial neural architecture of the critic, the actor and the adversary. " + ], + "image_footnote": [], + "bbox": [ + 207, + 166, + 787, + 542 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E PROOF OF SECTION 4.1 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 611, + 482, + 628 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, we provide a short proof for the result of the optimization problem in Section 4.1. We recall the result here: ", + "bbox": [ + 174, + 645, + 823, + 674 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2024813b42e36b4d88b1449320b8e80f8860fdc726554bfff7e58be829a6bb7a.jpg", + "text": "$$\n\\pi _ { k + 1 } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\pi } \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 679, + 656, + 718 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "with the objective function: ", + "bbox": [ + 173, + 724, + 354, + 739 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/af279849052942bf74464168820472c5207cabac51d9533eeb152ae0a1351302.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 223, + 748, + 774, + 767 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. We first consider a simpler optimization problem: a $\\because \\operatorname { g m a x } _ { \\pi } \\langle \\pi , Q _ { \\pi _ { k } } \\rangle + \\alpha \\mathcal { H } ( \\pi )$ , whose solution is known (Vieillard et al., 2020a, Appendix A). The expression for the maximizer is the $\\alpha$ -scaled softmax: ", + "bbox": [ + 174, + 789, + 823, + 830 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/af0be1f1ed8a2d87a91e4d6238ccdfe701dfda388c64ed4ae25abc89583ea5e2.jpg", + "text": "$$\n\\pi ^ { * } = \\frac { \\exp ( \\frac { Q _ { \\pi _ { k } } } { \\alpha } ) } { \\langle 1 , \\exp ( \\frac { Q _ { \\pi _ { k } } } { \\alpha } ) \\rangle } .\n$$", + "text_format": "latex", + "bbox": [ + 424, + 830, + 573, + 875 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We now turn towards the optimization problem of interest, which we can rewrite as: ", + "bbox": [ + 173, + 878, + 722, + 895 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b4274842b6d1fa563cf9fa451f48948e73a0d409b0a11240212631f8c058d056.jpg", + "text": "$$\n\\underset { \\pi } { \\arg \\operatorname* { m a x } } \\langle \\pi , Q _ { \\pi _ { k } } + c \\left( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } \\right) \\rangle + \\alpha \\mathcal { H } ( \\pi ) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 902, + 671, + 928 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By the simple change of variable $\\tilde { Q } _ { \\pi _ { k } } = Q _ { \\pi _ { k } } + c ( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } )$ , we can reuse the previous solution (replacing $Q _ { \\pi _ { k } }$ by $\\tilde { Q } _ { \\pi _ { k } }$ ). With the simplification: ", + "bbox": [ + 169, + 102, + 825, + 135 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2bc203fd3a59825715a246ddb6bbf5a7ef0cabd45b556e775d2e4892c0516cd1.jpg", + "text": "$$\n\\exp \\frac { Q _ { \\pi _ { k } } + c \\left( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } \\right) } { \\alpha } = \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 141, + 686, + 179 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "we obtain the result and conclude the proof. 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Our evaluation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 480, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 480, + 591 + ], + "score": 1.0, + "content": "domains will, therefore, be tasks with sparse reward in procedurally-generated environments.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "In this work, we propose Adversarially Guided Actor-Critic (AGAC), which reconsiders the actor-critic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "framework by introducing a third protagonist: the adversary. Its role is to predict the actor’s actions", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "correctly. Meanwhile, the actor must not only find the optimal actions to maximize the sum of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "expected returns, but also counteract the predictions of the adversary. This formulation is lightly", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "inspired by adversarial methods, specifically generative adversarial networks (GANs) (Goodfellow", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "et al., 2014). Such a link between GANs and actor-critic methods has been formalized by Pfau &", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Vinyals (2016); however, in the context of a third protagonist, we draw a different analogy. 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These methods", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 295, + 469, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 469, + 306 + ], + "score": 1.0, + "content": "consider a policy (the actor) and a value function (the critic) whose respective losses", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "score": 1.0, + "content": "are built using different motivations and approaches. This paper introduces a third", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 317, + 469, + 328 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 469, + 328 + ], + "score": 1.0, + "content": "protagonist: the adversary. While the adversary mimics the actor by minimizing the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 327, + 470, + 340 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 470, + 340 + ], + "score": 1.0, + "content": "KL-divergence between their respective action distributions, the actor, in addition to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 338, + 471, + 351 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 471, + 351 + ], + "score": 1.0, + "content": "learning to solve the task, tries to differentiate itself from the adversary predictions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 349, + 469, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 469, + 361 + ], + "score": 1.0, + "content": "This novel objective stimulates the actor to follow strategies that could not have", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 361, + 469, + 372 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 469, + 372 + ], + "score": 1.0, + "content": "been correctly predicted from previous trajectories, making its behavior innovative", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 371, + 469, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 371, + 469, + 383 + ], + "score": 1.0, + "content": "in tasks where the reward is extremely rare. Our experimental analysis shows that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 382, + 469, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 382, + 469, + 394 + ], + "score": 1.0, + "content": "the resulting Adversarially Guided Actor-Critic (AGAC) algorithm leads to more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 393, + 469, + 405 + ], + "spans": [ + { + "bbox": [ + 142, + 393, + 469, + 405 + ], + "score": 1.0, + "content": "exhaustive exploration. Notably, AGAC outperforms current state-of-the-art methods", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 405, + 423, + 416 + ], + "spans": [ + { + "bbox": [ + 142, + 405, + 423, + 416 + ], + "score": 1.0, + "content": "on a set of various hard-exploration and procedurally-generated tasks.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 23.5, + "bbox_fs": [ + 141, + 261, + 471, + 416 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 433, + 206, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "Research in deep reinforcement learning (RL) has proven to be successful across a wide range", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "of problems (Silver et al., 2014; Schulman et al., 2016; Lillicrap et al., 2016; Mnih et al., 2016).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "Nevertheless, generalization and exploration in RL still represent key challenges that leave most", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "score": 1.0, + "content": "current methods ineffective. First, a battery of recent studies (Farebrother et al., 2018; Zhang et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "2018a; Song et al., 2020; Cobbe et al., 2020) indicates that current RL methods fail to generalize", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "correctly even when agents have been trained in a diverse set of environments. Second, exploration has", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "been extensively studied in RL; however, most hard-exploration problems use the same environment", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "for training and evaluation. Hence, since a well-designed exploration strategy should maximize the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "information received from a trajectory about an environment, the exploration capabilities may not be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "appropriately assessed if that information is memorized. In this line of research, we choose to study", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "the exploration capabilities of our method and its ability to generalize to new scenarios. Our evaluation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 480, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 480, + 591 + ], + "score": 1.0, + "content": "domains will, therefore, be tasks with sparse reward in procedurally-generated environments.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 457, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "In this work, we propose Adversarially Guided Actor-Critic (AGAC), which reconsiders the actor-critic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "framework by introducing a third protagonist: the adversary. Its role is to predict the actor’s actions", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "correctly. Meanwhile, the actor must not only find the optimal actions to maximize the sum of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "expected returns, but also counteract the predictions of the adversary. This formulation is lightly", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "inspired by adversarial methods, specifically generative adversarial networks (GANs) (Goodfellow", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "et al., 2014). Such a link between GANs and actor-critic methods has been formalized by Pfau &", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Vinyals (2016); however, in the context of a third protagonist, we draw a different analogy. 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Moreover, because stability is a legitimate concern since specific instances of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "adversarial networks were shown to be prone to hyperparameter sensitivity issues (Arjovsky & Bottou,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 327, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 327, + 150 + ], + "score": 1.0, + "content": "2017), we also examine this aspect in our experiments.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "The contributions of this work are as follow: (i) we propose a novel actor-critic formulation inspired", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 178 + ], + "score": 1.0, + "content": "from adversarial learning (AGAC), (ii) we analyse empirically AGAC on key reinforcement learning as-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "pects such as diversity, exploration and stability, (iii) we demonstrate significant gains in performance", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 458, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 458, + 199 + ], + "score": 1.0, + "content": "on several sparse-reward hard-exploration tasks including procedurally-generated tasks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 216, + 211, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 213, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 213, + 231 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 255 + ], + "score": 1.0, + "content": "Actor-critic methods (Barto et al., 1983; Sutton, 1984) have been extended to the deep learning setting", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "by Mnih et al. 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Since then, many additions have been proposed, be it", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "architectural improvements (Vinyals et al., 2019), better advantage or value estimation (Schulman", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "et al., 2016; Flet-Berliac et al., 2021), or the incorporation of off-policy elements (Wang et al., 2017;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "Oh et al., 2018; Flet-Berliac & Preux, 2020). Regularization was shown to improve actor-critic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "methods, either by enforcing trust regions (Schulman et al., 2015; 2017; Wu et al., 2017), or by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "correcting for off-policiness (Munos et al., 2016; Gruslys et al., 2018); and recent works analyzed its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "impact from a theoretical standpoint (Geist et al., 2019; Ahmed et al., 2019; Vieillard et al., 2020a;b).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "Related to our work, Han & Sung (2020) use the entropy of the mixture between the policy induced", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "from a replay buffer and the current policy as a regularizer. To the best of our knowledge, none of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 423, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 423, + 374 + ], + "score": 1.0, + "content": "these methods explored the use of an adversarial objective to drive exploration.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 391 + ], + "score": 1.0, + "content": "While introduced in supervised learning, adversarial learning (Goodfellow et al., 2015; Miyato et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "2016; Kurakin et al., 2017) was leveraged in several RL works. Ho & Ermon (2016) propose an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "imitation learning method that uses a discriminator whose task is to distinguish between expert trajec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "tories and those of the agent while the agent tries to match expert behavior to fool the discriminator.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "Bahdanau et al. 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For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "instance, intrinsic motivation encourages exploratory behavior from the agent. Some works use", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 518, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 528 + ], + "score": 1.0, + "content": "state-visitation counts or pseudo-counts to promote exhaustive exploration (Bellemare et al., 2016a),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "while others use curiosity rewards, expressed in the magnitude of prediction error from the agent, to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "push it towards unfamiliar areas of the state space (Burda et al., 2018). Ecoffet et al. 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Finally, methods based on network", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "randomization (Igl et al., 2019), noise injection (Lee et al., 2020), and credit assignment (Ferret et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 654, + 403, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 403, + 667 + ], + "score": 1.0, + "content": "2020) have been proposed to reduce the generalization gap for RL agents.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 287, + 696 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 290, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 290, + 699 + ], + "score": 1.0, + "content": "3 BACKGROUND AND NOTATIONS", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We place ourselves in the Markov Decision Processes (Puterman, 1994) framework. 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Moreover, because stability is a legitimate concern since specific instances of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "adversarial networks were shown to be prone to hyperparameter sensitivity issues (Arjovsky & Bottou,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 327, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 327, + 150 + ], + "score": 1.0, + "content": "2017), we also examine this aspect in our experiments.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "The contributions of this work are as follow: (i) we propose a novel actor-critic formulation inspired", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 178 + ], + "score": 1.0, + "content": "from adversarial learning (AGAC), (ii) we analyse empirically AGAC on key reinforcement learning as-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "pects such as diversity, exploration and stability, (iii) we demonstrate significant gains in performance", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 458, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 458, + 199 + ], + "score": 1.0, + "content": "on several sparse-reward hard-exploration tasks including procedurally-generated tasks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 153, + 506, + 199 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 216, + 211, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 213, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 213, + 231 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 255 + ], + "score": 1.0, + "content": "Actor-critic methods (Barto et al., 1983; Sutton, 1984) have been extended to the deep learning setting", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "by Mnih et al. (2016), who combined deep neural networks and multiple distributed actors with an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "actor-critic setting, with strong results on Atari. Since then, many additions have been proposed, be it", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 287 + ], + "score": 1.0, + "content": "architectural improvements (Vinyals et al., 2019), better advantage or value estimation (Schulman", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "et al., 2016; Flet-Berliac et al., 2021), or the incorporation of off-policy elements (Wang et al., 2017;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "Oh et al., 2018; Flet-Berliac & Preux, 2020). Regularization was shown to improve actor-critic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "methods, either by enforcing trust regions (Schulman et al., 2015; 2017; Wu et al., 2017), or by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "correcting for off-policiness (Munos et al., 2016; Gruslys et al., 2018); and recent works analyzed its", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "impact from a theoretical standpoint (Geist et al., 2019; Ahmed et al., 2019; Vieillard et al., 2020a;b).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "Related to our work, Han & Sung (2020) use the entropy of the mixture between the policy induced", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "from a replay buffer and the current policy as a regularizer. To the best of our knowledge, none of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 423, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 423, + 374 + ], + "score": 1.0, + "content": "these methods explored the use of an adversarial objective to drive exploration.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 240, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 391 + ], + "score": 1.0, + "content": "While introduced in supervised learning, adversarial learning (Goodfellow et al., 2015; Miyato et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "2016; Kurakin et al., 2017) was leveraged in several RL works. Ho & Ermon (2016) propose an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "imitation learning method that uses a discriminator whose task is to distinguish between expert trajec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "tories and those of the agent while the agent tries to match expert behavior to fool the discriminator.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 436 + ], + "score": 1.0, + "content": "Bahdanau et al. (2019) use a discriminator to distinguish goal states from non-goal states based on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "a textual instruction, and use the resulting model as a reward function. Florensa et al. (2018) use a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "GAN to produce sub-goals at the right level of difficulty for the current agent, inducing a form of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "curriculum. Additionally, Pfau & Vinyals (2016) provide a parallel between GANs and the actor-critic", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 154, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 154, + 479 + ], + "score": 1.0, + "content": "framework.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 380, + 506, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 507, + 497 + ], + "score": 1.0, + "content": "While exploration is driven in part by the core RL algorithms (Fortunato et al., 2018; Han & Sung,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "2020; Ferret et al., 2021), it is often necessary to resort to exploration-specific techniques. For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "instance, intrinsic motivation encourages exploratory behavior from the agent. Some works use", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 518, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 528 + ], + "score": 1.0, + "content": "state-visitation counts or pseudo-counts to promote exhaustive exploration (Bellemare et al., 2016a),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "while others use curiosity rewards, expressed in the magnitude of prediction error from the agent, to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "push it towards unfamiliar areas of the state space (Burda et al., 2018). Ecoffet et al. (2019) propose a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "technique akin to tree traversal to explore while learning to come back to promising areas. Eysenbach", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "et al. (2018) show that encouraging diversity helps with exploration, even in the absence of reward.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 482, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Last but not least, generalization is a key challenge in RL. 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t } ( r _ { t ^ { \\prime } } + \\gamma V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } + 1 } ) - V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 302, + 436, + 312 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 156, + 313, + 175, + 325 + ], + "spans": [ + { + "bbox": [ + 156, + 313, + 175, + 325 + ], + "score": 0.91, + "content": "V _ { \\phi _ { \\mathrm { o l d } } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 324, + 507, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 382, + 342 + ], + "score": 1.0, + "content": "value function, we solve the non-linear regression problem minimizeφ", + "type": "text" + }, + { + "bbox": [ + 383, + 324, + 477, + 339 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { t ^ { \\prime } = t } ^ { t + N } ( V _ { \\phi } ( s _ { t ^ { \\prime } } ) - \\hat { V } _ { t ^ { \\prime } } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 324, + 507, + 342 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 194, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 190, + 352 + ], + "score": 0.91, + "content": "\\hat { V } _ { t } = A _ { t } + V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t ^ { \\prime } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 337, + 194, + 353 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 106, + 366, + 341, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 341, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 341, + 381 + ], + "score": 1.0, + "content": "4 ADVERSARIALLY GUIDED ACTOR-CRITIC", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "To foster diversified behavior in its trajectories, AGAC introduces a third protagonist to the actor-critic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "score": 1.0, + "content": "framework: the adversary. The role of the adversary is to accurately predict the actor’s actions,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 366, + 425 + ], + "score": 1.0, + "content": "by minimizing the discrepancy between its action distribution", + "type": "text" + }, + { + "bbox": [ + 366, + 415, + 384, + 424 + ], + "score": 0.87, + "content": "\\pi _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "and the distribution induced", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 164, + 437 + ], + "score": 1.0, + "content": "by the policy", + "type": "text" + }, + { + "bbox": [ + 164, + 426, + 172, + 434 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 424, + 506, + 437 + ], + "score": 1.0, + "content": ". 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This discrepancy, used as a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "form of exploration bonus, is defined as the difference of action log-probabilities (see Eq. 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This discrepancy, used as a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "form of exploration bonus, is defined as the difference of action log-probabilities (see Eq. (1)), whose", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 300, + 481 + ], + "score": 1.0, + "content": "expectation is the Kullback–Leibler divergence:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 390, + 506, + 481 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 482, + 428, + 496 + ], + "lines": [ + { + "bbox": [ + 183, + 482, + 428, + 496 + ], + "spans": [ + { + "bbox": [ + 183, + 482, + 428, + 496 + ], + "score": 0.88, + "content": "D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) = \\mathbb { E } _ { \\pi ( \\cdot | s ) } \\left[ \\log \\pi ( \\cdot | s ) - \\log \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) \\right] .", + "type": "interline_equation", + "image_path": "313661e94c3f88cf48c3210e2d6b7d59e59485b696566ea7ed195b46b5b4b5e7.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 183, + 482, + 428, + 496 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 504, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 246, + 518 + ], + "score": 1.0, + "content": "Formally, for each state-action pair", + "type": "text" + }, + { + "bbox": [ + 247, + 505, + 276, + 517 + ], + "score": 0.93, + "content": "( s _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 504, + 444, + 518 + ], + "score": 1.0, + "content": "in a trajectory, an action-dependent bonus", + "type": "text" + }, + { + "bbox": [ + 445, + 505, + 505, + 517 + ], + "score": 0.89, + "content": "\\log \\pi ( a _ { t } | s _ { t } ) -", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 107, + 516, + 166, + 528 + ], + "score": 0.91, + "content": "\\log \\pi _ { \\mathrm { a d v } } \\mathbf { \\dot { ( } } a _ { t } | s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "is added to the advantage. In addition, the value target of the critic is modified to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 387, + 540 + ], + "score": 1.0, + "content": "include the action-independent equivalent, which is the KL-divergence", + "type": "text" + }, + { + "bbox": [ + 387, + 527, + 485, + 539 + ], + "score": 0.92, + "content": "D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s _ { t } ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s _ { t } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 526, + 506, + 540 + ], + "score": 1.0, + "content": ". 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(1), (2) and (3) are the three equations that our method modifies (we color in blue the specific", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 497, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 338, + 158 + ], + "score": 1.0, + "content": "parts) in the traditional actor-critic framework. 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More precisely, our method favors", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 451, + 208 + ], + "score": 1.0, + "content": "transitions whose actions were less accurately predicted than the average action, i.e.", + "type": "text" + }, + { + "bbox": [ + 452, + 195, + 505, + 207 + ], + "score": 0.86, + "content": "\\log \\pi ( a | s ) -", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 265, + 218 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\log \\pi _ { \\mathrm { a d v } } ( a | s ) \\geq D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 206, + 397, + 218 + ], + "score": 1.0, + "content": ". 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This way, it represents a delayed and more steady version of the actor’s policy,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 500, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 500, + 333 + ], + "score": 1.0, + "content": "which prevents the agent from having to constantly adapt or focus solely on fooling the adversary.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 344, + 234, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 235, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 235, + 357 + ], + "score": 1.0, + "content": "4.1 BUILDING MOTIVATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "score": 1.0, + "content": "In the following, we provide an interpretation of AGAC by studying the dynamics of attraction and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "repulsion between the actor and the adversary. To simplify, we study the equivalent of AGAC in a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "policy iteration (PI) scheme. PI being the dynamic programming scheme underlying the standard", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "actor-critic, we have reasons to think that some of our findings translate to the original AGAC algorithm.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 424, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 424, + 422 + ], + "score": 1.0, + "content": "In PI, the quantity of interest is the action-value, which AGAC would modify as:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 422, + 381, + 438 + ], + "lines": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "spans": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "score": 0.92, + "content": "Q _ { \\pi _ { k } } ^ { \\mathtt { A G A C } } = Q _ { \\pi _ { k } } + c ( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } ) ,", + "type": "interline_equation", + "image_path": "7d6b08d78f936da16bf5b153c99ede87e47398c7e59016e9231adc066ebb1aa4.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 440, + 495, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 496, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 126, + 453 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 442, + 138, + 451 + ], + "score": 0.86, + "content": "\\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 438, + 227, + 453 + ], + "score": 1.0, + "content": "the policy at iteration", + "type": "text" + }, + { + "bbox": [ + 227, + 441, + 233, + 450 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 438, + 438, + 453 + ], + "score": 1.0, + "content": ". Incorporating the entropic penalty, the new policy", + "type": "text" + }, + { + "bbox": [ + 439, + 443, + 461, + 452 + ], + "score": 0.88, + "content": "\\pi _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 438, + 496, + 453 + ], + "score": 1.0, + "content": "verifies:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 455, + 460, + 475 + ], + "lines": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "spans": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "score": 0.92, + "content": "\\pi _ { k + 1 } = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot \\vert s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G A C } } } ( s , a ) - \\alpha \\log \\pi ( a \\vert s ) ] .", + "type": "interline_equation", + "image_path": "1f8c6698cba2a1b0d9b9ea9b2b1f56e50845c825d13115375ba0a7eeba66a55f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 226, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 226, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 226, + 490 + ], + "score": 1.0, + "content": "We can rewrite this objective:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 492, + 514, + 568 + ], + "lines": [ + { + "bbox": [ + 111, + 492, + 514, + 568 + ], + "spans": [ + { + "bbox": [ + 111, + 492, + 514, + 568 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathcal { T } _ { \\mathrm { { p I } } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G } A C } } ( s , a ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi ( a | s ) + \\log \\pi ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\mathrm { { g } } } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\Big [ \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) ] \\underbrace { - c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { k } ( \\cdot | s ) ) } _ { \\pi _ { k } \\mathrm { ~ i s ~ a t u r a c i t i v e } } \\underbrace { + c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\pi _ { \\mathrm { a d v } } \\mathrm { ~ i s ~ r e p u l s i v e } } \\underbrace { + \\alpha \\mathcal { H } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\mathrm { c n ~ t e r ~ i s r e p u l s i n c } } . } \\end{array}", + "type": "interline_equation", + "image_path": "c8e3a798f08d00f76eae423e4f37f046cdf18b7ecf46223fc27cf82bcda6e619.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 111, + 492, + 514, + 517.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 111, + 517.3333333333334, + 514, + 542.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 111, + 542.6666666666667, + 514, + 568.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 506, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 362, + 586 + ], + "score": 1.0, + "content": "Thus, in the PI scheme, AGAC finds a policy that maximizes", + "type": "text" + }, + { + "bbox": [ + 363, + 574, + 372, + 585 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "-values, while at the same time", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 455, + 599 + ], + "score": 1.0, + "content": "remaining close to the current policy and far from a mixture of the previous policies (i.e.,", + "type": "text" + }, + { + "bbox": [ + 455, + 586, + 477, + 596 + ], + "score": 0.63, + "content": "\\pi _ { k - 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Note that we experimentally observe (see Section 5.3) that our method performs better", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "with a smaller learning rate for the adversarial network than that of the other networks, which could", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 617, + 294, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 294, + 630 + ], + "score": 1.0, + "content": "imply that a stable repulsive term is beneficial.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 304, + 648 + ], + "score": 1.0, + "content": "This optimization problem is strongly concave in", + "type": "text" + }, + { + "bbox": [ + 305, + 636, + 312, + 644 + ], + "score": 0.78, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "(thanks to the entropy term), and is state-wise a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 645, + 481, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 481, + 658 + ], + "score": 1.0, + "content": "Legendre-Fenchel transform. Its solution is given by (see Appendix E for the full derivation):", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 660, + 365, + 690 + ], + "lines": [ + { + "bbox": [ + 246, + 660, + 365, + 690 + ], + "spans": [ + { + "bbox": [ + 246, + 660, + 365, + 690 + ], + "score": 0.94, + "content": "\\pi _ { k + 1 } \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } .", + "type": "interline_equation", + "image_path": "159b54c1a7faae5b408523e6b6ce2055023d3883ef1dede8bef99403f72c8612.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 660, + 365, + 675.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 246, + 675.0, + 365, + 690.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "This result gives us some insight into the behavior of the objective function. Notably, in our example,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 115, + 723 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 116, + 711, + 134, + 721 + ], + "score": 0.88, + "content": "\\pi _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 709, + 187, + 723 + ], + "score": 1.0, + "content": "is fixed and", + "type": "text" + }, + { + "bbox": [ + 187, + 711, + 215, + 720 + ], + "score": 0.86, + "content": "c = \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ", we recover a KL-regularized PI scheme (Geist et al., 2019) with the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 718, + 234, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 174, + 735 + ], + "score": 1.0, + "content": "modified reward", + "type": "text" + }, + { + "bbox": [ + 174, + 721, + 229, + 732 + ], + "score": 0.9, + "content": "r - c \\log \\pi _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 718, + 234, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 326, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 327, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 139, + 97 + ], + "score": 1.0, + "content": "Finally,", + "type": "text" + }, + { + "bbox": [ + 139, + 83, + 158, + 93 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 80, + 327, + 97 + ], + "score": 1.0, + "content": "is the objective function of the adversary:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 327, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 97, + 402, + 131 + ], + "lines": [ + { + "bbox": [ + 209, + 97, + 402, + 131 + ], + "spans": [ + { + "bbox": [ + 209, + 97, + 402, + 131 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { \\mathrm { a d v } } = \\frac { 1 } { N } \\sum _ { t = 0 } ^ { N } D _ { \\mathrm { K L } } \\big ( \\pi \\big ( \\cdot | s _ { t } , \\theta _ { \\mathrm { o l d } } \\big ) \\big | \\big | \\pi _ { \\mathrm { a d v } } \\big ( \\cdot | s _ { t } , \\psi \\big ) \\big ) .", + "type": "interline_equation", + "image_path": "9342fc161baad4a1f78266251ba83b09583ca4b3ea287f19105d8a4faedf6f1e.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 97, + 402, + 114.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 209, + 114.0, + 402, + 131.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 134, + 504, + 157 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 505, + 147 + ], + "score": 1.0, + "content": "Eqs. (1), (2) and (3) are the three equations that our method modifies (we color in blue the specific", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 497, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 338, + 158 + ], + "score": 1.0, + "content": "parts) in the traditional actor-critic framework. The terms", + "type": "text" + }, + { + "bbox": [ + 338, + 146, + 351, + 157 + ], + "score": 0.89, + "content": "\\beta _ { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 144, + 369, + 158 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 146, + 387, + 157 + ], + "score": 0.91, + "content": "\\beta _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 144, + 497, + 158 + ], + "score": 1.0, + "content": "are fixed hyperparameters.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 134, + 505, + 158 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 161, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 506, + 175 + ], + "score": 1.0, + "content": "Under the proposed actor-critic formulation, the probability of sampling an action is increased if", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "the modified advantage is positive, i.e. (i) the corresponding return is larger than the predicted", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 197 + ], + "score": 1.0, + "content": "value and/or (ii) the action log-probability difference is large. More precisely, our method favors", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 451, + 208 + ], + "score": 1.0, + "content": "transitions whose actions were less accurately predicted than the average action, i.e.", + "type": "text" + }, + { + "bbox": [ + 452, + 195, + 505, + 207 + ], + "score": 0.86, + "content": "\\log \\pi ( a | s ) -", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 265, + 218 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\log \\pi _ { \\mathrm { a d v } } ( a | s ) \\geq D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) \\| \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 206, + 397, + 218 + ], + "score": 1.0, + "content": ". This is particularly visible for", + "type": "text" + }, + { + "bbox": [ + 397, + 207, + 426, + 216 + ], + "score": 0.91, + "content": "\\lambda 1", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 206, + 505, + 218 + ], + "score": 1.0, + "content": ", in which case the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 209, + 231 + ], + "score": 1.0, + "content": "generalized advantage is", + "type": "text" + }, + { + "bbox": [ + 209, + 218, + 292, + 230 + ], + "score": 0.9, + "content": "A _ { t } = G _ { t } - V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 216, + 506, + 231 + ], + "score": 1.0, + "content": ", resulting in the appearance of both aforementioned", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 276, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 276, + 241 + ], + "score": 1.0, + "content": "mirrored terms in the modified advantage:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 161, + 506, + 241 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 243, + 478, + 264 + ], + "lines": [ + { + "bbox": [ + 131, + 243, + 478, + 264 + ], + "spans": [ + { + "bbox": [ + 131, + 243, + 478, + 264 + ], + "score": 0.9, + "content": "A _ { t } ^ { \\mathtt { A G A C } } = G _ { t } - \\hat { V } _ { t } ^ { \\phi _ { \\mathrm { o d d } } } + c \\left( \\log \\pi ( a _ { t } | s _ { t } ) - \\log \\pi _ { \\mathrm { a d v } } ( a _ { t } | s _ { t } ) - \\hat { D } _ { \\mathrm { K L } } ^ { \\phi _ { \\mathrm { o d d } } } ( \\pi ( \\cdot | s _ { t } ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s _ { t } ) ) \\right) ,", + "type": "interline_equation", + "image_path": "640255cc1738d9bd160976e173be1c2fe4381afdd9afec2513ecf5eff1dd157f.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 131, + 243, + 478, + 264 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 268, + 504, + 293 + ], + "lines": [ + { + "bbox": [ + 102, + 263, + 509, + 293 + ], + "spans": [ + { + "bbox": [ + 102, + 263, + 127, + 293 + ], + "score": 1.0, + "content": "with KL-", + "type": "text" + }, + { + "bbox": [ + 127, + 270, + 139, + 281 + ], + "score": 0.88, + "content": "G _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 263, + 224, + 293 + ], + "score": 1.0, + "content": "the observed return, rgence (estimated co", + "type": "text" + }, + { + "bbox": [ + 225, + 268, + 245, + 282 + ], + "score": 0.92, + "content": "\\hat { V } _ { t } ^ { \\phi _ { \\mathrm { o l d } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 263, + 274, + 293 + ], + "score": 1.0, + "content": "the estents of", + "type": "text" + }, + { + "bbox": [ + 309, + 263, + 347, + 293 + ], + "score": 1.0, + "content": "eturn and from Eq.", + "type": "text" + }, + { + "bbox": [ + 348, + 267, + 447, + 282 + ], + "score": 0.92, + "content": "\\hat { D } _ { \\mathrm { K L } } ^ { \\phi _ { \\mathrm { o l d } } } \\big ( \\pi \\big ( \\cdot | s _ { t } \\big ) \\big | \\big | \\pi _ { \\mathrm { a d v } } \\big ( \\cdot | s _ { t } \\big ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 263, + 509, + 293 + ], + "score": 1.0, + "content": "the estimated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 275, + 281, + 309, + 293 + ], + "spans": [ + { + "bbox": [ + 275, + 281, + 309, + 293 + ], + "score": 0.94, + "content": "V _ { \\phi _ { \\mathrm { o l d } } } ( s _ { t } )", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 102, + 263, + 509, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 506, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 311 + ], + "score": 1.0, + "content": "To avoid instability, in practice the adversary is a separate estimator, updated with a smaller learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 308, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 507, + 322 + ], + "score": 1.0, + "content": "rate than the actor. This way, it represents a delayed and more steady version of the actor’s policy,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 500, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 500, + 333 + ], + "score": 1.0, + "content": "which prevents the agent from having to constantly adapt or focus solely on fooling the adversary.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 296, + 507, + 333 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 344, + 234, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 235, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 235, + 357 + ], + "score": 1.0, + "content": "4.1 BUILDING MOTIVATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 378 + ], + "score": 1.0, + "content": "In the following, we provide an interpretation of AGAC by studying the dynamics of attraction and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "repulsion between the actor and the adversary. To simplify, we study the equivalent of AGAC in a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 386, + 506, + 399 + ], + "score": 1.0, + "content": "policy iteration (PI) scheme. PI being the dynamic programming scheme underlying the standard", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "actor-critic, we have reasons to think that some of our findings translate to the original AGAC algorithm.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 424, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 424, + 422 + ], + "score": 1.0, + "content": "In PI, the quantity of interest is the action-value, which AGAC would modify as:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 363, + 506, + 422 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 422, + 381, + 438 + ], + "lines": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "spans": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "score": 0.92, + "content": "Q _ { \\pi _ { k } } ^ { \\mathtt { A G A C } } = Q _ { \\pi _ { k } } + c ( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } ) ,", + "type": "interline_equation", + "image_path": "7d6b08d78f936da16bf5b153c99ede87e47398c7e59016e9231adc066ebb1aa4.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 228, + 422, + 381, + 438 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 440, + 495, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 496, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 126, + 453 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 442, + 138, + 451 + ], + "score": 0.86, + "content": "\\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 438, + 227, + 453 + ], + "score": 1.0, + "content": "the policy at iteration", + "type": "text" + }, + { + "bbox": [ + 227, + 441, + 233, + 450 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 438, + 438, + 453 + ], + "score": 1.0, + "content": ". Incorporating the entropic penalty, the new policy", + "type": "text" + }, + { + "bbox": [ + 439, + 443, + 461, + 452 + ], + "score": 0.88, + "content": "\\pi _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 438, + 496, + 453 + ], + "score": 1.0, + "content": "verifies:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 438, + 496, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 455, + 460, + 475 + ], + "lines": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "spans": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "score": 0.92, + "content": "\\pi _ { k + 1 } = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\ : \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot \\vert s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G A C } } } ( s , a ) - \\alpha \\log \\pi ( a \\vert s ) ] .", + "type": "interline_equation", + "image_path": "1f8c6698cba2a1b0d9b9ea9b2b1f56e50845c825d13115375ba0a7eeba66a55f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 149, + 455, + 460, + 475 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 226, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 226, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 226, + 490 + ], + "score": 1.0, + "content": "We can rewrite this objective:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 477, + 226, + 490 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 492, + 514, + 568 + ], + "lines": [ + { + "bbox": [ + 111, + 492, + 514, + 568 + ], + "spans": [ + { + "bbox": [ + 111, + 492, + 514, + 568 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathcal { T } _ { \\mathrm { { p I } } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ^ { \\mathrm { { A G } A C } } ( s , a ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi ( a | s ) + \\log \\pi ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\mathrm { { g } } } \\\\ & { \\qquad = \\mathbb { E } _ { s } \\Big [ \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) ] \\underbrace { - c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { k } ( \\cdot | s ) ) } _ { \\pi _ { k } \\mathrm { ~ i s ~ a t u r a c i t i v e } } \\underbrace { + c D _ { \\mathrm { K L } } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\pi _ { \\mathrm { a d v } } \\mathrm { ~ i s ~ r e p u l s i v e } } \\underbrace { + \\alpha \\mathcal { H } ( \\pi ( \\cdot | s ) | | \\pi _ { \\mathrm { a d v } } ( \\cdot | s ) ) } _ { \\mathrm { c n ~ t e r ~ i s r e p u l s i n c } } . } \\end{array}", + "type": "interline_equation", + "image_path": "c8e3a798f08d00f76eae423e4f37f046cdf18b7ecf46223fc27cf82bcda6e619.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 111, + 492, + 514, + 517.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 111, + 517.3333333333334, + 514, + 542.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 111, + 542.6666666666667, + 514, + 568.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 506, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 362, + 586 + ], + "score": 1.0, + "content": "Thus, in the PI scheme, AGAC finds a policy that maximizes", + "type": "text" + }, + { + "bbox": [ + 363, + 574, + 372, + 585 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "-values, while at the same time", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 455, + 599 + ], + "score": 1.0, + "content": "remaining close to the current policy and far from a mixture of the previous policies (i.e.,", + "type": "text" + }, + { + "bbox": [ + 455, + 586, + 477, + 596 + ], + "score": 0.63, + "content": "\\pi _ { k - 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Its solution is given by (see Appendix E for the full derivation):", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 632, + 506, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 660, + 365, + 690 + ], + "lines": [ + { + "bbox": [ + 246, + 660, + 365, + 690 + ], + "spans": [ + { + "bbox": [ + 246, + 660, + 365, + 690 + ], + "score": 0.94, + "content": "\\pi _ { k + 1 } \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } .", + "type": "interline_equation", + "image_path": "159b54c1a7faae5b408523e6b6ce2055023d3883ef1dede8bef99403f72c8612.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 660, + 365, + 675.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 246, + 675.0, + 365, + 690.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "This result gives us some insight into the behavior of the objective function. 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The actor, critic and adversary", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "use the convolutional architecture of the Nature paper of DQN (Mnih et al., 2015) with different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "hidden sizes (see Appendix D for architecture details). The three neural networks are optimized", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "using Adam (Kingma & Ba, 2015). 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Note that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "the parameters are not shared between the policy, the critic and the adversary and that we did not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "observe any noticeable difference in computational complexity when using AGAC compared to PPO.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 429, + 306 + ], + "score": 1.0, + "content": "We direct the reader to Appendix C for a list of hyperparameters. 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(c) MiniGrid-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 601, + 327, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 327, + 613 + ], + "score": 1.0, + "content": "KeyCorridorS6R3. (d) MiniGrid-ObstructedMazeFull.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 32.25 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "Environments. 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Hence,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 102, + 505, + 127 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 131, + 483, + 165 + ], + "lines": [ + { + "bbox": [ + 125, + 131, + 483, + 165 + ], + "spans": [ + { + "bbox": [ + 125, + 131, + 483, + 165 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { P G } = - \\frac { 1 } { N } \\sum _ { t ^ { \\prime } = t } ^ { t + N } \\operatorname* { m i n } \\left( \\frac { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta ) } { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta _ { \\mathrm { o l d } } ) } A _ { t ^ { \\prime } } ^ { \\mathrm { R G K } } , \\mathrm { c l i p } \\left( \\frac { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta ) } { \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } , \\theta _ { \\mathrm { o l d } } ) } , 1 - \\epsilon , 1 + \\epsilon \\right) A _ { t ^ { \\prime } } ^ { \\mathrm { R G K } } \\right) ,", + "type": "interline_equation", + "image_path": "512051f4cddcfb03938253cc36c2f458c14301b0bf117b1b82988a099e86e84d.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 125, + 131, + 483, + 142.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 125, + 142.33333333333334, + 483, + 153.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 125, + 153.66666666666669, + 483, + 165.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 506, + 315 + ], + "lines": [ + { + "bbox": [ + 104, + 169, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 127, + 186 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 171, + 151, + 184 + ], + "score": 0.91, + "content": "A _ { t ^ { \\prime } } ^ { \\tt A G A C }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 169, + 221, + 186 + ], + "score": 1.0, + "content": "given in Eq. 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The actor, critic and adversary", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "use the convolutional architecture of the Nature paper of DQN (Mnih et al., 2015) with different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "hidden sizes (see Appendix D for architecture details). The three neural networks are optimized", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "using Adam (Kingma & Ba, 2015). Our method does not use RNNs in its architecture; instead,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 506, + 251 + ], + "score": 1.0, + "content": "in all our experiments, we use frame stacking. Indeed, Hausknecht & Stone (2015) interestingly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "demonstrate that although recurrence is a reliable method for processing state observation, it does not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 260, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 271 + ], + "score": 1.0, + "content": "confer any systematic advantage over stacking observations in the input layer of a CNN. Note that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "the parameters are not shared between the policy, the critic and the adversary and that we did not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "observe any noticeable difference in computational complexity when using AGAC compared to PPO.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 429, + 306 + ], + "score": 1.0, + "content": "We direct the reader to Appendix C for a list of hyperparameters. 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(c) MiniGrid-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 601, + 327, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 327, + 613 + ], + "score": 1.0, + "content": "KeyCorridorS6R3. (d) MiniGrid-ObstructedMazeFull.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 32.25 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "Environments. 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For a fair assessment of our method, we compare to some of the most prominent methods", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "score": 1.0, + "content": "specialized in hard-exploration tasks: RIDE (Raileanu & Rocktäschel, 2019), based on an intrinsic", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "reward associated with the magnitude of change between two consecutive state representations and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 205 + ], + "score": 1.0, + "content": "state visitation, Count as Count-Based Exploration (Bellemare et al., 2016b), which we couple with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "IMPALA (Espeholt et al., 2018), RND (Burda et al., 2018) in which an exploration bonus is positively", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "correlated to the error of predicting features from the observations and ICM (Pathak et al., 2017),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "where a module only predicts the changes in the environment that are produced by the actions of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "agent. Finally, we compare to most the recent and best performing method at the time of writing in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "procedurally-generated environments: AMIGo (Campero et al., 2021) in which a goal-generating", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 286, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 286, + 272 + ], + "score": 1.0, + "content": "teacher provides count-based intrinsic goals.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 108, + 284, + 401, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 403, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 403, + 297 + ], + "score": 1.0, + "content": "5.1 ADVERSARIALLY-BASED EXPLORATION (NO EPISODIC COUNT)", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table", + "bbox": [ + 112, + 329, + 497, + 405 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 187, + 308, + 423, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 186, + 307, + 424, + 322 + ], + "spans": [ + { + "bbox": [ + 186, + 307, + 424, + 322 + ], + "score": 1.0, + "content": "Table 1: Average return in VizDoom at different timesteps.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 112, + 329, + 497, + 405 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 329, + 497, + 405 + ], + "spans": [ + { + "bbox": [ + 112, + 329, + 497, + 405 + ], + "score": 0.981, + "html": "
Nb.of Timesteps2M4M6M8M10M
AGAC0.74± 0.050.96 ± 0.0010.96 ± 0.0010.97 ± 0.0010.97 ± 0.001
RIDE0.0.0.95 ± 0.0010.97 ± 0.0010.97 ± 0.001
ICM0.0.0.95 ±0.0010.97 ± 0.0010.97 ± 0.001
AMIGo0.0.0.0.0.
RND0.0.0.0.0.
Count0.0.0.0.0.
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In order to provide a comparison to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "score": 1.0, + "content": "state-of-the-art methods, we choose VizDoom, a hard-exploration problem used in prior work. In this", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "game, the map consists of 9 rooms connected by corridors where 270 steps separate the initial position", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "of the agent and the goal under an optimal policy. 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Results in Table 1", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "indicate that AGAC clearly outperforms other methods in sample-efficiency. Only the methods ICM", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 477, + 526 + ], + "score": 1.0, + "content": "and RIDE succeed in matching the score of AGAC, and with about twice as much transitions", + "type": "text" + }, + { + "bbox": [ + 478, + 514, + 505, + 525 + ], + "score": 0.74, + "content": "( \\sim 3 \\mathbf { M }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 523, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 506, + 540 + ], + "score": 1.0, + "content": "vs. 6M). Interestingly, AMIGo performs similarly to Count and RND. We find this result surprising", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "because AMIGo has proven to perform well in the MiniGrid environments. 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However, the VizDoom task is not procedurally-generated;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "hence we have not evaluated the generalization to new states yet. 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Nb.of Timesteps2M4M6M8M10M
AGAC0.74± 0.050.96 ± 0.0010.96 ± 0.0010.97 ± 0.0010.97 ± 0.001
RIDE0.0.0.95 ± 0.0010.97 ± 0.0010.97 ± 0.001
ICM0.0.0.95 ±0.0010.97 ± 0.0010.97 ± 0.001
AMIGo0.0.0.0.0.
RND0.0.0.0.0.
Count0.0.0.0.0.
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In order to provide a comparison to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "score": 1.0, + "content": "state-of-the-art methods, we choose VizDoom, a hard-exploration problem used in prior work. In this", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "game, the map consists of 9 rooms connected by corridors where 270 steps separate the initial position", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "of the agent and the goal under an optimal policy. Episodes are terminated either when the agent finds", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "the goal or if the episode exceeds 2100 timesteps. Importantly, while other algorithms (Raileanu", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "score": 1.0, + "content": "& Rocktäschel, 2019; Campero et al., 2021) benefit from count-based exploration, this study has", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "been conducted with our method not benefiting from episodic count whatsoever. Results in Table 1", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "indicate that AGAC clearly outperforms other methods in sample-efficiency. Only the methods ICM", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 477, + 526 + ], + "score": 1.0, + "content": "and RIDE succeed in matching the score of AGAC, and with about twice as much transitions", + "type": "text" + }, + { + "bbox": [ + 478, + 514, + 505, + 525 + ], + "score": 0.74, + "content": "( \\sim 3 \\mathbf { M }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 523, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 506, + 540 + ], + "score": 1.0, + "content": "vs. 6M). Interestingly, AMIGo performs similarly to Count and RND. We find this result surprising", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "because AMIGo has proven to perform well in the MiniGrid environments. Nevertheless, it appears", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "that concurrent works to ours experienced similar issues with the accompanying implementation1.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "The results of AGAC support the capabilities of the adversarial bonus and show that it can, on its own,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "achieve significant gains in performance. However, the VizDoom task is not procedurally-generated;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "hence we have not evaluated the generalization to new states yet. In the following section, we use", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 591, + 221, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 221, + 603 + ], + "score": 1.0, + "content": "MiniGrid to investigate this.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 415, + 506, + 603 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 615, + 460, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 462, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 462, + 628 + ], + "score": 1.0, + "content": "5.2 HARD-EXPLORATION TASKS WITH PARTIALLY-OBSERVABLE ENVIRONMENTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 507, + 648 + ], + "score": 1.0, + "content": "We now evaluate our method on multiple hard-exploration procedurally-generated tasks from Mini-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "Grid. Details about MiniGrid can be found in Appendix C.1. Fig. 2 indicates that AGAC significantly", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "outperforms other methods on these tasks in sample-efficiency and performance. AGAC also outper-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 670, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 681 + ], + "score": 1.0, + "content": "forms the current state-of-the-art method, AMIGo, despite the fact that it uses the fully-observable", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "version of MiniGrid. Note that we find the same poor performance results when training AMIGo in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "MiniGrid, similar to Vizdoom results. For completeness, we also report in Table 2 of Appendix A.1", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "the performance results with the scores reported in the original papers Raileanu & Rocktäschel", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "(2019) and Campero et al. (2021). We draw similar conclusions: AGAC clearly outperforms the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 300, + 326, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 326, + 310 + ], + "score": 1.0, + "content": "state-of-the-art RIDE, AMIGo, Count, RND and ICM.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 636, + 507, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 78, + 502, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 78, + 502, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 78, + 502, + 248 + ], + "spans": [ + { + "bbox": [ + 108, + 78, + 502, + 248 + ], + "score": 0.966, + "type": "image", + "image_path": "c694a663b7b25ee719de7eb448621e3cb5826c21e451c62ee01f34f8d0c5792f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 78, + 502, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 502, + 191.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 191.33333333333331, + 502, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 220, + 256, + 392, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 255, + 392, + 268 + ], + "spans": [ + { + "bbox": [ + 218, + 255, + 392, + 268 + ], + "score": 1.0, + "content": "Figure 2: Performance evaluation of AGAC.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "(2019) and Campero et al. (2021). We draw similar conclusions: AGAC clearly outperforms the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 300, + 326, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 326, + 310 + ], + "score": 1.0, + "content": "state-of-the-art RIDE, AMIGo, Count, RND and ICM.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "In all the considered tasks, the agent must learn to generalize across a very large state space because", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "the layouts are generated procedurally. We consider three main arguments to explain why our method", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 338, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 507, + 351 + ], + "score": 1.0, + "content": "is successful: (i) our method makes use of partial observations: in this context, the adversary has a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "harder time predicting the actor’s actions; nevertheless, the mistakes of the former benefit the latter in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "the form of an exploration bonus, which pushes the agent to explore further in order to deceive the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "adversary, (ii) the exploration bonus (i.e. intrinsic reward) does not dissipate compared to most other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "methods, as observed in Fig. 9 in Appendix A.4, (iii) our method does not make assumptions about", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "the environment dynamics (e.g., changes in the environment produced by an action as in Raileanu &", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "Rocktäschel (2019)) since this can hinder learning when the space of state changes induced by an", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 415, + 414, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 414, + 427 + ], + "score": 1.0, + "content": "action is too large (such as the action of moving a block in ObstructedMaze).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "In Appendix A.3, we also include experiments in two environments with extremely sparse reward", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "signals: KeyCorridorS8R3 and ObstructedMazeFull. Despite the challenge, AGAC still manages to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "find rewards and can perform well by taking advantage of the diversified behaviour induced by our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 454, + 479 + ], + "score": 1.0, + "content": "method. To the best of our knowledge, no other method ever succeeded to perform well", + "type": "text" + }, + { + "bbox": [ + 454, + 465, + 471, + 476 + ], + "score": 0.85, + "content": "\\mathit { \\Theta } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 463, + 505, + 479 + ], + "score": 1.0, + "content": "average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 476, + 486, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 486, + 489 + ], + "score": 1.0, + "content": "return) in those tasks. We think that given more computing time, AGAC’s score could go higher.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 501, + 223, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 225, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 225, + 513 + ], + "score": 1.0, + "content": "5.3 TRAINING STABILITY", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "image", + "bbox": [ + 146, + 523, + 464, + 618 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 523, + 464, + 618 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 146, + 523, + 464, + 618 + ], + "spans": [ + { + "bbox": [ + 146, + 523, + 464, + 618 + ], + "score": 0.966, + "type": "image", + "image_path": "df1a9efb68034f4a5f52a20d9d5ef22a9514eee7144535790fe25105fc795095.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 146, + 523, + 464, + 554.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 146, + 554.6666666666666, + 464, + 586.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 146, + 586.3333333333333, + 464, + 617.9999999999999 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 185, + 627, + 425, + 640 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 185, + 627, + 425, + 641 + ], + "spans": [ + { + "bbox": [ + 185, + 627, + 425, + 641 + ], + "score": 1.0, + "content": "Figure 3: Sensitivity analysis of AGAC in KeyCorridorS4R3.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "Here we want to analyse the stability of the method when changing hyperparameters. The most", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 243, + 677 + ], + "score": 1.0, + "content": "important parameters in AGAC are", + "type": "text" + }, + { + "bbox": [ + 243, + 666, + 249, + 675 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 664, + 505, + 677 + ], + "score": 1.0, + "content": ", the coefficient for the adversarial bonus, and the learning rates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 128, + 686 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 128, + 675, + 162, + 690 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nu = \\frac { \\eta _ { 2 } } { \\eta _ { 1 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 675, + 505, + 687 + ], + "score": 1.0, + "content": ". We choose KeyCorridorS4R3 as the evaluation task because among all the tasks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "considered, its difficulty is at a medium level. Fig. 3 shows the learning curves. For readability, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "plot the average return only; the standard deviation is sensibly the same for all curves. We observe", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "that deviating from the hyperparameter values found using grid search results in a slower training.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 721, + 403, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 235, + 734 + ], + "score": 1.0, + "content": "Moreover, although reasonable,", + "type": "text" + }, + { + "bbox": [ + 235, + 723, + 241, + 730 + ], + "score": 0.66, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 721, + 392, + 734 + ], + "score": 1.0, + "content": "appears to have more sensitivity than", + "type": "text" + }, + { + "bbox": [ + 392, + 723, + 398, + 730 + ], + "score": 0.7, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 721, + 403, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 78, + 502, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 78, + 502, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 78, + 502, + 248 + ], + "spans": [ + { + "bbox": [ + 108, + 78, + 502, + 248 + ], + "score": 0.966, + "type": "image", + "image_path": "c694a663b7b25ee719de7eb448621e3cb5826c21e451c62ee01f34f8d0c5792f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 78, + 502, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 502, + 191.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 191.33333333333331, + 502, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 220, + 256, + 392, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 255, + 392, + 268 + ], + "spans": [ + { + "bbox": [ + 218, + 255, + 392, + 268 + ], + "score": 1.0, + "content": "Figure 2: Performance evaluation of AGAC.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 311 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 105, + 288, + 505, + 310 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "In all the considered tasks, the agent must learn to generalize across a very large state space because", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "the layouts are generated procedurally. We consider three main arguments to explain why our method", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 338, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 507, + 351 + ], + "score": 1.0, + "content": "is successful: (i) our method makes use of partial observations: in this context, the adversary has a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "harder time predicting the actor’s actions; nevertheless, the mistakes of the former benefit the latter in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "the form of an exploration bonus, which pushes the agent to explore further in order to deceive the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "adversary, (ii) the exploration bonus (i.e. intrinsic reward) does not dissipate compared to most other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "methods, as observed in Fig. 9 in Appendix A.4, (iii) our method does not make assumptions about", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "the environment dynamics (e.g., changes in the environment produced by an action as in Raileanu &", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "Rocktäschel (2019)) since this can hinder learning when the space of state changes induced by an", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 415, + 414, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 414, + 427 + ], + "score": 1.0, + "content": "action is too large (such as the action of moving a block in ObstructedMaze).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 316, + 507, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 445 + ], + "score": 1.0, + "content": "In Appendix A.3, we also include experiments in two environments with extremely sparse reward", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "signals: KeyCorridorS8R3 and ObstructedMazeFull. Despite the challenge, AGAC still manages to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "find rewards and can perform well by taking advantage of the diversified behaviour induced by our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 463, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 454, + 479 + ], + "score": 1.0, + "content": "method. To the best of our knowledge, no other method ever succeeded to perform well", + "type": "text" + }, + { + "bbox": [ + 454, + 465, + 471, + 476 + ], + "score": 0.85, + "content": "\\mathit { \\Theta } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 463, + 505, + 479 + ], + "score": 1.0, + "content": "average", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 476, + 486, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 486, + 489 + ], + "score": 1.0, + "content": "return) in those tasks. We think that given more computing time, AGAC’s score could go higher.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 431, + 505, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 501, + 223, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 225, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 225, + 513 + ], + "score": 1.0, + "content": "5.3 TRAINING STABILITY", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "image", + "bbox": [ + 146, + 523, + 464, + 618 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 523, + 464, + 618 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 146, + 523, + 464, + 618 + ], + "spans": [ + { + "bbox": [ + 146, + 523, + 464, + 618 + ], + "score": 0.966, + "type": "image", + "image_path": "df1a9efb68034f4a5f52a20d9d5ef22a9514eee7144535790fe25105fc795095.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 146, + 523, + 464, + 554.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 146, + 554.6666666666666, + 464, + 586.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 146, + 586.3333333333333, + 464, + 617.9999999999999 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 185, + 627, + 425, + 640 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 185, + 627, + 425, + 641 + ], + "spans": [ + { + "bbox": [ + 185, + 627, + 425, + 641 + ], + "score": 1.0, + "content": "Figure 3: Sensitivity analysis of AGAC in KeyCorridorS4R3.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "Here we want to analyse the stability of the method when changing hyperparameters. The most", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 243, + 677 + ], + "score": 1.0, + "content": "important parameters in AGAC are", + "type": "text" + }, + { + "bbox": [ + 243, + 666, + 249, + 675 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 664, + 505, + 677 + ], + "score": 1.0, + "content": ", the coefficient for the adversarial bonus, and the learning rates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 675, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 128, + 686 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 128, + 675, + 162, + 690 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nu = \\frac { \\eta _ { 2 } } { \\eta _ { 1 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 675, + 505, + 687 + ], + "score": 1.0, + "content": ". We choose KeyCorridorS4R3 as the evaluation task because among all the tasks", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "considered, its difficulty is at a medium level. Fig. 3 shows the learning curves. For readability, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "plot the average return only; the standard deviation is sensibly the same for all curves. We observe", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "that deviating from the hyperparameter values found using grid search results in a slower training.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 721, + 403, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 235, + 734 + ], + "score": 1.0, + "content": "Moreover, although reasonable,", + "type": "text" + }, + { + "bbox": [ + 235, + 723, + 241, + 730 + ], + "score": 0.66, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 721, + 392, + 734 + ], + "score": 1.0, + "content": "appears to have more sensitivity than", + "type": "text" + }, + { + "bbox": [ + 392, + 723, + 398, + 730 + ], + "score": 0.7, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 721, + 403, + 734 + ], + "score": 1.0, + "content": ".", + 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455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 317, + 455 + ], + "score": 1.0, + "content": "Beyond the good performance of our method when", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 453, + 318, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 318, + 466 + ], + "score": 1.0, + "content": "extrinsic rewards are given to the agent, Fig. 4 in-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 464, + 317, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 317, + 476 + ], + "score": 1.0, + "content": "dicates that the exploration induced by our method", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 475, + 317, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 317, + 487 + ], + "score": 1.0, + "content": "makes the agent succeed in a significant proportion", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 486, + 318, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 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In singleton environments, an agent has to solve the same task in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "same environment/maze in every episode. Fig. 5 shows the state visitation heatmaps (darker areas", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "correspond to more visits) after a training of 10M timesteps. We observe that most of the methods", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "explore inefficiently in a singleton environment and that only RIDE succeeds in reaching the fifth", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "room while AGAC reaches the last (tenth) room. After training the agents in procedurally-generated", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 734 + ], + "score": 1.0, + "content": "environments, the methods explore even less efficiently while AGAC succeeds in exploring all rooms.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 644, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 138, + 90, + 460, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 90, + 460, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 90, + 460, + 232 + ], + "spans": [ + { + "bbox": [ + 138, + 90, + 460, + 232 + ], + "score": 0.974, + "type": "image", + "image_path": "79b4e4dfaa987603232a227680bcdf155166caf72f8802e24284152dfcbbd4e8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 138, + 90, + 460, + 137.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 138, + 137.33333333333334, + 460, + 184.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 184.66666666666669, + 460, + 232.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 244, + 506, + 277 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "Figure 6: State visitation heatmaps of the last ten episodes of an agent trained in procedurally-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "generated environments without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 265, + 322, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 322, + 279 + ], + "score": 1.0, + "content": "The agent is continuously engaging in new strategies.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 315, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "We now qualitatively study the diversity of an agent’s behavior when trained with AGAC. Fig. 6", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "presents the state visitation heatmaps of the last ten episodes for an agent trained in procedurally-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "generated environments in the MultiRoomN10S6 task without extrinsic reward. The heatmaps", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "score": 1.0, + "content": "correspond to the behavior of the resulting policy, which is still learning from the AGAC objective.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 358, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 507, + 372 + ], + "score": 1.0, + "content": "Looking at the figure, we can see that the strategies vary at each update with, for example, back-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "and-forth and back-to-start behaviors. Although there are no extrinsic reward, the strategies seem", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "to diversify from one update to the next. Finally, Fig. 7 in Appendix A.2 shows the state visitation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 392, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "heatmaps in a different configuration: when the agent has been trained on a singleton environment", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "in the MultiRoomN10S6 task without extrinsic reward. Same as previously, the agent is updated", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 415, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 426 + ], + "score": 1.0, + "content": "between each episode. Looking at the figure, we can make essentially the same observations as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "previously, with a noteworthy behavior in the fourth heatmap of the bottom row where it appears the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "agent went to the fourth room to remain inside it. Those episodes indicate that, although the agent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "sees the same environment repeatedly, the successive adversarial updates force it to continuously", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 458, + 271, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 271, + 471 + ], + "score": 1.0, + "content": "adapt its behavior and try new strategies.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 503, + 190, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 192, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 192, + 519 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "This paper introduced AGAC, a modification to the traditional actor-critic framework: an adversary", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "network is added as a third protagonist. The mechanics of AGAC have been discussed from a policy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "iteration point of view, and we provided theoretical insight into the inner workings of the proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "algorithm: the adversary forces the agent to remain close to the current policy while moving away", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "from the previous ones. In a nutshell, the influence of the adversary makes the actor conservatively", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 153, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 153, + 606 + ], + "score": 1.0, + "content": "diversified.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "In the experimental study, we have evaluated the adversarially-based bonus in VizDoom and em-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "pirically demonstrated its effectiveness and superiority compared to other relevant methods (some", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "benefiting from count-based exploration). Then, we have conducted several performance experiments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "using AGAC and have shown a significant performance improvement over some of the most popular", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "exploration methods (RIDE, AMIGo, Count, RND and ICM) on a set of various challenging tasks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "from MiniGrid. These procedurally-generated environments have served another purpose which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "to validate the capacity of our method to generalize to unseen scenarios. In addition, the training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 479, + 700 + ], + "score": 1.0, + "content": "stability of our method has been studied, showing a greater but acceptable sensitivity for", + "type": "text" + }, + { + "bbox": [ + 480, + 690, + 485, + 698 + ], + "score": 0.65, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 688, + 505, + 700 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "adversarial bonus coefficient. Finally, we have investigated the exploration capabilities of AGAC in a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "reward-free setting where the agent demonstrated exhaustive exploration through various strategic", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 487, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 487, + 732 + ], + "score": 1.0, + "content": "choices, confirming that the adversary successfully drives diversity in the behavior of the actor.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 138, + 90, + 460, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 90, + 460, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 90, + 460, + 232 + ], + "spans": [ + { + "bbox": [ + 138, + 90, + 460, + 232 + ], + "score": 0.974, + "type": "image", + "image_path": "79b4e4dfaa987603232a227680bcdf155166caf72f8802e24284152dfcbbd4e8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 138, + 90, + 460, + 137.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 138, + 137.33333333333334, + 460, + 184.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 184.66666666666669, + 460, + 232.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 244, + 506, + 277 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 506, + 256 + ], + "score": 1.0, + "content": "Figure 6: State visitation heatmaps of the last ten episodes of an agent trained in procedurally-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "generated environments without extrinsic reward for 10M timesteps in the MultiRoomN10S6 task.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 265, + 322, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 322, + 279 + ], + "score": 1.0, + "content": "The agent is continuously engaging in new strategies.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 315, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "We now qualitatively study the diversity of an agent’s behavior when trained with AGAC. Fig. 6", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "presents the state visitation heatmaps of the last ten episodes for an agent trained in procedurally-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "generated environments in the MultiRoomN10S6 task without extrinsic reward. The heatmaps", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 507, + 361 + ], + "score": 1.0, + "content": "correspond to the behavior of the resulting policy, which is still learning from the AGAC objective.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 358, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 507, + 372 + ], + "score": 1.0, + "content": "Looking at the figure, we can see that the strategies vary at each update with, for example, back-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "and-forth and back-to-start behaviors. Although there are no extrinsic reward, the strategies seem", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "to diversify from one update to the next. Finally, Fig. 7 in Appendix A.2 shows the state visitation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 392, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "heatmaps in a different configuration: when the agent has been trained on a singleton environment", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "in the MultiRoomN10S6 task without extrinsic reward. Same as previously, the agent is updated", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 415, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 426 + ], + "score": 1.0, + "content": "between each episode. Looking at the figure, we can make essentially the same observations as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "previously, with a noteworthy behavior in the fourth heatmap of the bottom row where it appears the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "agent went to the fourth room to remain inside it. Those episodes indicate that, although the agent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "sees the same environment repeatedly, the successive adversarial updates force it to continuously", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 458, + 271, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 271, + 471 + ], + "score": 1.0, + "content": "adapt its behavior and try new strategies.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 315, + 507, + 471 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 503, + 190, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 192, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 192, + 519 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "This paper introduced AGAC, a modification to the traditional actor-critic framework: an adversary", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "network is added as a third protagonist. The mechanics of AGAC have been discussed from a policy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "iteration point of view, and we provided theoretical insight into the inner workings of the proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "algorithm: the adversary forces the agent to remain close to the current policy while moving away", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "from the previous ones. In a nutshell, the influence of the adversary makes the actor conservatively", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 594, + 153, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 153, + 606 + ], + "score": 1.0, + "content": "diversified.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 538, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "In the experimental study, we have evaluated the adversarially-based bonus in VizDoom and em-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "pirically demonstrated its effectiveness and superiority compared to other relevant methods (some", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "benefiting from count-based exploration). Then, we have conducted several performance experiments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "using AGAC and have shown a significant performance improvement over some of the most popular", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "exploration methods (RIDE, AMIGo, Count, RND and ICM) on a set of various challenging tasks", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "from MiniGrid. These procedurally-generated environments have served another purpose which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "to validate the capacity of our method to generalize to unseen scenarios. In addition, the training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 479, + 700 + ], + "score": 1.0, + "content": "stability of our method has been studied, showing a greater but acceptable sensitivity for", + "type": "text" + }, + { + "bbox": [ + 480, + 690, + 485, + 698 + ], + "score": 0.65, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 688, + 505, + 700 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "adversarial bonus coefficient. Finally, we have investigated the exploration capabilities of AGAC in a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "reward-free setting where the agent demonstrated exhaustive exploration through various strategic", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 487, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 487, + 732 + ], + "score": 1.0, + "content": "choices, confirming that the adversary successfully drives diversity in the behavior of the actor.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 611, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 506, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 505, + 112 + ], + "score": 1.0, + "content": "Zafarali Ahmed, Nicolas Le Roux, Mohammad Norouzi, and Dale Schuurmans. 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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 269, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 269, + 160 + ], + "score": 1.0, + "content": "methods have a budget of 200M frames.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 126, + 505, + 160 + ] + }, + { + "type": "table", + "bbox": [ + 140, + 189, + 469, + 265 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 132, + 169, + 476, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 169, + 478, + 182 + ], + "spans": [ + { + "bbox": [ + 133, + 169, + 478, + 182 + ], + "score": 1.0, + "content": "Table 2: Final average performance of all methods on several MiniGrid environments.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 140, + 189, + 469, + 265 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 189, + 469, + 265 + ], + "spans": [ + { + "bbox": [ + 140, + 189, + 469, + 265 + ], + "score": 0.981, + "html": "
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Please note that, in all considered experiments,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 236, + 322 + ], + "score": 1.0, + "content": "the adversarial bonus coefficient", + "type": "text" + }, + { + "bbox": [ + 237, + 311, + 243, + 320 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "in Eq. 2 and 3 is linearly annealed throughout the training since it", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 319, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 319, + 506, + 334 + ], + "score": 1.0, + "content": "is mainly useful at the beginning of learning when the rewards have not yet been met. 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In order to go from one room to another, the agent must perform a specific action to open", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 398, + 200 + ], + "score": 1.0, + "content": "a door. 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In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "ObstructedMaze, keys are hidden in boxes, and doors are obstructed by balls the agent has to get out", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "of its way. 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ParameterValue
Horizon T2048
Nb. epochs4
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NonlinearityELU (Clevert et al., 2016)
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C c anneal schedule4·10-4 (4·10-5 in VizDoom)
linear
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ParameterValue
Horizon T2048
Nb. epochs4
Nb.minibatches8
Nb.frames stacked4
NonlinearityELU (Clevert et al., 2016)
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GAE parameter 入0.95
PPO clipping parameter e0.2
βv0.5
C c anneal schedule4·10-4 (4·10-5 in VizDoom)
linear
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We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 523, + 192, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 192, + 534 + ], + "score": 1.0, + "content": "recall the result here:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 538, + 402, + 569 + ], + "lines": [ + { + "bbox": [ + 208, + 538, + 402, + 569 + ], + "spans": [ + { + "bbox": [ + 208, + 538, + 402, + 569 + ], + "score": 0.94, + "content": "\\pi _ { k + 1 } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\pi } \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } ,", + "type": "interline_equation", + "image_path": "2024813b42e36b4d88b1449320b8e80f8860fdc726554bfff7e58be829a6bb7a.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 538, + 402, + 553.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 208, + 553.5, + 402, + 569.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 217, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 218, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 218, + 587 + ], + "score": 1.0, + "content": "with the objective function:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 593, + 474, + 608 + ], + "lines": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "spans": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "af279849052942bf74464168820472c5207cabac51d9533eeb152ae0a1351302.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 625, + 504, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 360, + 639 + ], + "score": 1.0, + "content": "Proof. 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We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 523, + 192, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 192, + 534 + ], + "score": 1.0, + "content": "recall the result here:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 511, + 505, + 534 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 538, + 402, + 569 + ], + "lines": [ + { + "bbox": [ + 208, + 538, + 402, + 569 + ], + "spans": [ + { + "bbox": [ + 208, + 538, + 402, + 569 + ], + "score": 0.94, + "content": "\\pi _ { k + 1 } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\pi } \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) \\propto \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } ,", + "type": "interline_equation", + "image_path": "2024813b42e36b4d88b1449320b8e80f8860fdc726554bfff7e58be829a6bb7a.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 538, + 402, + 553.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 208, + 553.5, + 402, + 569.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 217, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 218, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 218, + 587 + ], + "score": 1.0, + "content": "with the objective function:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 574, + 218, + 587 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 593, + 474, + 608 + ], + "lines": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "spans": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathcal { I } _ { \\mathrm { P I } } ( \\pi ) = \\mathbb { E } _ { s } \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ Q _ { \\pi _ { k } } ( s , a ) + c ( \\log \\pi _ { k } ( a | s ) - \\log \\pi _ { \\mathrm { a d v } } ( a | s ) ) - \\alpha \\log \\pi ( a | s ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "af279849052942bf74464168820472c5207cabac51d9533eeb152ae0a1351302.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 137, + 593, + 474, + 608 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 625, + 504, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 360, + 639 + ], + "score": 1.0, + "content": "Proof. We first consider a simpler optimization problem: a", + "type": "text" + }, + { + "bbox": [ + 360, + 626, + 471, + 638 + ], + "score": 0.89, + "content": "\\because \\operatorname { g m a x } _ { \\pi } \\langle \\pi , Q _ { \\pi _ { k } } \\rangle + \\alpha \\mathcal { H } ( \\pi )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 624, + 506, + 639 + ], + "score": 1.0, + "content": ", whose", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "solution is known (Vieillard et al., 2020a, Appendix A). The expression for the maximizer is the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 648, + 182, + 660 + ], + "spans": [ + { + "bbox": [ + 107, + 650, + 114, + 658 + ], + "score": 0.7, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 648, + 182, + 660 + ], + "score": 1.0, + "content": "-scaled softmax:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 624, + 506, + 660 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 658, + 351, + 693 + ], + "lines": [ + { + "bbox": [ + 260, + 658, + 351, + 693 + ], + "spans": [ + { + "bbox": [ + 260, + 658, + 351, + 693 + ], + "score": 0.94, + "content": "\\pi ^ { * } = \\frac { \\exp ( \\frac { Q _ { \\pi _ { k } } } { \\alpha } ) } { \\langle 1 , \\exp ( \\frac { Q _ { \\pi _ { k } } } { \\alpha } ) \\rangle } .", + "type": "interline_equation", + "image_path": "af0be1f1ed8a2d87a91e4d6238ccdfe701dfda388c64ed4ae25abc89583ea5e2.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 260, + 658, + 351, + 675.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 260, + 675.5, + 351, + 693.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 442, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 444, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 444, + 710 + ], + "score": 1.0, + "content": "We now turn towards the optimization problem of interest, which we can rewrite as:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 696, + 444, + 710 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 715, + 411, + 735 + ], + "lines": [ + { + "bbox": [ + 199, + 715, + 411, + 735 + ], + "spans": [ + { + "bbox": [ + 199, + 715, + 411, + 735 + ], + "score": 0.88, + "content": "\\underset { \\pi } { \\arg \\operatorname* { m a x } } \\langle \\pi , Q _ { \\pi _ { k } } + c \\left( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } \\right) \\rangle + \\alpha \\mathcal { H } ( \\pi ) .", + "type": "interline_equation", + "image_path": "b4274842b6d1fa563cf9fa451f48948e73a0d409b0a11240212631f8c058d056.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 199, + 715, + 411, + 735 + ], + "spans": [], + "index": 19 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 107 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 245, + 97 + ], + "score": 1.0, + "content": "By the simple change of variable", + "type": "text" + }, + { + "bbox": [ + 245, + 81, + 392, + 95 + ], + "score": 0.64, + "content": "\\tilde { Q } _ { \\pi _ { k } } = Q _ { \\pi _ { k } } + c ( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 80, + 506, + 97 + ], + "score": 1.0, + "content": ", we can reuse the previous", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 340, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 184, + 109 + ], + "score": 1.0, + "content": "solution (replacing", + "type": "text" + }, + { + "bbox": [ + 184, + 95, + 202, + 107 + ], + "score": 0.91, + "content": "Q _ { \\pi _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 94, + 216, + 109 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 217, + 93, + 235, + 107 + ], + "score": 0.9, + "content": "\\tilde { Q } _ { \\pi _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 94, + 340, + 109 + ], + "score": 1.0, + "content": "). 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With the simplification:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 80, + 506, + 109 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 112, + 420, + 142 + ], + "lines": [ + { + "bbox": [ + 189, + 112, + 420, + 142 + ], + "spans": [ + { + "bbox": [ + 189, + 112, + 420, + 142 + ], + "score": 0.93, + "content": "\\exp \\frac { Q _ { \\pi _ { k } } + c \\left( \\log \\pi _ { k } - \\log \\pi _ { \\mathrm { a d v } } \\right) } { \\alpha } = \\left( \\frac { \\pi _ { k } } { \\pi _ { \\mathrm { a d v } } } \\right) ^ { \\frac { c } { \\alpha } } \\exp \\frac { Q _ { \\pi _ { k } } } { \\alpha } ,", + "type": "interline_equation", + "image_path": "2bc203fd3a59825715a246ddb6bbf5a7ef0cabd45b556e775d2e4892c0516cd1.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 189, + 112, + 420, + 142 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 283, + 159 + ], + "lines": 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ParameterValue
Horizon T2048
Nb. epochs4
Nb.minibatches8
Nb.frames stacked4
NonlinearityELU (Clevert et al., 2016)
Discount γ0.99
GAE parameter 入0.95
PPO clipping parameter e0.2
βv0.5
C c anneal schedule4·10-4 (4·10-5 in VizDoom)
linear
βadv4·10-5
Adam stepsize m13·10-4
Adam stepsize m29.10-5 = 0.3· m1
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b/parse/train/l3YcqzaPlx0/l3YcqzaPlx0.md new file mode 100644 index 0000000000000000000000000000000000000000..af7c25f200cf12f4ea66c4546dc3f6b3b93ca2b8 --- /dev/null +++ b/parse/train/l3YcqzaPlx0/l3YcqzaPlx0.md @@ -0,0 +1,330 @@ +# NEIGHBOR2SEQ: DEEP LEARNING ON MASSIVE GRAPHS BY TRANSFORMING NEIGHBORS TO SEQUENCES + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Modern graph neural networks (GNNs) use a message passing scheme and have achieved great success in many fields. However, this recursive design inherently leads to excessive computation and memory requirements, making it not applicable to massive real-world graphs. In this work, we propose the Neighbor2Seq to transform the hierarchical neighborhood of each node into a sequence. This novel transformation enables the subsequent use of general deep learning operations, such as convolution and attention, that are designed for grid-like data. Therefore, our Neighbor2Seq naturally endows GNNs with the efficiency and advantages of deep learning operations on grid-like data by precomputing the Neighbor2Seq transformations. In addition, our Neighbor2Seq can alleviate the over-squashing issue suffered by GNNs based on message passing. We evaluate our method on a massive graph, with more than 111 million nodes and 1.6 billion edges, as well as several medium-scale graphs. Results show that our proposed method is scalable to massive graphs and achieves superior performance across massive and mediumscale graphs. + +# 1 INTRODUCTION + +Graph neural networks (GNNs) have shown effectiveness in many fields with rich relational structures, such as citation networks (Kipf & Welling, 2016; Velickovi ˇ c et al., 2018), social net- ´ works (Hamilton et al., 2017), drug discovery (Gilmer et al., 2017; Stokes et al., 2020), physical systems (Battaglia et al., 2016), and point clouds (Wang et al., 2019). Most current GNNs follow a message passing scheme (Gilmer et al., 2017; Battaglia et al., 2018), in which the representation of each node is recursively updated by aggregating the representations of its neighbors. Various GNNs (Li et al., 2016; Kipf & Welling, 2016; Velickovi ˇ c et al., 2018; Xu et al., 2019) mainly differ ´ in the forms of aggregation functions. + +Real-world applications usually generate massive graphs, such as social networks. However, message passing methods have difficulties in handling such large graphs as the recursive message passing mechanism leads to prohibitive computation and memory requirements. To date, sampling methods (Hamilton et al., 2017; Ying et al., 2018; Chen et al., 2018a;b; Huang et al., 2018; Zou et al., 2019; Zeng et al., 2020; Gao et al., 2018; Chiang et al., 2019; Zeng et al., 2020) and precomputing methods (Wu et al., 2019; Rossi et al., 2020; Bojchevski et al., 2020) have been proposed to scale GNNs on large graphs. While the sampling methods can speed up training, they might result in redundancy, still incur high computational complexity, lead to loss of performance, or introduce bias (see Section 2.2). Generally, precomputing methods can scale to larger graphs as compared to sampling methods as recursive message passing is still required in sampling methods. + +In this work, we propose the Neighbor2Seq that transforms the hierarchical neighborhood of each node to a sequence in a precomputing step. After the Neighbor2Seq transformation, each node and its associated neighborhood tree are converted to an ordered sequence. Therefore, each node can be viewed as an independent sample and is no longer constrained by the topological structure. This novel transformation from graphs to grid-like data enables the use of mini-batch training for subsequent models. As a result, our models can be used on extremely large graphs, as long as the Neighbor2Seq step can be precomputed. + +As a radical departure from existing precomputing methods, we consider the hierarchical neighborhood of each node as an ordered sequence. The order information corresponds to hops between nodes. As a result of our Neighbor2Seq transformation, generic deep learning operations for gridlike data, such as convolution and attention, can be applied in subsequent models. In addition, our Neighbor2Seq can alleviate the over-squashing issue (Alon & Yahav, 2020) suffered by current GNNs. Experimental results indicate that our proposed method can be used on a massive graph, where most current methods cannot be applied. Furthermore, our method achieves superior performance as compared with previous sampling and precomputing methods. + +# 2 ANALYSIS OF CURRENT GRAPH NEURAL NETWORK METHODS + +We start by defining necessary notations. A graph is formally defined as $\mathcal { G } = ( V , E )$ , where $V$ is the set of nodes and $E \subseteq V \times V$ is the set of edges. We use $n = | V |$ and $m = | E |$ to denote the numbers of nodes and edges, respectively. The nodes are indexed from 1 to $n$ . We consider a node feature matrix $\ b { X } \in \mathbb { R } ^ { n \times \breve { d } }$ , where each row $\pmb { x } _ { i } \in \mathbb { R } ^ { d }$ is the $d$ -dimensional feature vector associated with node $i$ . The topology information of the graph is encoded in the adjacency matrix $\ b { A } \in \mathbb { R } ^ { n \times n }$ , where $A _ { ( i , j ) } = 1$ if an edge exists between node $i$ and node $j$ , and $A _ { ( i , j ) } = 0$ otherwise. + +# 2.1 GRAPH NEURAL NETWORKS VIA MESSAGE PASSING + +There are two primary deep learning methods on graphs (Bronstein et al.); those are, spectral methods and spatial methods. The spectral method in Bruna et al. (2014) extends convolutional neural networks (LeCun et al., 1989) to the graph domain based on the spectrum of the graph Laplacian. The main limitation of spectral methods is the high complexity. ChebNet (Defferrard et al., 2016) and GCN (Kipf & Welling, 2016) simplify the spectral methods and can be understood from the spatial perspective. In this work, we focus on the analysis of the current mainstream spatial methods. Generally, most existing spatial methods, such as ChebNet (Defferrard et al., 2016), GCN (Kipf & Welling, 2016), GG-NN (Li et al., 2016), GAT (Velickovi ˇ c et al., 2018), and GIN (Xu et al., ´ 2019), can be understood from the message passing perspective (Gilmer et al., 2017; Battaglia et al., 2018). Specifically, we iteratively update node representations by aggregating representations from its immediate neighbors. These message passing methods have been shown to be effective in many fields. However, they have inherent difficulties when applied on large graphs due to their excessive computation and memory requirements, as described in Section 2.2. + +# 2.2 GRAPH NEURAL NETWORKS ON LARGE GRAPHS + +The above message passing methods are often trained in full batch. This requires the whole graph, i.e., all the node representations and edge connections, to be in memory to allow recursive message passing on the whole graph. Usually, the number of neighbors grows very rapidly with the increase of receptive field. Hence, these methods cannot be applied directly on large-scale graphs due to the prohibitive requirements on computation and memory. To enable deep learning on large graphs, two families of methods have been proposed; those are methods based on sampling and precomputing. + +To circumvent the recursive expansion of neighbors across layers, sampling methods apply GNNs on a sampled subset of nodes with mini-batch training. Sampling methods can be further divided into three categories. First, node-wise sampling methods perform message passing for each node in its sampled neighborhood. This strategy is first proposed in GraphSAGE (Hamilton et al., 2017), where neighbors are randomly sampled. This is extended by PinSAGE (Ying et al., 2018), which selects neighbors based on random walks. VR-GCN (Chen et al., 2018a) further proposes to use variance reduction techniques to obtain a convergence guarantee. Although these node-wise sampling methods can reduce computation, the remaining computation is still very expensive and some redundancy might have been introduced, as described in Huang et al. (2018). Second, layer-wise sampling methods sample a fixed number of nodes for each layer. In particular, FastGCN (Chen et al., 2018b) samples a fixed number of nodes for each layer independently based on the degree of each node. AS-GCN (Huang et al., 2018) and LADIES (Zou et al., 2019) introduce between-layer dependencies during sampling, thus alleviating the loss of information. Layer-wise sampling methods can avoid the redundancy introduced by node-wise sampling methods. However, the expensive sampling algorithms that aim to ensure performance may themselves incur high computational cost, as pointed out in Zeng et al. (2020). Third, graph-wise sampling methods build mini-batches on sampled subgraphs. Specifically, LGCN (Gao et al., 2018) proposes to leverage mini-batch training on subgraphs selected by Breadth-First-Search algorithms. ClusterGCN (Chiang et al., 2019) conducts mini-batch training on sampled subgraphs that are obtained by a graph clustering algorithm. GraphSAINT (Zeng et al., 2020) proposes to derive subgraphs by importance-sampling and introduces normalization techniques to eliminate biases. These graph-wise sampling methods usually have high efficiency. The main limitation is that the nodes in a sampled subgraph are usually clustered together. This implies that two distant nodes in the original graph usually cannot be feeded into the GNNs in the same mini-batch during training, potentially leading to bias in the trained models. + +The second family of methods for enabling GNNs training on large graphs are based on procomputing. Specifically, SGC (Wu et al., 2019) removes the non-linearity between GCN layers, resulting in a simplification as $Y = \mathrm { s o f t m a x } ( \hat { A } ^ { L } X W )$ . In this formulation, $\hat { \pmb { A } } = \tilde { \pmb { D } } ^ { - \frac 1 2 } \tilde { \pmb { A } } \tilde { \pmb { D } } ^ { - \frac 1 2 }$ is the symmetrically normalized adjacency matrix, ${ \tilde { A } } = A + I$ is the adjacency matrix with self-loops, $\tilde { D }$ is the corresponding diagonal node degree matrix with $\begin{array} { r } { \tilde { D } _ { ( i , i ) } = \mathrm { \bar { \sum } } _ { j } \tilde { A } _ { ( i , j ) } } \end{array}$ , $L$ is the size of receptive field (i.e., the number of considered neighboring hops), which is the same as a $L$ -layer GCN, $\mathbf { Y }$ is the output of the softmax classifier. Since there is no learnable parameters in $\hat { A } ^ { L } X$ , this term can be precomputed as a feature pre-processing step. Similarly, SIGN (Rossi et al., 2020) applies an inception-like model to the precomputed features ${ \hat { A } } ^ { \ell } X$ for $\ell \in \{ 1 , \cdots , L \}$ , where $L$ is the predefined size of receptive field. Instead of precomputing the smoothing features as in SGC and SIGN, PPRGo (Bojchevski et al., 2020) extends the idea of PPNP (Klicpera et al., 2018) by approximately precomputing the personalized PageRank (Page et al., 1999) matrix, thereby enabling model training on large graphs using mini-batches. Generally, the precomputing methods can scale to larger graphs than sampling methods because the latter still needs to perform the recursive message passing during training. Differing from these precomputing methods, we consider the hierarchical neighborhood of each node as an ordered sequence, thus retaining the useful information about hops between nodes and enabling subsequent powerful and efficient operations. + +# 3 THE PROPOSED NEIGHBOR2SEQ METHOD AND ANALYSIS + +In this section, we describe our proposed method, known as Neighbor2Seq, which transforms the hierarchical neighborhood of each node into an ordered sequence, thus enabling the subsequent use of general deep learning operations. We analyze the scalability of our method (See Section 3.5) and describe how our method can alleviate the over-squashing issue suffered by current message passing methods (See Section 3.6). + +# 3.1 OVERVIEW + +As described in Section 2.1, existing message passing methods recursively update each node’s representation by aggregating information from its immediate neighbors. Hence, what these methods aim at capturing for each node is essentially its corresponding hierarchical neighborhood, i.e., the neighborhood tree rooted at current node, as illustrated in Figure 1 (b). In this work, we attempt to go beyond the message passing scheme to overcome the limitations mentioned in Section 2. We propose to capture the information of this hierarchical neighborhood by transforming it into an ordered sequence, instead of recursively squashing it into a fixed-length vector. Our proposed method is composed of three steps. First, we transform a neighborhood to a sequence for each node. Second, we apply a normalization technique to the derived sequence features. Third, we use general deep learning operations, i.e., convolution and attention, to learn from these sequence features and then make predictions for nodes. In the following, we describe these three steps in detail. + +# 3.2 NEIGHBOR2SEQ: TRANSFORMING NEIGHBORHOODS TO SEQUENCES + +The basic idea of Neighbor2Seq is to transform the hierarchical neighborhood of each node to an ordered sequence by integrating the features of nodes in each layer of the neighborhood tree. Following the notations defined in Section 2, we let $z _ { 0 } ^ { i } , z _ { 1 } ^ { i } , \cdots , z _ { L } ^ { i }$ denote the resulting sequence for node $i$ , where $L$ is the height (i.e., the number of hops we consider) of the neighborhood tree rooted at node $i$ . $z _ { \ell } ^ { i } \in \mathbb { R } ^ { d }$ denotes the $\ell$ -th feature of the sequence. The length of the resulting sequence for each node is $L + 1$ . Formally, for each node $i$ , our Neighbor2Seq can be expressed as + +![](images/875bf8963e4d313f0401ac7c554728624d2b6b93f1de7b23e96dfc7730e414ea.jpg) +Figure 1: (a) An illustration of the original graph. The current node is denoted as two concentric circles. (b) Message passing in the neighborhood tree. (c) Our proposed Neighbor2Seq. (d) Our proposed models: Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn. + +$$ +z _ { \ell } ^ { i } = \sum _ { j = 1 } ^ { n } w ( i , j , \ell ) { \pmb x } _ { j } , \quad \forall \ell \in \{ 0 , 1 , 2 , \cdot \cdot \cdot , L \} , +$$ + +where $w ( i , j , \ell )$ denotes the number of walks with length $\ell$ between node $i$ and $j , n$ is the number of nodes in the graph. We define $w ( i , j , 0 )$ as 1 for $j = i$ and 0 otherwise. Hence, $z _ { 0 } ^ { i }$ is the original node feature $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . Intuitively, $\boldsymbol { z } _ { \ell } ^ { i }$ is obtained by computing a weighted sum of features of all nodes with walks of length $\ell$ to $i$ , and the numbers of qualified walks are used as the corresponding weights. Our Neighbor2Seq is illustrated in Figure 1 (c). Note that the derived sequence has meaningful order information, indicating the hops between nodes. After we obtain ordered sequences from the original hierarchical neighborhoods, we can use generic deep learning operations to learn from these sequences, as detailed below. + +# 3.3 NORMALIZATION + +Since the number of nodes in the hierarchical neighborhood grows exponentially as the hop number increases, different layers in the neighborhood tree have drastically different numbers of nodes. Hence, feature vectors of a sequence computed by Equation (1) have very different scales. In order to make the subsequent learning easier, we propose a layer to normalize the sequence features. We use a normalization technique similar to layer normalization (Ba et al., 2016). In particular, each feature of a sequence is normalized based on the mean and the standard deviation of its own feature values. Formally, our normalization process for each node $i$ can be written as + +$$ +\begin{array} { r l } & { { \boldsymbol y } _ { \ell } ^ { i } = W _ { \ell } \boldsymbol z _ { \ell } ^ { i } , \quad { \boldsymbol o } _ { \ell } ^ { i } = \frac { \boldsymbol y _ { \ell } ^ { i } - { \boldsymbol \mu } _ { \ell } ^ { i } } { \sigma _ { \ell } ^ { i } } \odot \gamma _ { \ell } + \beta _ { \ell } , \quad \forall \ell \in \{ 0 , 1 , 2 , \cdots , L \} } \\ & { \boldsymbol \mu _ { \ell } ^ { i } = \frac { 1 } { d ^ { \prime } } \displaystyle \sum _ { c = 1 } ^ { d ^ { \prime } } { \boldsymbol y } _ { \ell } ^ { i } [ \boldsymbol c ] , \quad { \boldsymbol \sigma } _ { \ell } ^ { i } = \sqrt { \frac { 1 } { d ^ { \prime } } \displaystyle \sum _ { c = 1 } ^ { d ^ { \prime } } ( { \boldsymbol y } _ { \ell } ^ { i } [ \boldsymbol c ] - { \boldsymbol \mu } _ { \ell } ^ { i } ) ^ { 2 } } . } \end{array} +$$ + +We first apply a linear transformation $W _ { \ell } \in \mathbb { R } ^ { d ^ { \prime } \times d }$ to produce a low-dimensional representation $\boldsymbol { y } _ { \ell } ^ { i } \in \mathbb { R } ^ { d ^ { \prime } }$ for the $\ell$ -th feature of the sequence, since the original feature dimension $d$ is usually large. $\mu _ { \ell } ^ { i } \in \mathbb { R }$ and $\sigma _ { \ell } ^ { i } \in \mathbb { R }$ are the mean and standard deviation of the corresponding representation $\boldsymbol { y } _ { \ell } ^ { i }$ . $\gamma _ { \ell } \in \mathbb { R } ^ { d ^ { \prime } }$ and $\beta _ { \ell } \in \mathbb { R } ^ { d ^ { \prime } }$ denote the learnable affine transformation parameters. $\odot$ denotes the element-wise multiplication. Note that the learnable parameters in this normalization layer is associated with $\ell$ , implying that each feature of the sequence is normalized separately. Using this normalization layer, we obtain the normalized feature vector $o _ { \ell } ^ { i } \in \mathbb { R } ^ { d ^ { \prime } }$ for every $\ell \in \{ 0 , 1 , 2 , \cdot \cdot \cdot , L \}$ . + +# 3.4 NEIGHBOR2SEQ $^ +$ CONV AND NEIGHBOR2SEQ $^ +$ ATTN + +After obtaining an ordered sequence for each node, we can view each node in the graph as a sequence of feature vectors. We can use general deep learning techniques to learn from these sequences. In this work, we propose two models, namely Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn, in which convolution and attention are applied on the sequences of each node. + +As illustrated in Figure 1 (d), Neighbor2Seq+Conv applies a 1-D convolutional neural network to the sequence features and then use an average pooling to yield a representation for the sequence. Formally, for each node $i$ , + +$$ +\left( \hat { \sigma } _ { 0 } ^ { i } , \hat { \sigma } _ { 1 } ^ { i } , \cdots , \hat { \sigma } _ { L } ^ { i } \right) = { \bf C } { \bf N } { \bf N } \left( o _ { 0 } ^ { i } , o _ { 1 } ^ { i } , \cdots , o _ { L } ^ { i } \right) , \quad r ^ { i } = { \frac { 1 } { L + 1 } } \sum _ { \ell = 0 } ^ { L } \hat { \sigma } _ { \ell } ^ { i } , +$$ + +where $\mathrm { C N N } ( \cdot )$ denotes a 1-D convolutional neural network. $r ^ { i }$ denotes the obtained representation of node $i$ that is used as the input to a linear classifier to make predictions for this node. Specifically, we implement $\mathrm { C N N } ( \cdot )$ as a 2-layer convolutional neural network composed of two 1-D convolutions. The kernel size is set according to the length of input sequence. The activation function between layers is ReLU (Krizhevsky et al., 2012). + +Incorporating attention is another natural idea to learn from sequences. As shown in Figure 1 (d), Neighbor2Seq+Attn uses an attention mechanism (Bahdanau et al., 2015) to integrate sequential feature vectors in order to derive a representation. Unlike convolutional neural networks, the vanilla attention mechanism cannot make use of the order of the sequence. Hence, we add positional encodings (Vaswani et al., 2017) to the features such that the position information of different features in the sequence can be incorporated. Formally, for each node $i$ , we add positional encoding for each feature in the sequence as + +$$ +k _ { \ell } ^ { i } = o _ { \ell } ^ { i } + p _ { \ell } ^ { i } , \quad p _ { \ell } ^ { i } [ m ] = \left\{ \begin{array} { l l } { { \sin \left( \displaystyle { \frac { \ell } { 1 0 0 0 0 ^ { \frac { 2 n } { d ^ { \prime } } } } } \right) } } & { { m = 2 n } } \\ { { \cos \left( \displaystyle { \frac { \ell } { 1 0 0 0 0 ^ { \frac { 2 n } { d ^ { \prime } } } } } \right) } } & { { m = 2 n + 1 } } \end{array} \right. . +$$ + +The positional encoding for $\ell$ -th feature of node $i$ is denoted as $\pmb { p } _ { \ell } ^ { i } \in \mathbb { R } ^ { d ^ { \prime } }$ . $m \in \{ 1 , 2 , \cdots , d ^ { \prime } \}$ is the dimensional index. Intuitively, a position-dependent vector is added to each feature such that the order information can be captured. Then we use the attention mechanism with learnable query (Yang et al., 2016) to combine these sequential feature vectors to obtain the final representations $r ^ { i }$ for each node $i$ . Formally, + +$$ +\pmb { r } ^ { i } = \sum _ { \ell = 0 } ^ { L } \alpha _ { \ell } ^ { i } \pmb { k } _ { \ell } ^ { i } , \quad \alpha _ { \ell } ^ { i } = \frac { \exp ( \pmb { k } _ { \ell } ^ { i ^ { T } } \pmb { q } ) } { \sum _ { \ell = 0 } ^ { L } \exp ( \pmb { k } _ { \ell } ^ { i ^ { T } } \pmb { q } ) } . +$$ + +$\boldsymbol { q } \in \mathbb { R } ^ { d ^ { \prime } }$ is the learnable query vector that is trained along with other model parameters. The derived representation $r ^ { i }$ will be taken as the input to a linear classifier to make prediction for node $i$ . + +# 3.5 ANALYSIS OF SCALABILITY + +Precomputing Neighbor2Seq. A well-known fact is that the value of $w ( i , j , \ell )$ in Equation (1) can be obtained by computing the power of the original adjacency matrix $\pmb { A }$ . Following GCN, we add self-loops to make each node connected to itself. Concretely, $w ( i , j , \ell ) = \tilde { A } _ { ( i , j ) } ^ { \ell }$ Hence, the Neighbor2Seq can be implemented by computing the matrix multiplications $\tilde { A } ^ { \ell } X$ for $\forall \ell \in$ $\{ 0 , 1 , 2 , \bar { \cdot } \cdot \cdot , L \}$ . Since there is no learnable parameters in the Neighbor2Seq step, these matrix multiplications can be precomputed sequentially for large graphs on CPU platforms with large memory. + +This can be easily precomputed because the matrix $\tilde { A }$ is usually sparse. For extremely large graphs, this precomputation can even be performed on distributed systems. + +Enabling mini-batch training. After we obtain the precomputed sequence features, each node in the graph corresponds to a sequence of feature vectors. Therefore, each node can be viewed as an independent sample. That is, we are no longer restricted by the original graph connectivity anymore. Then, we can randomly sample from all the training nodes to conduct mini-batch training. This is more flexible and unbiased than sampling methods as reviewed in Section 2.2. Our mini-batches can be randomly extracted over all nodes, opening the possibility that any pair of nodes can be sampled in the same mini-batch. In contrast, mini-batches in sampling methods are usually restricted by the fixed sampling strategies. This advantage opens the door for subsequent model training on extremely large graphs, as long as the corresponding Neighbor2Seq step can be precomputed. + +Computational complexity comparison. We compare our methods with several existing sampling and precomputing methods in terms of computational complexity. We let $L$ denote the number of hops we consider. For simplicity, we assume the feature dimension $d$ is fixed for all layers. For sampling methods, $s$ is the number of sampled neighbors for each node. The computation of Neighbor2Seq $^ +$ Conv mainly lies in the linear transformation (i.e., $\mathcal { O } ( L d ^ { 2 } \bar { n } ) )$ in the normalization step and the 1-D convolutional neural networks (i.e., + +Table 1: Comparison of computational complexity for precomputing and forward pass corresponding to an entire epoch. + +
MethodPrecomputingForward Pass
GCNO(Ldm + Ld²n)
GraphSAGEO(sLn)O(sLd²n)
ClusterGCNO(m)O(Ldm + Ld²n)
GraphSAINTO(sn)O(Ldm +Ld²n)
SGCO(Ldm)O(d²n)
SIGNO(Ldm)O(Ld²n)
Neighbor2Seq+ConvO(Ldm)O(Ld²+Lkd²)n)
Neighbor2Seq+AttnO(Ldm)O(Ld²+Ld)n)
+ +$O ( L k d ^ { 2 } n )$ , where $k$ is the kernel size). Hence, the computational complexity for the forward pass of Neighbo $_ { 1 2 \mathrm { S e q + C o n v } }$ is $\mathcal { O } ( ( L d ^ { 2 } + L k d ^ { 2 } ) n )$ . Neighbor2Seq+Attn has a computational complexity of $\mathcal { \bar { O } } ( ( L d ^ { 2 } + \bar { L } d ) n )$ because the attention mechanism is more efficient than 1-D convolutional neural networks. As shown in Table 1, the forward pass complexities of precomputing methods, including our Neighbor2Seq+Conv and Neighbor2Seq+Attn, are all linear with respect to the number of nodes $n$ and do not depend on the number of edges $m$ . Hence, the training processes of our models are computationally efficient. + +# 3.6 ALLEVIATING THE OVER-SQUASHING ISSUE + +An inherent problem in message passing methods is known as the over-squashing (Alon & Yahav, 2020). In particular, recursively propagating information between neighbors creates a bottleneck because the number of nodes in the receptive field grows exponentially with the number of layers. This bottleneck causes the over-squashing issue; that is, information from the exponentially-growing receptive field is compressed into fixed-length vectors. Consequently, message passing methods fail to capture the message flowing from distant nodes and performs poorly when long-range information is essential for the prediction tasks. Note that the over-squashing issue is not identical to the oversmoothing issue. Over-smoothing is related to the phenomenon that node representations converge to indistinguishable limits when the number of layers increases (Li et al., 2018; Wu et al., 2019; NT & Maehara, 2019; Liu et al., 2020a; Oono & Suzuki, 2020; Cai & Wang, 2020; Chen et al., 2020). The virtual edges added in Gilmer et al. (2017) and recent non-local aggregations (Pei et al., 2020; Liu et al., 2020b) can be viewed as attempts to alleviate the over-squashing issue by incorporating distant nodes. Another study (Ma et al., 2019) considers message passing along all possible paths between two nodes, instead of propagating information between neighbors. + +Our Neighbor2Seq can alleviate the over-squashing issue because we transform the exponentiallygrowing nodes in hierarchical neighborhoods into an ordered sequence, instead of recursively squashing them into a fixed-size vector. With our Neighbor2Seq, capturing long-range information on graphs becomes similar to achieving this on sequence data, such as texts. + +# 4 DISCUSSIONS + +# 4.1 INFORMATION LOSS IN NEIGHBOR2SEQ + +As shown in Figure 1 (c), Neighbor2Seq obtains the sequence by integrating features of nodes in each layer of the neighborhood tree. This transformation may lose the cross-layer dependency information in the tree. Specifically, the Neighbor2Seq ignores the identities of nodes that each walk passes through and only considers what are the nodes in each layer of the neighborhood tree. Nevertheless, this information can neither be captured by message passing methods because the aggregation is usually permutation-invariant. This implies that messages from different neighbors cannot be distinguished, as pointed in Pei et al. (2020). According to our experimental results in Table 5, our models without this information can outperform message passing methods, such as GCN. It is intriguing to have an in-depth exploration of whether such information is useful and how it can be captured. + +# 4.2 RELATIONS WITH THE WEISFEILER-LEHMAN HIERARCHY + +As shown in Xu et al. (2019), most of current GNNs are at most as powerful as the WeisfeilerLehman (WL) graph isomorphism test (Weisfeiler & Lehman, 1968) in distinguishing graph structures. Our Neighbor2Seq is still under the WL hierarchy because the neighborhood tree used to obtain the sequence is indeed the one that the WL test uses to distinguish different graphs. We would be interested in exploring how Neighbor2Seq can be extended to go beyond the WL hierarchy as a future direction. + +# 4.3 BRIDGING THE GAP BETWEEN GRAPH AND GRID-LIKE DATA + +The main difference between graph and grid-like data lies in the notion and properties of locality. Specifically, the numbers of neighbors differ for different nodes, and there is no order information among the neighbors of a node in graphs. These are the main obstacles preventing the use of generic deep learning operations on graphs. Our Neighbor2Seq is an attempt to bridge the gap between graph and grid-like data. Base on our Neighbor2Seq, many effective strategies for grid-like data can be naturally transferred to graph data. These include self-supervised learning and pre-training on graphs (Hu et al., 2019; Velickovic et al., 2019; Sun et al., 2019; Hassani & Khasahmadi, 2020; You et al., 2020; Hu et al., 2020b; Qiu et al., 2020; Jin et al., 2020). + +We notice an existing work AWE Ivanov & Burnaev (2018) which also embed the information in graph as a sequence. In order to avoid confusion, we make a clarification about the fundamental and significant differences between AWE and our Neighbor2Seq. First, AWE produces a sequence embedding for the entire graph, while our Neighbor2Seq yields a sequence embedding for each node in the graph. Second, each element in the obtained sequence in AWE is the probability of an anonymous walk embedding. In our Neighbor2Seq, each feature vector in the obtained sequence for one node is computed by summing up the features of all nodes in the corresponding layer of the neighborhood tree. This point distinguishes these two methods fundamentally. + +# 5 EXPERIMENTAL STUDIES + +# 5.1 EXPERIMENTAL SETUP + +Datasets. We evaluate our proposed models on 1 massive-scale graph and 4 medium-scale graphs using node classification tasks. The massive-scale graph ogbn-papers100M provided by the Open Graph Benchmark (OGB) (Hu et al., 2020a) is the existing largest benchmark dataset for node classification. Medium-scale graphs include ogbn-products (Hu et al., 2020a), Reddit (Hamilton et al., 2017), Yelp Zeng et al. (2020), and Flickr Zeng et al. (2020). These tasks cover inductive and transductive settings. The statistics of these datasets are summarized in Table 2. The detailed description of these datasets are provided in Appendix A.1. + +Table 2: Statistics of datasets. “m” denotes multi-label classification. + +
DatasetScale#Nodes#EdgesAvg. Deg.#Features#ClassesTrain/Val/Test
ogbn-papers100MMassive111,059,9561,615,685,872291281720.78/0.08/0.14
ogbn-productsMedium2,449,02961,859,14051100470.08/0.02/0.90
RedditMedium232,96511,606,91950602410.66/0.10/0.24
YelpMedium716,8576,997,41010300100(m)0.75/0.10/0.15
FlickrMedium89,250899,7561050070.50/0.25/0.25
+ +Implementation. We implemented our methods using Pytorch (Paszke et al., 2017) and Pytorch Geometric (Fey & Lenssen, 2019). For our proposed methods, we conduct the precomputation on + +the CPU, after which we train our models on a GeForce RTX 2080 Ti GPU. We perform a grid search for the following hyperparameters: number of hops $L$ , batch size, learning rate, hidden dimension $d ^ { \prime }$ , dropout rate, weight decay, and convolutional kernel size $k$ . The chosen hyperparameters for our Neighbor2Seq+Conv and Neighbor2Seq+Attn are summarized in Appendix A.2 for reproducibility. + +# 5.2 RESULTS ON MASSIVE-SCALE GRAPHS + +Since ogbn-papers100M is a massive graph with more than $1 1 1 \ \mathrm { m i l }$ - lion nodes and 1.6 billion edges, most existing methods have difficulty handling such a graph. We consider three baselines that have available results evaluated by OGB: Multilayer Perceptron (MLP), Node2Vec (Grover & Leskovec, 2016), and SGC (Wu et al., 2019). The results under transductive setting is reported in Table 3. Following OGB, we report accuracies for all models on training, validation, and test sets. The previous state-of-theart result on ogbn-papers100M is ob + +Table 3: Results on ogbn-papers100M in terms of classification accuracy (in percent). The reported accuracy is averaged over 10 random runs. Note that existing sampling methods cannot scale to this massive graph. During precomputation, both SGC and our models have to randomly remove $4 0 \%$ edges to avoid a memory overflow on CPU. This implies that the performance could be further improved if more advanced precomuting platform is used. + +
MethodTrainingValidationTest
MLP54.84±0.4349.60±0.2947.24±0.31
Node2vec158.07±0.2855.60±0.23
SGC67.54±0.4366.48±0.2063.29±0.19
Neighbor2Seq+Conv69.87±0.8167.46±0.1664.04±0.22
Neighbor2Seq+Attn68.83±0.3066.90±0.1063.59±0.17
+ +tained by the precomputing method SGC. Our models outperform the baselines consistently in terms of training, validation, and test, which demonstrates the expressive power and the generalization ability of our method on massive graphs. + +# 5.3 RESULTS ON MEDIUM-SCALE GRAPHS + +We also evaluate our models on medium-scale graphs, thus enabling comparison with more existing works. We conduct transductive learning on ogbn-products, a medium-scale graph from OGB. We also conduct inductive learning on Reddit, Yelp, and Flickr, which are frequently used for inductive learning by the community. The following baselines are considered: MLP, Node2Vec (Grover & Leskovec, 2016), GCN (Kipf & Welling, 2016), SGC Wu et al. (2019), GraphSAGE (Hamilton et al., 2017), FastGCN (Chen et al., 2018b), VR-GCN (Chen et al., 2018a), AS-GCN (Huang et al., 2018), ClusterGCN (Chiang et al., 2019), GraphSAINT (Zeng et al., 2020), and SIGN (Rossi et al., 2020). + +Table 4: Results on ogbn-products in terms of classification accuracy (in percent). The reported accuracy is averaged over 10 random runs. Obtaining the results of GCN requires a GPU with 33GB of memory. + +
MethodTrainingValidationTest
MLP84.03±0.9375.54±0.1461.06±0.08
Node2vec93.39±0.1090.32±0.0672.49±0.10
GCN93.56±0.0992.00±0.0375.64±0.21
GraphSAGE92.96±0.0791.70±0.0978.70±0.36
ClusterGCN93.75±0.1392.12±0.0978.97±0.33
GraphSAINT92.71±0.1491.62±0.0879.08±0.24
SGC92.60±0.1091.19±0.0672.46±0.27
SIGN96.92±0.4693.10±0.0877.60±0.13
Neighbor2Seq+Conv95.32±0.1092.92±0.0579.67±0.16
Neighbor2Seq+Attn92.82±0.1492.20±0.0279.35±0.17
+ +Table 5: Results for inductive learning on three datasets in terms of F1-micro score. The reported score is averaged over 10 random runs. The results of baselines are partially obtained from Zeng et al. (2020); Rossi et al. (2020). + +
MethodRedditFlickrYelp
GCN0.933±0.0000.492±0.0030.378±0.001
FastGCN0.924±0.0010.504±0.0010.265±0.053
VR-GCN0.964±0.0010.482±0.0030.640±0.002
AS-GCN0.958±0.0010.504±0.002-
GraphSAGE0.953±0.0010.501±0.0130.634±0.006
ClusterGCN0.954±0.0010.481±0.0050.609±0.005
GraphSAINT0.966±0.0010.511±0.0010.653±0.003
SGC0.949±0.0000.502±0.0010.358±0.006
SIGN0.968±0.0000.514±0.0010.631±0.003
Neighbor2Seq+Conv0.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn0.967±0.0000.523±0.0020.647±0.001
+ +The ogbn-products dataset is challenging because the splitting is not random. The splitting procedure is more realistic. The nodes (i.e., products) are sorted according to their sales ranking and the top $8 \%$ nodes are used for training, next $2 \%$ for validation, and the rest $9 0 \%$ for testing. This matches the real-world application where manual labeling is prioritized to important nodes and models are subsequently used to make prediction on less important nodes. Hence, ogbn-products is an ideal benchmark dataset to improve out-of-distribution prediction. As shown in Table 4, our Neighbor2Seq+Conv and Neighbor2Seq+Attn outperfom baselines on test set (i.e., $9 0 \%$ nodes), which further demonstrates the generalization ability of our method. + +The results on inductive tasks are summarized in Table 5. On Reddit, our models perform better than all sampling methods and achieve the competitive result as SIGN. On Flickr, our models obtain significantly better results. Specifically, our Neighbor2Seq $+$ Conv outperforms the previous state-ofthe-art models by an obvious margin. Although our models perform not as good as GraphSAINT on $Y e l p$ , we outperform other sampling methods and the precomputing model SIGN consistently on this dataset. + +# 5.4 COMPARISONS OF COMPUTIONAL EFFICIENCY + +In order to show the computational efficiency, we conduct an empirical comparison with existing methods in terms of time comsuming during preprocessing, training, and inference. We consider the following representative sampling methods and precom + +Table 6: Computational efficiency in terms of preprocessing, training (per epoch), and inference times (in seconds) on ogbn-products. The reported time is averaged over 10 runs. + +
MethodPreprocessing (↓)Training (↓)Inferenece (↓)Test Accuracy (↑)
ClusterGCN44.15±0.7711.87±0.8487.03±0.2478.97±0.33
GraphSAINT80.78±3.54.29±0.48107.26±0.9479.08±0.24
SGC153.36±3.60.15±0.011.08±0.0172.46±0.27
SIGN151.47±3.51.22±0.022.93±0.0677.60±0.13
Neighbor2Seq+Conv153.42±3.24.09±0.1231.52±1.4479.67±0.16
Neighbor2Seq+Attn153.42±3.22.67±0.0831.24±0.6179.35±0.17
+ +puting methods: ClusterGCN Chiang et al. (2019), GraphSAINT Zeng et al. (2020), SGC Wu et al. (2019), and SIGN Rossi et al. (2020). The comparison is performed on ogbn-products and the similar trend can be observed on other datasets. As demonstrated in Table 6, our approaches, like existing precomputing methods, are more computationally efficient than sampling methods in terms of training and inference. Compared with existing precomputing methods, our methods achieve a better balance between performance and efficiency. + +# 5.5 ABLATION STUDY ON ORDER INFORMATION + +Table 7: Comparison of models with and without capturing order information. Neighbor2Seq+Attn w/o PE denotes the Neighbor2Seq+Attn without adding positional encoding. + +
ModelOrderogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Conv：64.04±0.2279.67±0.160.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn63.59±0.1779.35±0.170.967±0.0000.523±0.0020.647±0.001
Neighbor2Seq+Attn w/o PEX63.61±0.0978.54±0.250.965±0.0000.521±0.0030.646±0.001
+ +Intuitively, the order information in the sequence obtained by Neighbor2Seq indicates the hops between nodes. Hence, we conduct an ablation study to verify the significance of this order information. We remove the positional encoding in Neighbor2Seq $+ .$ Attn, leading to a model without the ability to capture the order information. The comparison is demonstrated in Table 7. Note that Neighbor2Seq $^ +$ Attn and Neighbor2Seq $+ .$ Attn w/o PE have the same number of parameters. Hence, Comparing the results of these two models, we can conclude that the order information is usually necessary. Neighbor2Seq+Conv and Neighbor2Seq $+ .$ Attn both can capture the order information. There are two possible reasons why Neighbor2Seq $\Vdash$ Conv performs better. First, Neighbor2Seq+Conv has more learnable parameters than Neighbor2Se+Attn, which only has a learnable query. Second, the convolutional neural network in Neighbor2Seq $^ +$ Conv can additionally investigate the dependencies between feature dimensions because each feature dimension of the output depends on every feature dimension of the input. + +# 6 CONCLUSIONS AND OUTLOOK + +In this work, we propose Neighbor2Seq, for transforming the heirarchical neighborhoods to ordered sequences. Neighbor2Seq enables the subsequent use of powerful general deep learning operations, leading to the proposed Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn. Our models can be deployed on massive graphs and trained efficiently. The extensive expriments demonstrate the scalability and the promising performance of our method. As discussed in Section 4, based on our Neighbor2Seq, several significant directions can be further explored in the future research. + +# REFERENCES + +Uri Alon and Eran Yahav. 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In International Conference on Machine Learning, 2020. + +Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. Graphsaint: Graph sampling based inductive learning method. In International Conference on Learning Representations, 2020. + +Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. Layer-dependent importance sampling for training deep and large graph convolutional networks. In Advances in Neural Information Processing Systems, pp. 11249–11259, 2019. + +# A APPENDIX + +# A.1 DATASET DESCRIPTIONS + +ogbn-papers100M (Hu et al., 2020a) is the existing largest benckmark dataset for node classification. It is a directed citation graph of 111 million papers indexed by Microsoft Academic Graph (MAG) (Wang et al., 2020). For simplicity, it is converted to an undirected graph in baselines and our method. Each node is a paper and each directed edge indicates that one paper cites another one. Each node is associated with a 128-dimensional feature vector obtained by averaging the word2vec (Mikolov et al., 2013) embeddings of words in its title and abstract. Among the node set, approximately 1.5 millione of them are ARXIV papers, each of which has a label with one of ARXIV’s subject areas. The rest nodes (i.e., non-ARXIV papers) are not associated with label information. The task is to leverage the entire citation graph to infer the labels of the ARXIV papers. The time-based splitting is used as the splitting strategy. To be more specifical, the training nodes are all ARXIV papers published until 2017, while the validation nodes are the ARXIV papers published in 2018, and the ARXIV papers published since 2019 are treated as test nodes. + +ogbn-products (Hu et al., 2020a) is an undirected Amazon product co-purchasing graph (Bhatia et al., 2016). Nodes denote products and edges between two nodes indicate that the corresponding products are purchased together. Node features are derived by extracting bag-of-words representations from the product descriptions. Further, a Principal Component Analysis is applied to these features to reduce the dimension to 100. The task is to predict the category of a product. A realistic splitting scheme is used in this data. Specifically, the products are firstly sorted according to their sales ranking, and then the top $1 0 \%$ products are used for training, next $\mathrm { \bar { 2 } \% }$ for validation, and the rest for testing. This strategy matches the real-world situation where manual labeling is prioritized to important nodes and models are subsequently deployed to predict the less important ones. + +Reddit (Hamilton et al., 2017), Yelp (Zeng et al., 2020), and Flickr (Zeng et al., 2020) are widely used datasets for inductive learning. During training, only the node features of training nodes and the edges between training nodes are visible. Reddit is a social netowork extracted from Reddit forum. Nodes represent posts and edges between two posts indicate the same user comments on both posts. Node features are fromed by GloVe CommonCrawl word vectors Pennington et al. (2014) of the posts. The task is to predict which community different posts belong to. The splitting is also time-based. Yelp is a social netowork constructed from Yelp website. Nodes are users and edges between two nodes indicate they are friends. Node features of users are obtained by the word2vec embeddings of their corresponding reviews. The task is to predict the categories of businesses reviewed by different users, which is multi-label classification task. Flickr is a social network based on Flickr, a photo sharing website. Nodes represent images and there is an edge between two nodes if two images share some common properties. The node features are fromed by the bag-of-words representations of the images. The task is to predict the category each image belongs to. + +# A.2 HYPERPARAMETER CONFIGURATIONS + +We conduct a grid search for hyperparameters. Table 8 summarizes the chosen hyperparameters for our models. + +Table 8: The chosen hyperparameters for our models on all datasets. + +
ModelHyperparameterogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Convnumber of hopsL373102
hidden dimension d'512512256256512
convolutional kernel size k57573
learning rate5e-42e-58e-58e-45e-4
batch size122886432768245768192
weight decay5e-55e-505e-50
dropout rate0.250.50.50.50
Neighbor2Seq+Attnnumber of hopsL373102
hidden dimension d'512512256256512
learning rate5e-41e-32e-32e-35e-4
batch size122883072327682568192
weight decay5e-65e-505e-50
dropout rate0.250.50.50.50
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However, this recursive design inherently leads to excessive computation and memory requirements, making it not applicable to massive real-world graphs. In this work, we propose the Neighbor2Seq to transform the hierarchical neighborhood of each node into a sequence. This novel transformation enables the subsequent use of general deep learning operations, such as convolution and attention, that are designed for grid-like data. Therefore, our Neighbor2Seq naturally endows GNNs with the efficiency and advantages of deep learning operations on grid-like data by precomputing the Neighbor2Seq transformations. In addition, our Neighbor2Seq can alleviate the over-squashing issue suffered by GNNs based on message passing. We evaluate our method on a massive graph, with more than 111 million nodes and 1.6 billion edges, as well as several medium-scale graphs. Results show that our proposed method is scalable to massive graphs and achieves superior performance across massive and mediumscale graphs. ", + "bbox": [ + 233, + 295, + 764, + 502 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 530, + 336, + 546 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph neural networks (GNNs) have shown effectiveness in many fields with rich relational structures, such as citation networks (Kipf & Welling, 2016; Velickovi ˇ c et al., 2018), social net- ´ works (Hamilton et al., 2017), drug discovery (Gilmer et al., 2017; Stokes et al., 2020), physical systems (Battaglia et al., 2016), and point clouds (Wang et al., 2019). Most current GNNs follow a message passing scheme (Gilmer et al., 2017; Battaglia et al., 2018), in which the representation of each node is recursively updated by aggregating the representations of its neighbors. Various GNNs (Li et al., 2016; Kipf & Welling, 2016; Velickovi ˇ c et al., 2018; Xu et al., 2019) mainly differ ´ in the forms of aggregation functions. ", + "bbox": [ + 174, + 561, + 823, + 672 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Real-world applications usually generate massive graphs, such as social networks. However, message passing methods have difficulties in handling such large graphs as the recursive message passing mechanism leads to prohibitive computation and memory requirements. To date, sampling methods (Hamilton et al., 2017; Ying et al., 2018; Chen et al., 2018a;b; Huang et al., 2018; Zou et al., 2019; Zeng et al., 2020; Gao et al., 2018; Chiang et al., 2019; Zeng et al., 2020) and precomputing methods (Wu et al., 2019; Rossi et al., 2020; Bojchevski et al., 2020) have been proposed to scale GNNs on large graphs. While the sampling methods can speed up training, they might result in redundancy, still incur high computational complexity, lead to loss of performance, or introduce bias (see Section 2.2). Generally, precomputing methods can scale to larger graphs as compared to sampling methods as recursive message passing is still required in sampling methods. ", + "bbox": [ + 174, + 681, + 823, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we propose the Neighbor2Seq that transforms the hierarchical neighborhood of each node to a sequence in a precomputing step. After the Neighbor2Seq transformation, each node and its associated neighborhood tree are converted to an ordered sequence. Therefore, each node can be viewed as an independent sample and is no longer constrained by the topological structure. This novel transformation from graphs to grid-like data enables the use of mini-batch training for subsequent models. As a result, our models can be used on extremely large graphs, as long as the Neighbor2Seq step can be precomputed. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As a radical departure from existing precomputing methods, we consider the hierarchical neighborhood of each node as an ordered sequence. The order information corresponds to hops between nodes. As a result of our Neighbor2Seq transformation, generic deep learning operations for gridlike data, such as convolution and attention, can be applied in subsequent models. In addition, our Neighbor2Seq can alleviate the over-squashing issue (Alon & Yahav, 2020) suffered by current GNNs. Experimental results indicate that our proposed method can be used on a massive graph, where most current methods cannot be applied. Furthermore, our method achieves superior performance as compared with previous sampling and precomputing methods. ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 ANALYSIS OF CURRENT GRAPH NEURAL NETWORK METHODS ", + "text_level": 1, + "bbox": [ + 178, + 236, + 730, + 252 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We start by defining necessary notations. A graph is formally defined as $\\mathcal { G } = ( V , E )$ , where $V$ is the set of nodes and $E \\subseteq V \\times V$ is the set of edges. We use $n = | V |$ and $m = | E |$ to denote the numbers of nodes and edges, respectively. The nodes are indexed from 1 to $n$ . We consider a node feature matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times \\breve { d } }$ , where each row $\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }$ is the $d$ -dimensional feature vector associated with node $i$ . The topology information of the graph is encoded in the adjacency matrix $\\ b { A } \\in \\mathbb { R } ^ { n \\times n }$ , where $A _ { ( i , j ) } = 1$ if an edge exists between node $i$ and node $j$ , and $A _ { ( i , j ) } = 0$ otherwise. ", + "bbox": [ + 174, + 270, + 825, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GRAPH NEURAL NETWORKS VIA MESSAGE PASSING ", + "text_level": 1, + "bbox": [ + 174, + 372, + 581, + 386 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There are two primary deep learning methods on graphs (Bronstein et al.); those are, spectral methods and spatial methods. The spectral method in Bruna et al. (2014) extends convolutional neural networks (LeCun et al., 1989) to the graph domain based on the spectrum of the graph Laplacian. The main limitation of spectral methods is the high complexity. ChebNet (Defferrard et al., 2016) and GCN (Kipf & Welling, 2016) simplify the spectral methods and can be understood from the spatial perspective. In this work, we focus on the analysis of the current mainstream spatial methods. Generally, most existing spatial methods, such as ChebNet (Defferrard et al., 2016), GCN (Kipf & Welling, 2016), GG-NN (Li et al., 2016), GAT (Velickovi ˇ c et al., 2018), and GIN (Xu et al., ´ 2019), can be understood from the message passing perspective (Gilmer et al., 2017; Battaglia et al., 2018). Specifically, we iteratively update node representations by aggregating representations from its immediate neighbors. These message passing methods have been shown to be effective in many fields. However, they have inherent difficulties when applied on large graphs due to their excessive computation and memory requirements, as described in Section 2.2. ", + "bbox": [ + 174, + 398, + 825, + 579 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 GRAPH NEURAL NETWORKS ON LARGE GRAPHS ", + "text_level": 1, + "bbox": [ + 174, + 598, + 555, + 612 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The above message passing methods are often trained in full batch. This requires the whole graph, i.e., all the node representations and edge connections, to be in memory to allow recursive message passing on the whole graph. Usually, the number of neighbors grows very rapidly with the increase of receptive field. Hence, these methods cannot be applied directly on large-scale graphs due to the prohibitive requirements on computation and memory. To enable deep learning on large graphs, two families of methods have been proposed; those are methods based on sampling and precomputing. ", + "bbox": [ + 174, + 625, + 825, + 708 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To circumvent the recursive expansion of neighbors across layers, sampling methods apply GNNs on a sampled subset of nodes with mini-batch training. Sampling methods can be further divided into three categories. First, node-wise sampling methods perform message passing for each node in its sampled neighborhood. This strategy is first proposed in GraphSAGE (Hamilton et al., 2017), where neighbors are randomly sampled. This is extended by PinSAGE (Ying et al., 2018), which selects neighbors based on random walks. VR-GCN (Chen et al., 2018a) further proposes to use variance reduction techniques to obtain a convergence guarantee. Although these node-wise sampling methods can reduce computation, the remaining computation is still very expensive and some redundancy might have been introduced, as described in Huang et al. (2018). Second, layer-wise sampling methods sample a fixed number of nodes for each layer. In particular, FastGCN (Chen et al., 2018b) samples a fixed number of nodes for each layer independently based on the degree of each node. AS-GCN (Huang et al., 2018) and LADIES (Zou et al., 2019) introduce between-layer dependencies during sampling, thus alleviating the loss of information. Layer-wise sampling methods can avoid the redundancy introduced by node-wise sampling methods. However, the expensive sampling algorithms that aim to ensure performance may themselves incur high computational cost, as pointed out in Zeng et al. (2020). Third, graph-wise sampling methods build mini-batches on sampled subgraphs. Specifically, LGCN (Gao et al., 2018) proposes to leverage mini-batch training on subgraphs selected by Breadth-First-Search algorithms. ClusterGCN (Chiang et al., 2019) conducts mini-batch training on sampled subgraphs that are obtained by a graph clustering algorithm. GraphSAINT (Zeng et al., 2020) proposes to derive subgraphs by importance-sampling and introduces normalization techniques to eliminate biases. These graph-wise sampling methods usually have high efficiency. The main limitation is that the nodes in a sampled subgraph are usually clustered together. This implies that two distant nodes in the original graph usually cannot be feeded into the GNNs in the same mini-batch during training, potentially leading to bias in the trained models. ", + "bbox": [ + 174, + 715, + 825, + 922 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 229 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The second family of methods for enabling GNNs training on large graphs are based on procomputing. Specifically, SGC (Wu et al., 2019) removes the non-linearity between GCN layers, resulting in a simplification as $Y = \\mathrm { s o f t m a x } ( \\hat { A } ^ { L } X W )$ . In this formulation, $\\hat { \\pmb { A } } = \\tilde { \\pmb { D } } ^ { - \\frac 1 2 } \\tilde { \\pmb { A } } \\tilde { \\pmb { D } } ^ { - \\frac 1 2 }$ is the symmetrically normalized adjacency matrix, ${ \\tilde { A } } = A + I$ is the adjacency matrix with self-loops, $\\tilde { D }$ is the corresponding diagonal node degree matrix with $\\begin{array} { r } { \\tilde { D } _ { ( i , i ) } = \\mathrm { \\bar { \\sum } } _ { j } \\tilde { A } _ { ( i , j ) } } \\end{array}$ , $L$ is the size of receptive field (i.e., the number of considered neighboring hops), which is the same as a $L$ -layer GCN, $\\mathbf { Y }$ is the output of the softmax classifier. Since there is no learnable parameters in $\\hat { A } ^ { L } X$ , this term can be precomputed as a feature pre-processing step. Similarly, SIGN (Rossi et al., 2020) applies an inception-like model to the precomputed features ${ \\hat { A } } ^ { \\ell } X$ for $\\ell \\in \\{ 1 , \\cdots , L \\}$ , where $L$ is the predefined size of receptive field. Instead of precomputing the smoothing features as in SGC and SIGN, PPRGo (Bojchevski et al., 2020) extends the idea of PPNP (Klicpera et al., 2018) by approximately precomputing the personalized PageRank (Page et al., 1999) matrix, thereby enabling model training on large graphs using mini-batches. Generally, the precomputing methods can scale to larger graphs than sampling methods because the latter still needs to perform the recursive message passing during training. Differing from these precomputing methods, we consider the hierarchical neighborhood of each node as an ordered sequence, thus retaining the useful information about hops between nodes and enabling subsequent powerful and efficient operations. ", + "bbox": [ + 174, + 236, + 825, + 483 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THE PROPOSED NEIGHBOR2SEQ METHOD AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 506, + 691, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we describe our proposed method, known as Neighbor2Seq, which transforms the hierarchical neighborhood of each node into an ordered sequence, thus enabling the subsequent use of general deep learning operations. We analyze the scalability of our method (See Section 3.5) and describe how our method can alleviate the over-squashing issue suffered by current message passing methods (See Section 3.6). ", + "bbox": [ + 174, + 539, + 825, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 OVERVIEW ", + "text_level": 1, + "bbox": [ + 174, + 628, + 290, + 642 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As described in Section 2.1, existing message passing methods recursively update each node’s representation by aggregating information from its immediate neighbors. Hence, what these methods aim at capturing for each node is essentially its corresponding hierarchical neighborhood, i.e., the neighborhood tree rooted at current node, as illustrated in Figure 1 (b). In this work, we attempt to go beyond the message passing scheme to overcome the limitations mentioned in Section 2. We propose to capture the information of this hierarchical neighborhood by transforming it into an ordered sequence, instead of recursively squashing it into a fixed-length vector. Our proposed method is composed of three steps. First, we transform a neighborhood to a sequence for each node. Second, we apply a normalization technique to the derived sequence features. Third, we use general deep learning operations, i.e., convolution and attention, to learn from these sequence features and then make predictions for nodes. In the following, we describe these three steps in detail. ", + "bbox": [ + 174, + 655, + 825, + 808 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 NEIGHBOR2SEQ: TRANSFORMING NEIGHBORHOODS TO SEQUENCES ", + "text_level": 1, + "bbox": [ + 176, + 827, + 692, + 842 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The basic idea of Neighbor2Seq is to transform the hierarchical neighborhood of each node to an ordered sequence by integrating the features of nodes in each layer of the neighborhood tree. Following the notations defined in Section 2, we let $z _ { 0 } ^ { i } , z _ { 1 } ^ { i } , \\cdots , z _ { L } ^ { i }$ denote the resulting sequence for node $i$ , where $L$ is the height (i.e., the number of hops we consider) of the neighborhood tree rooted at node $i$ . $z _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d }$ denotes the $\\ell$ -th feature of the sequence. The length of the resulting sequence for each node is $L + 1$ . Formally, for each node $i$ , our Neighbor2Seq can be expressed as ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/875bf8963e4d313f0401ac7c554728624d2b6b93f1de7b23e96dfc7730e414ea.jpg", + "image_caption": [ + "Figure 1: (a) An illustration of the original graph. The current node is denoted as two concentric circles. (b) Message passing in the neighborhood tree. (c) Our proposed Neighbor2Seq. (d) Our proposed models: Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn. " + ], + "image_footnote": [], + "bbox": [ + 205, + 95, + 790, + 396 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 469, + 733, + 486 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9b79ed03df632124b451e1a3ba89e28b2cbd52e8eb5d286cb9ffed6d273e4209.jpg", + "text": "$$\nz _ { \\ell } ^ { i } = \\sum _ { j = 1 } ^ { n } w ( i , j , \\ell ) { \\pmb x } _ { j } , \\quad \\forall \\ell \\in \\{ 0 , 1 , 2 , \\cdot \\cdot \\cdot , L \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 491, + 653, + 535 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $w ( i , j , \\ell )$ denotes the number of walks with length $\\ell$ between node $i$ and $j , n$ is the number of nodes in the graph. We define $w ( i , j , 0 )$ as 1 for $j = i$ and 0 otherwise. Hence, $z _ { 0 } ^ { i }$ is the original node feature $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ . Intuitively, $\\boldsymbol { z } _ { \\ell } ^ { i }$ is obtained by computing a weighted sum of features of all nodes with walks of length $\\ell$ to $i$ , and the numbers of qualified walks are used as the corresponding weights. Our Neighbor2Seq is illustrated in Figure 1 (c). Note that the derived sequence has meaningful order information, indicating the hops between nodes. After we obtain ordered sequences from the original hierarchical neighborhoods, we can use generic deep learning operations to learn from these sequences, as detailed below. ", + "bbox": [ + 173, + 540, + 825, + 652 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 NORMALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 667, + 336, + 683 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since the number of nodes in the hierarchical neighborhood grows exponentially as the hop number increases, different layers in the neighborhood tree have drastically different numbers of nodes. Hence, feature vectors of a sequence computed by Equation (1) have very different scales. In order to make the subsequent learning easier, we propose a layer to normalize the sequence features. We use a normalization technique similar to layer normalization (Ba et al., 2016). In particular, each feature of a sequence is normalized based on the mean and the standard deviation of its own feature values. Formally, our normalization process for each node $i$ can be written as ", + "bbox": [ + 174, + 694, + 825, + 792 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/29b9ff063872a2061eb4ac605b217c810dc5615a0bbed1db894dadf3fc573510.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\boldsymbol y } _ { \\ell } ^ { i } = W _ { \\ell } \\boldsymbol z _ { \\ell } ^ { i } , \\quad { \\boldsymbol o } _ { \\ell } ^ { i } = \\frac { \\boldsymbol y _ { \\ell } ^ { i } - { \\boldsymbol \\mu } _ { \\ell } ^ { i } } { \\sigma _ { \\ell } ^ { i } } \\odot \\gamma _ { \\ell } + \\beta _ { \\ell } , \\quad \\forall \\ell \\in \\{ 0 , 1 , 2 , \\cdots , L \\} } \\\\ & { \\boldsymbol \\mu _ { \\ell } ^ { i } = \\frac { 1 } { d ^ { \\prime } } \\displaystyle \\sum _ { c = 1 } ^ { d ^ { \\prime } } { \\boldsymbol y } _ { \\ell } ^ { i } [ \\boldsymbol c ] , \\quad { \\boldsymbol \\sigma } _ { \\ell } ^ { i } = \\sqrt { \\frac { 1 } { d ^ { \\prime } } \\displaystyle \\sum _ { c = 1 } ^ { d ^ { \\prime } } ( { \\boldsymbol y } _ { \\ell } ^ { i } [ \\boldsymbol c ] - { \\boldsymbol \\mu } _ { \\ell } ^ { i } ) ^ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 281, + 797, + 717, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We first apply a linear transformation $W _ { \\ell } \\in \\mathbb { R } ^ { d ^ { \\prime } \\times d }$ to produce a low-dimensional representation $\\boldsymbol { y } _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ for the $\\ell$ -th feature of the sequence, since the original feature dimension $d$ is usually large. $\\mu _ { \\ell } ^ { i } \\in \\mathbb { R }$ and $\\sigma _ { \\ell } ^ { i } \\in \\mathbb { R }$ are the mean and standard deviation of the corresponding representation $\\boldsymbol { y } _ { \\ell } ^ { i }$ . $\\gamma _ { \\ell } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ and $\\beta _ { \\ell } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ denote the learnable affine transformation parameters. $\\odot$ denotes the element-wise multiplication. Note that the learnable parameters in this normalization layer is associated with $\\ell$ , implying that each feature of the sequence is normalized separately. Using this normalization layer, we obtain the normalized feature vector $o _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ for every $\\ell \\in \\{ 0 , 1 , 2 , \\cdot \\cdot \\cdot , L \\}$ . ", + "bbox": [ + 173, + 892, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 179 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 NEIGHBOR2SEQ $^ +$ CONV AND NEIGHBOR2SEQ $^ +$ ATTN ", + "text_level": 1, + "bbox": [ + 173, + 194, + 583, + 210 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After obtaining an ordered sequence for each node, we can view each node in the graph as a sequence of feature vectors. We can use general deep learning techniques to learn from these sequences. In this work, we propose two models, namely Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn, in which convolution and attention are applied on the sequences of each node. ", + "bbox": [ + 173, + 220, + 825, + 277 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As illustrated in Figure 1 (d), Neighbor2Seq+Conv applies a 1-D convolutional neural network to the sequence features and then use an average pooling to yield a representation for the sequence. Formally, for each node $i$ , ", + "bbox": [ + 174, + 284, + 825, + 327 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b323aa69666fa28c8ae742d6f46e62483a32d28e7facf3ef173455147cbeae5b.jpg", + "text": "$$\n\\left( \\hat { \\sigma } _ { 0 } ^ { i } , \\hat { \\sigma } _ { 1 } ^ { i } , \\cdots , \\hat { \\sigma } _ { L } ^ { i } \\right) = { \\bf C } { \\bf N } { \\bf N } \\left( o _ { 0 } ^ { i } , o _ { 1 } ^ { i } , \\cdots , o _ { L } ^ { i } \\right) , \\quad r ^ { i } = { \\frac { 1 } { L + 1 } } \\sum _ { \\ell = 0 } ^ { L } \\hat { \\sigma } _ { \\ell } ^ { i } ,\n$$", + "text_format": "latex", + "bbox": [ + 277, + 333, + 722, + 377 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathrm { C N N } ( \\cdot )$ denotes a 1-D convolutional neural network. $r ^ { i }$ denotes the obtained representation of node $i$ that is used as the input to a linear classifier to make predictions for this node. Specifically, we implement $\\mathrm { C N N } ( \\cdot )$ as a 2-layer convolutional neural network composed of two 1-D convolutions. The kernel size is set according to the length of input sequence. The activation function between layers is ReLU (Krizhevsky et al., 2012). ", + "bbox": [ + 173, + 383, + 825, + 455 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Incorporating attention is another natural idea to learn from sequences. As shown in Figure 1 (d), Neighbor2Seq+Attn uses an attention mechanism (Bahdanau et al., 2015) to integrate sequential feature vectors in order to derive a representation. Unlike convolutional neural networks, the vanilla attention mechanism cannot make use of the order of the sequence. Hence, we add positional encodings (Vaswani et al., 2017) to the features such that the position information of different features in the sequence can be incorporated. Formally, for each node $i$ , we add positional encoding for each feature in the sequence as ", + "bbox": [ + 173, + 460, + 826, + 559 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a8d764b29b380ea5970f6076db24a83373f9976d1fba2cc51aa5a129e43d705f.jpg", + "text": "$$\nk _ { \\ell } ^ { i } = o _ { \\ell } ^ { i } + p _ { \\ell } ^ { i } , \\quad p _ { \\ell } ^ { i } [ m ] = \\left\\{ \\begin{array} { l l } { { \\sin \\left( \\displaystyle { \\frac { \\ell } { 1 0 0 0 0 ^ { \\frac { 2 n } { d ^ { \\prime } } } } } \\right) } } & { { m = 2 n } } \\\\ { { \\cos \\left( \\displaystyle { \\frac { \\ell } { 1 0 0 0 0 ^ { \\frac { 2 n } { d ^ { \\prime } } } } } \\right) } } & { { m = 2 n + 1 } } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 565, + 712, + 637 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The positional encoding for $\\ell$ -th feature of node $i$ is denoted as $\\pmb { p } _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ . $m \\in \\{ 1 , 2 , \\cdots , d ^ { \\prime } \\}$ is the dimensional index. Intuitively, a position-dependent vector is added to each feature such that the order information can be captured. Then we use the attention mechanism with learnable query (Yang et al., 2016) to combine these sequential feature vectors to obtain the final representations $r ^ { i }$ for each node $i$ . Formally, ", + "bbox": [ + 173, + 643, + 826, + 714 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2f7d6802d2122f35927e768b610a02f08b246cc689cbd8a7b1a6f6abf9ce9a03.jpg", + "text": "$$\n\\pmb { r } ^ { i } = \\sum _ { \\ell = 0 } ^ { L } \\alpha _ { \\ell } ^ { i } \\pmb { k } _ { \\ell } ^ { i } , \\quad \\alpha _ { \\ell } ^ { i } = \\frac { \\exp ( \\pmb { k } _ { \\ell } ^ { i ^ { T } } \\pmb { q } ) } { \\sum _ { \\ell = 0 } ^ { L } \\exp ( \\pmb { k } _ { \\ell } ^ { i ^ { T } } \\pmb { q } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 712, + 642, + 756 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\boldsymbol { q } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ is the learnable query vector that is trained along with other model parameters. The derived representation $r ^ { i }$ will be taken as the input to a linear classifier to make prediction for node $i$ . ", + "bbox": [ + 173, + 761, + 826, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 ANALYSIS OF SCALABILITY ", + "text_level": 1, + "bbox": [ + 176, + 806, + 408, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Precomputing Neighbor2Seq. A well-known fact is that the value of $w ( i , j , \\ell )$ in Equation (1) can be obtained by computing the power of the original adjacency matrix $\\pmb { A }$ . Following GCN, we add self-loops to make each node connected to itself. Concretely, $w ( i , j , \\ell ) = \\tilde { A } _ { ( i , j ) } ^ { \\ell }$ Hence, the Neighbor2Seq can be implemented by computing the matrix multiplications $\\tilde { A } ^ { \\ell } X$ for $\\forall \\ell \\in$ $\\{ 0 , 1 , 2 , \\bar { \\cdot } \\cdot \\cdot , L \\}$ . Since there is no learnable parameters in the Neighbor2Seq step, these matrix multiplications can be precomputed sequentially for large graphs on CPU platforms with large memory. ", + "bbox": [ + 173, + 832, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This can be easily precomputed because the matrix $\\tilde { A }$ is usually sparse. For extremely large graphs, this precomputation can even be performed on distributed systems. ", + "bbox": [ + 173, + 102, + 821, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Enabling mini-batch training. After we obtain the precomputed sequence features, each node in the graph corresponds to a sequence of feature vectors. Therefore, each node can be viewed as an independent sample. That is, we are no longer restricted by the original graph connectivity anymore. Then, we can randomly sample from all the training nodes to conduct mini-batch training. This is more flexible and unbiased than sampling methods as reviewed in Section 2.2. Our mini-batches can be randomly extracted over all nodes, opening the possibility that any pair of nodes can be sampled in the same mini-batch. In contrast, mini-batches in sampling methods are usually restricted by the fixed sampling strategies. This advantage opens the door for subsequent model training on extremely large graphs, as long as the corresponding Neighbor2Seq step can be precomputed. ", + "bbox": [ + 174, + 138, + 825, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Computational complexity comparison. We compare our methods with several existing sampling and precomputing methods in terms of computational complexity. We let $L$ denote the number of hops we consider. For simplicity, we assume the feature dimension $d$ is fixed for all layers. For sampling methods, $s$ is the number of sampled neighbors for each node. The computation of Neighbor2Seq $^ +$ Conv mainly lies in the linear transformation (i.e., $\\mathcal { O } ( L d ^ { 2 } \\bar { n } ) )$ in the normalization step and the 1-D convolutional neural networks (i.e., ", + "bbox": [ + 174, + 271, + 514, + 422 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/54349962c508f86d4ef5fdbdbdc1d389ed999225fc0dfc0a04923d637b69ad13.jpg", + "table_caption": [ + "Table 1: Comparison of computational complexity for precomputing and forward pass corresponding to an entire epoch. " + ], + "table_footnote": [], + "table_body": "
MethodPrecomputingForward Pass
GCNO(Ldm + Ld²n)
GraphSAGEO(sLn)O(sLd²n)
ClusterGCNO(m)O(Ldm + Ld²n)
GraphSAINTO(sn)O(Ldm +Ld²n)
SGCO(Ldm)O(d²n)
SIGNO(Ldm)O(Ld²n)
Neighbor2Seq+ConvO(Ldm)O(Ld²+Lkd²)n)
Neighbor2Seq+AttnO(Ldm)O(Ld²+Ld)n)
", + "bbox": [ + 531, + 314, + 821, + 417 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$O ( L k d ^ { 2 } n )$ , where $k$ is the kernel size). Hence, the computational complexity for the forward pass of Neighbo $_ { 1 2 \\mathrm { S e q + C o n v } }$ is $\\mathcal { O } ( ( L d ^ { 2 } + L k d ^ { 2 } ) n )$ . Neighbor2Seq+Attn has a computational complexity of $\\mathcal { \\bar { O } } ( ( L d ^ { 2 } + \\bar { L } d ) n )$ because the attention mechanism is more efficient than 1-D convolutional neural networks. As shown in Table 1, the forward pass complexities of precomputing methods, including our Neighbor2Seq+Conv and Neighbor2Seq+Attn, are all linear with respect to the number of nodes $n$ and do not depend on the number of edges $m$ . Hence, the training processes of our models are computationally efficient. ", + "bbox": [ + 174, + 424, + 825, + 520 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.6 ALLEVIATING THE OVER-SQUASHING ISSUE ", + "text_level": 1, + "bbox": [ + 174, + 536, + 524, + 551 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "An inherent problem in message passing methods is known as the over-squashing (Alon & Yahav, 2020). In particular, recursively propagating information between neighbors creates a bottleneck because the number of nodes in the receptive field grows exponentially with the number of layers. This bottleneck causes the over-squashing issue; that is, information from the exponentially-growing receptive field is compressed into fixed-length vectors. Consequently, message passing methods fail to capture the message flowing from distant nodes and performs poorly when long-range information is essential for the prediction tasks. Note that the over-squashing issue is not identical to the oversmoothing issue. Over-smoothing is related to the phenomenon that node representations converge to indistinguishable limits when the number of layers increases (Li et al., 2018; Wu et al., 2019; NT & Maehara, 2019; Liu et al., 2020a; Oono & Suzuki, 2020; Cai & Wang, 2020; Chen et al., 2020). The virtual edges added in Gilmer et al. (2017) and recent non-local aggregations (Pei et al., 2020; Liu et al., 2020b) can be viewed as attempts to alleviate the over-squashing issue by incorporating distant nodes. Another study (Ma et al., 2019) considers message passing along all possible paths between two nodes, instead of propagating information between neighbors. ", + "bbox": [ + 173, + 563, + 825, + 756 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our Neighbor2Seq can alleviate the over-squashing issue because we transform the exponentiallygrowing nodes in hierarchical neighborhoods into an ordered sequence, instead of recursively squashing them into a fixed-size vector. With our Neighbor2Seq, capturing long-range information on graphs becomes similar to achieving this on sequence data, such as texts. ", + "bbox": [ + 176, + 763, + 823, + 819 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 DISCUSSIONS ", + "text_level": 1, + "bbox": [ + 176, + 839, + 320, + 856 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 INFORMATION LOSS IN NEIGHBOR2SEQ ", + "text_level": 1, + "bbox": [ + 176, + 869, + 493, + 883 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As shown in Figure 1 (c), Neighbor2Seq obtains the sequence by integrating features of nodes in each layer of the neighborhood tree. This transformation may lose the cross-layer dependency information in the tree. Specifically, the Neighbor2Seq ignores the identities of nodes that each walk passes through and only considers what are the nodes in each layer of the neighborhood tree. Nevertheless, this information can neither be captured by message passing methods because the aggregation is usually permutation-invariant. This implies that messages from different neighbors cannot be distinguished, as pointed in Pei et al. (2020). According to our experimental results in Table 5, our models without this information can outperform message passing methods, such as GCN. It is intriguing to have an in-depth exploration of whether such information is useful and how it can be captured. ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 RELATIONS WITH THE WEISFEILER-LEHMAN HIERARCHY ", + "text_level": 1, + "bbox": [ + 174, + 232, + 617, + 246 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As shown in Xu et al. (2019), most of current GNNs are at most as powerful as the WeisfeilerLehman (WL) graph isomorphism test (Weisfeiler & Lehman, 1968) in distinguishing graph structures. Our Neighbor2Seq is still under the WL hierarchy because the neighborhood tree used to obtain the sequence is indeed the one that the WL test uses to distinguish different graphs. We would be interested in exploring how Neighbor2Seq can be extended to go beyond the WL hierarchy as a future direction. ", + "bbox": [ + 174, + 257, + 825, + 342 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 BRIDGING THE GAP BETWEEN GRAPH AND GRID-LIKE DATA ", + "text_level": 1, + "bbox": [ + 176, + 358, + 640, + 373 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The main difference between graph and grid-like data lies in the notion and properties of locality. Specifically, the numbers of neighbors differ for different nodes, and there is no order information among the neighbors of a node in graphs. These are the main obstacles preventing the use of generic deep learning operations on graphs. Our Neighbor2Seq is an attempt to bridge the gap between graph and grid-like data. Base on our Neighbor2Seq, many effective strategies for grid-like data can be naturally transferred to graph data. These include self-supervised learning and pre-training on graphs (Hu et al., 2019; Velickovic et al., 2019; Sun et al., 2019; Hassani & Khasahmadi, 2020; You et al., 2020; Hu et al., 2020b; Qiu et al., 2020; Jin et al., 2020). ", + "bbox": [ + 174, + 385, + 825, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We notice an existing work AWE Ivanov & Burnaev (2018) which also embed the information in graph as a sequence. In order to avoid confusion, we make a clarification about the fundamental and significant differences between AWE and our Neighbor2Seq. First, AWE produces a sequence embedding for the entire graph, while our Neighbor2Seq yields a sequence embedding for each node in the graph. Second, each element in the obtained sequence in AWE is the probability of an anonymous walk embedding. In our Neighbor2Seq, each feature vector in the obtained sequence for one node is computed by summing up the features of all nodes in the corresponding layer of the neighborhood tree. This point distinguishes these two methods fundamentally. ", + "bbox": [ + 174, + 502, + 825, + 614 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 635, + 415, + 651 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 666, + 375, + 681 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Datasets. We evaluate our proposed models on 1 massive-scale graph and 4 medium-scale graphs using node classification tasks. The massive-scale graph ogbn-papers100M provided by the Open Graph Benchmark (OGB) (Hu et al., 2020a) is the existing largest benchmark dataset for node classification. Medium-scale graphs include ogbn-products (Hu et al., 2020a), Reddit (Hamilton et al., 2017), Yelp Zeng et al. (2020), and Flickr Zeng et al. (2020). These tasks cover inductive and transductive settings. The statistics of these datasets are summarized in Table 2. The detailed description of these datasets are provided in Appendix A.1. ", + "bbox": [ + 174, + 693, + 825, + 791 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/07c5d4fd00c048603152714a14f71d37fd3f69f80b64e9f931f9e898613bc207.jpg", + "table_caption": [ + "Table 2: Statistics of datasets. “m” denotes multi-label classification. " + ], + "table_footnote": [ + "Implementation. We implemented our methods using Pytorch (Paszke et al., 2017) and Pytorch Geometric (Fey & Lenssen, 2019). For our proposed methods, we conduct the precomputation on " + ], + "table_body": "
DatasetScale#Nodes#EdgesAvg. Deg.#Features#ClassesTrain/Val/Test
ogbn-papers100MMassive111,059,9561,615,685,872291281720.78/0.08/0.14
ogbn-productsMedium2,449,02961,859,14051100470.08/0.02/0.90
RedditMedium232,96511,606,91950602410.66/0.10/0.24
YelpMedium716,8576,997,41010300100(m)0.75/0.10/0.15
FlickrMedium89,250899,7561050070.50/0.25/0.25
", + "bbox": [ + 176, + 814, + 823, + 893 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "the CPU, after which we train our models on a GeForce RTX 2080 Ti GPU. We perform a grid search for the following hyperparameters: number of hops $L$ , batch size, learning rate, hidden dimension $d ^ { \\prime }$ , dropout rate, weight decay, and convolutional kernel size $k$ . The chosen hyperparameters for our Neighbor2Seq+Conv and Neighbor2Seq+Attn are summarized in Appendix A.2 for reproducibility. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 RESULTS ON MASSIVE-SCALE GRAPHS ", + "text_level": 1, + "bbox": [ + 178, + 176, + 486, + 189 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Since ogbn-papers100M is a massive graph with more than $1 1 1 \\ \\mathrm { m i l }$ - lion nodes and 1.6 billion edges, most existing methods have difficulty handling such a graph. We consider three baselines that have available results evaluated by OGB: Multilayer Perceptron (MLP), Node2Vec (Grover & Leskovec, 2016), and SGC (Wu et al., 2019). The results under transductive setting is reported in Table 3. Following OGB, we report accuracies for all models on training, validation, and test sets. The previous state-of-theart result on ogbn-papers100M is ob", + "bbox": [ + 174, + 202, + 423, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Results on ogbn-papers100M in terms of classification accuracy (in percent). The reported accuracy is averaged over 10 random runs. Note that existing sampling methods cannot scale to this massive graph. During precomputation, both SGC and our models have to randomly remove $4 0 \\%$ edges to avoid a memory overflow on CPU. This implies that the performance could be further improved if more advanced precomuting platform is used. ", + "bbox": [ + 437, + 204, + 823, + 315 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8231c469d959eeb37180f0fd515f142db613000891510b0e7b65999a6aa05a52.jpg", + "table_caption": [], + "table_footnote": [ + "tained by the precomputing method SGC. Our models outperform the baselines consistently in terms of training, validation, and test, which demonstrates the expressive power and the generalization ability of our method on massive graphs. " + ], + "table_body": "
MethodTrainingValidationTest
MLP54.84±0.4349.60±0.2947.24±0.31
Node2vec158.07±0.2855.60±0.23
SGC67.54±0.4366.48±0.2063.29±0.19
Neighbor2Seq+Conv69.87±0.8167.46±0.1664.04±0.22
Neighbor2Seq+Attn68.83±0.3066.90±0.1063.59±0.17
", + "bbox": [ + 439, + 315, + 825, + 409 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 RESULTS ON MEDIUM-SCALE GRAPHS ", + "text_level": 1, + "bbox": [ + 176, + 468, + 483, + 482 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also evaluate our models on medium-scale graphs, thus enabling comparison with more existing works. We conduct transductive learning on ogbn-products, a medium-scale graph from OGB. We also conduct inductive learning on Reddit, Yelp, and Flickr, which are frequently used for inductive learning by the community. The following baselines are considered: MLP, Node2Vec (Grover & Leskovec, 2016), GCN (Kipf & Welling, 2016), SGC Wu et al. (2019), GraphSAGE (Hamilton et al., 2017), FastGCN (Chen et al., 2018b), VR-GCN (Chen et al., 2018a), AS-GCN (Huang et al., 2018), ClusterGCN (Chiang et al., 2019), GraphSAINT (Zeng et al., 2020), and SIGN (Rossi et al., 2020). ", + "bbox": [ + 173, + 494, + 825, + 604 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/13e6ab87cd4ea0a977ccc7d5ab16c1388f68a89d6bdfa052b236c1f0c4dc45b1.jpg", + "table_caption": [ + "Table 4: Results on ogbn-products in terms of classification accuracy (in percent). The reported accuracy is averaged over 10 random runs. Obtaining the results of GCN requires a GPU with 33GB of memory. " + ], + "table_footnote": [], + "table_body": "
MethodTrainingValidationTest
MLP84.03±0.9375.54±0.1461.06±0.08
Node2vec93.39±0.1090.32±0.0672.49±0.10
GCN93.56±0.0992.00±0.0375.64±0.21
GraphSAGE92.96±0.0791.70±0.0978.70±0.36
ClusterGCN93.75±0.1392.12±0.0978.97±0.33
GraphSAINT92.71±0.1491.62±0.0879.08±0.24
SGC92.60±0.1091.19±0.0672.46±0.27
SIGN96.92±0.4693.10±0.0877.60±0.13
Neighbor2Seq+Conv95.32±0.1092.92±0.0579.67±0.16
Neighbor2Seq+Attn92.82±0.1492.20±0.0279.35±0.17
", + "bbox": [ + 176, + 681, + 486, + 806 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/d68d115d4580102e48ef6908c7941fa591fe53c0d659d95f60b47604f136dfd3.jpg", + "table_caption": [ + "Table 5: Results for inductive learning on three datasets in terms of F1-micro score. The reported score is averaged over 10 random runs. The results of baselines are partially obtained from Zeng et al. (2020); Rossi et al. (2020). " + ], + "table_footnote": [], + "table_body": "
MethodRedditFlickrYelp
GCN0.933±0.0000.492±0.0030.378±0.001
FastGCN0.924±0.0010.504±0.0010.265±0.053
VR-GCN0.964±0.0010.482±0.0030.640±0.002
AS-GCN0.958±0.0010.504±0.002-
GraphSAGE0.953±0.0010.501±0.0130.634±0.006
ClusterGCN0.954±0.0010.481±0.0050.609±0.005
GraphSAINT0.966±0.0010.511±0.0010.653±0.003
SGC0.949±0.0000.502±0.0010.358±0.006
SIGN0.968±0.0000.514±0.0010.631±0.003
Neighbor2Seq+Conv0.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn0.967±0.0000.523±0.0020.647±0.001
", + "bbox": [ + 516, + 679, + 825, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The ogbn-products dataset is challenging because the splitting is not random. The splitting procedure is more realistic. The nodes (i.e., products) are sorted according to their sales ranking and the top $8 \\%$ nodes are used for training, next $2 \\%$ for validation, and the rest $9 0 \\%$ for testing. This matches the real-world application where manual labeling is prioritized to important nodes and models are subsequently used to make prediction on less important nodes. Hence, ogbn-products is an ideal benchmark dataset to improve out-of-distribution prediction. As shown in Table 4, our Neighbor2Seq+Conv and Neighbor2Seq+Attn outperfom baselines on test set (i.e., $9 0 \\%$ nodes), which further demonstrates the generalization ability of our method. ", + "bbox": [ + 173, + 813, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The results on inductive tasks are summarized in Table 5. On Reddit, our models perform better than all sampling methods and achieve the competitive result as SIGN. On Flickr, our models obtain significantly better results. Specifically, our Neighbor2Seq $+$ Conv outperforms the previous state-ofthe-art models by an obvious margin. Although our models perform not as good as GraphSAINT on $Y e l p$ , we outperform other sampling methods and the precomputing model SIGN consistently on this dataset. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.4 COMPARISONS OF COMPUTIONAL EFFICIENCY ", + "text_level": 1, + "bbox": [ + 176, + 204, + 534, + 218 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to show the computational efficiency, we conduct an empirical comparison with existing methods in terms of time comsuming during preprocessing, training, and inference. We consider the following representative sampling methods and precom", + "bbox": [ + 173, + 231, + 352, + 369 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/0f1e10909ea1ba7d3f33d89e9ae536b9cb0c62e796c5c4da30ff48608b3a4f76.jpg", + "table_caption": [ + "Table 6: Computational efficiency in terms of preprocessing, training (per epoch), and inference times (in seconds) on ogbn-products. The reported time is averaged over 10 runs. " + ], + "table_footnote": [], + "table_body": "
MethodPreprocessing (↓)Training (↓)Inferenece (↓)Test Accuracy (↑)
ClusterGCN44.15±0.7711.87±0.8487.03±0.2478.97±0.33
GraphSAINT80.78±3.54.29±0.48107.26±0.9479.08±0.24
SGC153.36±3.60.15±0.011.08±0.0172.46±0.27
SIGN151.47±3.51.22±0.022.93±0.0677.60±0.13
Neighbor2Seq+Conv153.42±3.24.09±0.1231.52±1.4479.67±0.16
Neighbor2Seq+Attn153.42±3.22.67±0.0831.24±0.6179.35±0.17
", + "bbox": [ + 367, + 279, + 821, + 364 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "puting methods: ClusterGCN Chiang et al. (2019), GraphSAINT Zeng et al. (2020), SGC Wu et al. (2019), and SIGN Rossi et al. (2020). The comparison is performed on ogbn-products and the similar trend can be observed on other datasets. As demonstrated in Table 6, our approaches, like existing precomputing methods, are more computationally efficient than sampling methods in terms of training and inference. Compared with existing precomputing methods, our methods achieve a better balance between performance and efficiency. ", + "bbox": [ + 173, + 368, + 825, + 452 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.5 ABLATION STUDY ON ORDER INFORMATION ", + "text_level": 1, + "bbox": [ + 174, + 469, + 524, + 484 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/0fbfa0237630fb6342455c6d3624311ff6b9d100cd4a9f554cf9986e544df362.jpg", + "table_caption": [ + "Table 7: Comparison of models with and without capturing order information. Neighbor2Seq+Attn w/o PE denotes the Neighbor2Seq+Attn without adding positional encoding. " + ], + "table_footnote": [], + "table_body": "
ModelOrderogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Conv：64.04±0.2279.67±0.160.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn63.59±0.1779.35±0.170.967±0.0000.523±0.0020.647±0.001
Neighbor2Seq+Attn w/o PEX63.61±0.0978.54±0.250.965±0.0000.521±0.0030.646±0.001
", + "bbox": [ + 174, + 523, + 823, + 582 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Intuitively, the order information in the sequence obtained by Neighbor2Seq indicates the hops between nodes. Hence, we conduct an ablation study to verify the significance of this order information. We remove the positional encoding in Neighbor2Seq $+ .$ Attn, leading to a model without the ability to capture the order information. The comparison is demonstrated in Table 7. Note that Neighbor2Seq $^ +$ Attn and Neighbor2Seq $+ .$ Attn w/o PE have the same number of parameters. Hence, Comparing the results of these two models, we can conclude that the order information is usually necessary. Neighbor2Seq+Conv and Neighbor2Seq $+ .$ Attn both can capture the order information. There are two possible reasons why Neighbor2Seq $\\Vdash$ Conv performs better. First, Neighbor2Seq+Conv has more learnable parameters than Neighbor2Se+Attn, which only has a learnable query. Second, the convolutional neural network in Neighbor2Seq $^ +$ Conv can additionally investigate the dependencies between feature dimensions because each feature dimension of the output depends on every feature dimension of the input. ", + "bbox": [ + 173, + 588, + 825, + 753 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS AND OUTLOOK ", + "text_level": 1, + "bbox": [ + 176, + 775, + 459, + 790 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we propose Neighbor2Seq, for transforming the heirarchical neighborhoods to ordered sequences. Neighbor2Seq enables the subsequent use of powerful general deep learning operations, leading to the proposed Neighbor2Seq $^ +$ Conv and Neighbor2Seq+Attn. Our models can be deployed on massive graphs and trained efficiently. The extensive expriments demonstrate the scalability and the promising performance of our method. 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", + "bbox": [ + 173, + 502, + 825, + 545 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 DATASET DESCRIPTIONS ", + "text_level": 1, + "bbox": [ + 176, + 133, + 390, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "ogbn-papers100M (Hu et al., 2020a) is the existing largest benckmark dataset for node classification. It is a directed citation graph of 111 million papers indexed by Microsoft Academic Graph (MAG) (Wang et al., 2020). For simplicity, it is converted to an undirected graph in baselines and our method. Each node is a paper and each directed edge indicates that one paper cites another one. Each node is associated with a 128-dimensional feature vector obtained by averaging the word2vec (Mikolov et al., 2013) embeddings of words in its title and abstract. Among the node set, approximately 1.5 millione of them are ARXIV papers, each of which has a label with one of ARXIV’s subject areas. The rest nodes (i.e., non-ARXIV papers) are not associated with label information. The task is to leverage the entire citation graph to infer the labels of the ARXIV papers. The time-based splitting is used as the splitting strategy. To be more specifical, the training nodes are all ARXIV papers published until 2017, while the validation nodes are the ARXIV papers published in 2018, and the ARXIV papers published since 2019 are treated as test nodes. ", + "bbox": [ + 174, + 160, + 825, + 325 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "ogbn-products (Hu et al., 2020a) is an undirected Amazon product co-purchasing graph (Bhatia et al., 2016). Nodes denote products and edges between two nodes indicate that the corresponding products are purchased together. Node features are derived by extracting bag-of-words representations from the product descriptions. Further, a Principal Component Analysis is applied to these features to reduce the dimension to 100. The task is to predict the category of a product. A realistic splitting scheme is used in this data. Specifically, the products are firstly sorted according to their sales ranking, and then the top $1 0 \\%$ products are used for training, next $\\mathrm { \\bar { 2 } \\% }$ for validation, and the rest for testing. This strategy matches the real-world situation where manual labeling is prioritized to important nodes and models are subsequently deployed to predict the less important ones. ", + "bbox": [ + 174, + 333, + 825, + 458 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Reddit (Hamilton et al., 2017), Yelp (Zeng et al., 2020), and Flickr (Zeng et al., 2020) are widely used datasets for inductive learning. During training, only the node features of training nodes and the edges between training nodes are visible. Reddit is a social netowork extracted from Reddit forum. Nodes represent posts and edges between two posts indicate the same user comments on both posts. Node features are fromed by GloVe CommonCrawl word vectors Pennington et al. (2014) of the posts. The task is to predict which community different posts belong to. The splitting is also time-based. Yelp is a social netowork constructed from Yelp website. Nodes are users and edges between two nodes indicate they are friends. Node features of users are obtained by the word2vec embeddings of their corresponding reviews. The task is to predict the categories of businesses reviewed by different users, which is multi-label classification task. Flickr is a social network based on Flickr, a photo sharing website. Nodes represent images and there is an edge between two nodes if two images share some common properties. The node features are fromed by the bag-of-words representations of the images. The task is to predict the category each image belongs to. ", + "bbox": [ + 173, + 465, + 825, + 645 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 HYPERPARAMETER CONFIGURATIONS ", + "text_level": 1, + "bbox": [ + 178, + 662, + 482, + 676 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We conduct a grid search for hyperparameters. Table 8 summarizes the chosen hyperparameters for our models. ", + "bbox": [ + 174, + 689, + 823, + 717 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/56fc9e7751da96600b101beed5af73277f616af86ebe46fb979b2e3884f8d0c4.jpg", + "table_caption": [ + "Table 8: The chosen hyperparameters for our models on all datasets. " + ], + "table_footnote": [], + "table_body": "
ModelHyperparameterogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Convnumber of hopsL373102
hidden dimension d'512512256256512
convolutional kernel size k57573
learning rate5e-42e-58e-58e-45e-4
batch size122886432768245768192
weight decay5e-55e-505e-50
dropout rate0.250.50.50.50
Neighbor2Seq+Attnnumber of hopsL373102
hidden dimension d'512512256256512
learning rate5e-41e-32e-32e-35e-4
batch size122883072327682568192
weight decay5e-65e-505e-50
dropout rate0.250.50.50.50
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This novel", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "score": 1.0, + "content": "transformation enables the subsequent use of general deep learning operations,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "score": 1.0, + "content": "such as convolution and attention, that are designed for grid-like data. 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However, mes-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "sage passing methods have difficulties in handling such large graphs as the recursive message pass-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "ing mechanism leads to prohibitive computation and memory requirements. 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While the sampling methods can speed up training, they might result", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "in redundancy, still incur high computational complexity, lead to loss of performance, or introduce", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "bias (see Section 2.2). Generally, precomputing methods can scale to larger graphs as compared to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 450, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 450, + 651 + ], + "score": 1.0, + "content": "sampling methods as recursive message passing is still required in sampling methods.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 538, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this work, we propose the Neighbor2Seq that transforms the hierarchical neighborhood of each", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "node to a sequence in a precomputing step. After the Neighbor2Seq transformation, each node", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and its associated neighborhood tree are converted to an ordered sequence. Therefore, each node", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "can be viewed as an independent sample and is no longer constrained by the topological structure.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "This novel transformation from graphs to grid-like data enables the use of mini-batch training for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "subsequent models. As a result, our models can be used on extremely large graphs, as long as the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 270, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 270, + 734 + ], + "score": 1.0, + "content": "Neighbor2Seq step can be precomputed.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 655, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "As a radical departure from existing precomputing methods, we consider the hierarchical neighbor-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "hood of each node as an ordered sequence. The order information corresponds to hops between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "nodes. As a result of our Neighbor2Seq transformation, generic deep learning operations for grid-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "like data, such as convolution and attention, can be applied in subsequent models. In addition,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "our Neighbor2Seq can alleviate the over-squashing issue (Alon & Yahav, 2020) suffered by current", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "GNNs. Experimental results indicate that our proposed method can be used on a massive graph,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "where most current methods cannot be applied. Furthermore, our method achieves superior perfor-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 397, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 397, + 172 + ], + "score": 1.0, + "content": "mance as compared with previous sampling and precomputing methods.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 109, + 187, + 447, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 449, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 449, + 202 + ], + "score": 1.0, + "content": "2 ANALYSIS OF CURRENT GRAPH NEURAL NETWORK METHODS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 403, + 226 + ], + "score": 1.0, + "content": "We start by defining necessary notations. A graph is formally defined as", + "type": "text" + }, + { + "bbox": [ + 403, + 214, + 453, + 225 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 213, + 484, + 226 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 485, + 214, + 494, + 224 + ], + "score": 0.79, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 190, + 237 + ], + "score": 1.0, + "content": "the set of nodes and", + "type": "text" + }, + { + "bbox": [ + 191, + 225, + 243, + 236 + ], + "score": 0.92, + "content": "E \\subseteq V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 225, + 356, + 237 + ], + "score": 1.0, + "content": "is the set of edges. We use", + "type": "text" + }, + { + "bbox": [ + 356, + 225, + 392, + 237 + ], + "score": 0.92, + "content": "n = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 225, + 410, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 226, + 448, + 237 + ], + "score": 0.91, + "content": "m = | E |", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 411, + 248 + ], + "score": 1.0, + "content": "numbers of nodes and edges, respectively. The nodes are indexed from 1 to", + "type": "text" + }, + { + "bbox": [ + 412, + 238, + 419, + 246 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 235, + 505, + 248 + ], + "score": 1.0, + "content": ". We consider a node", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 507, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 165, + 260 + ], + "score": 1.0, + "content": "feature matrix", + "type": "text" + }, + { + "bbox": [ + 165, + 246, + 212, + 257 + ], + "score": 0.91, + "content": "\\ b { X } \\in \\mathbb { R } ^ { n \\times \\breve { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 244, + 282, + 260 + ], + "score": 1.0, + "content": ", where each row", + "type": "text" + }, + { + "bbox": [ + 282, + 246, + 318, + 258 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 244, + 344, + 260 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 344, + 248, + 351, + 257 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 244, + 507, + 260 + ], + "score": 1.0, + "content": "-dimensional feature vector associated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 255, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 148, + 271 + ], + "score": 1.0, + "content": "with node", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 153, + 268 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 255, + 455, + 271 + ], + "score": 1.0, + "content": ". The topology information of the graph is encoded in the adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 456, + 258, + 501, + 268 + ], + "score": 0.91, + "content": "\\ b { A } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 255, + 506, + 271 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 269, + 463, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 133, + 282 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 269, + 177, + 282 + ], + "score": 0.93, + "content": "A _ { ( i , j ) } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 269, + 302, + 282 + ], + "score": 1.0, + "content": "if an edge exists between node", + "type": "text" + }, + { + "bbox": [ + 303, + 270, + 308, + 279 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 269, + 347, + 282 + ], + "score": 1.0, + "content": "and node", + "type": "text" + }, + { + "bbox": [ + 347, + 270, + 353, + 281 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 269, + 374, + 282 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 374, + 269, + 418, + 282 + ], + "score": 0.93, + "content": "A _ { ( i , j ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 269, + 463, + 282 + ], + "score": 1.0, + "content": "otherwise.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 295, + 356, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 357, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 357, + 307 + ], + "score": 1.0, + "content": "2.1 GRAPH NEURAL NETWORKS VIA MESSAGE PASSING", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 504, + 329 + ], + "score": 1.0, + "content": "There are two primary deep learning methods on graphs (Bronstein et al.); those are, spectral meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "ods and spatial methods. The spectral method in Bruna et al. (2014) extends convolutional neural", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "networks (LeCun et al., 1989) to the graph domain based on the spectrum of the graph Laplacian.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "The main limitation of spectral methods is the high complexity. ChebNet (Defferrard et al., 2016)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "and GCN (Kipf & Welling, 2016) simplify the spectral methods and can be understood from the spa-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "tial perspective. In this work, we focus on the analysis of the current mainstream spatial methods.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "Generally, most existing spatial methods, such as ChebNet (Defferrard et al., 2016), GCN (Kipf", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "& Welling, 2016), GG-NN (Li et al., 2016), GAT (Velickovi ˇ c et al., 2018), and GIN (Xu et al., ´", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "2019), can be understood from the message passing perspective (Gilmer et al., 2017; Battaglia et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "2018). Specifically, we iteratively update node representations by aggregating representations from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "its immediate neighbors. These message passing methods have been shown to be effective in many", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "fields. However, they have inherent difficulties when applied on large graphs due to their excessive", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 380, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 380, + 460 + ], + "score": 1.0, + "content": "computation and memory requirements, as described in Section 2.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 340, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 341, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 341, + 487 + ], + "score": 1.0, + "content": "2.2 GRAPH NEURAL NETWORKS ON LARGE GRAPHS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "The above message passing methods are often trained in full batch. This requires the whole graph,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "i.e., all the node representations and edge connections, to be in memory to allow recursive message", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "passing on the whole graph. Usually, the number of neighbors grows very rapidly with the increase", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "of receptive field. Hence, these methods cannot be applied directly on large-scale graphs due to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "prohibitive requirements on computation and memory. To enable deep learning on large graphs, two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 501, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 501, + 564 + ], + "score": 1.0, + "content": "families of methods have been proposed; those are methods based on sampling and precomputing.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "To circumvent the recursive expansion of neighbors across layers, sampling methods apply GNNs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "on a sampled subset of nodes with mini-batch training. Sampling methods can be further divided", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "into three categories. First, node-wise sampling methods perform message passing for each node in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "its sampled neighborhood. This strategy is first proposed in GraphSAGE (Hamilton et al., 2017),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "where neighbors are randomly sampled. This is extended by PinSAGE (Ying et al., 2018), which", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "selects neighbors based on random walks. VR-GCN (Chen et al., 2018a) further proposes to use", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "variance reduction techniques to obtain a convergence guarantee. Although these node-wise sam-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 504, + 657 + ], + "score": 1.0, + "content": "pling methods can reduce computation, the remaining computation is still very expensive and some", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "redundancy might have been introduced, as described in Huang et al. (2018). Second, layer-wise", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "sampling methods sample a fixed number of nodes for each layer. In particular, FastGCN (Chen", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "et al., 2018b) samples a fixed number of nodes for each layer independently based on the degree of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "each node. AS-GCN (Huang et al., 2018) and LADIES (Zou et al., 2019) introduce between-layer", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "dependencies during sampling, thus alleviating the loss of information. Layer-wise sampling meth-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ods can avoid the redundancy introduced by node-wise sampling methods. However, the expensive", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "sampling algorithms that aim to ensure performance may themselves incur high computational cost,", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 43 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "As a radical departure from existing precomputing methods, we consider the hierarchical neighbor-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "hood of each node as an ordered sequence. The order information corresponds to hops between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "nodes. As a result of our Neighbor2Seq transformation, generic deep learning operations for grid-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "like data, such as convolution and attention, can be applied in subsequent models. In addition,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "our Neighbor2Seq can alleviate the over-squashing issue (Alon & Yahav, 2020) suffered by current", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "GNNs. Experimental results indicate that our proposed method can be used on a massive graph,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "where most current methods cannot be applied. Furthermore, our method achieves superior perfor-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 397, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 397, + 172 + ], + "score": 1.0, + "content": "mance as compared with previous sampling and precomputing methods.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 172 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 187, + 447, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 449, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 449, + 202 + ], + "score": 1.0, + "content": "2 ANALYSIS OF CURRENT GRAPH NEURAL NETWORK METHODS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 403, + 226 + ], + "score": 1.0, + "content": "We start by defining necessary notations. A graph is formally defined as", + "type": "text" + }, + { + "bbox": [ + 403, + 214, + 453, + 225 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 213, + 484, + 226 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 485, + 214, + 494, + 224 + ], + "score": 0.79, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 190, + 237 + ], + "score": 1.0, + "content": "the set of nodes and", + "type": "text" + }, + { + "bbox": [ + 191, + 225, + 243, + 236 + ], + "score": 0.92, + "content": "E \\subseteq V \\times V", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 225, + 356, + 237 + ], + "score": 1.0, + "content": "is the set of edges. We use", + "type": "text" + }, + { + "bbox": [ + 356, + 225, + 392, + 237 + ], + "score": 0.92, + "content": "n = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 225, + 410, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 226, + 448, + 237 + ], + "score": 0.91, + "content": "m = | E |", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 411, + 248 + ], + "score": 1.0, + "content": "numbers of nodes and edges, respectively. The nodes are indexed from 1 to", + "type": "text" + }, + { + "bbox": [ + 412, + 238, + 419, + 246 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 235, + 505, + 248 + ], + "score": 1.0, + "content": ". We consider a node", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 507, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 165, + 260 + ], + "score": 1.0, + "content": "feature matrix", + "type": "text" + }, + { + "bbox": [ + 165, + 246, + 212, + 257 + ], + "score": 0.91, + "content": "\\ b { X } \\in \\mathbb { R } ^ { n \\times \\breve { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 244, + 282, + 260 + ], + "score": 1.0, + "content": ", where each row", + "type": "text" + }, + { + "bbox": [ + 282, + 246, + 318, + 258 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 244, + 344, + 260 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 344, + 248, + 351, + 257 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 244, + 507, + 260 + ], + "score": 1.0, + "content": "-dimensional feature vector associated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 255, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 148, + 271 + ], + "score": 1.0, + "content": "with node", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 153, + 268 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 255, + 455, + 271 + ], + "score": 1.0, + "content": ". The topology information of the graph is encoded in the adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 456, + 258, + 501, + 268 + ], + "score": 0.91, + "content": "\\ b { A } \\in \\mathbb { R } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 255, + 506, + 271 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 269, + 463, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 133, + 282 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 269, + 177, + 282 + ], + "score": 0.93, + "content": "A _ { ( i , j ) } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 269, + 302, + 282 + ], + "score": 1.0, + "content": "if an edge exists between node", + "type": "text" + }, + { + "bbox": [ + 303, + 270, + 308, + 279 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 269, + 347, + 282 + ], + "score": 1.0, + "content": "and node", + "type": "text" + }, + { + "bbox": [ + 347, + 270, + 353, + 281 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 269, + 374, + 282 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 374, + 269, + 418, + 282 + ], + "score": 0.93, + "content": "A _ { ( i , j ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 269, + 463, + 282 + ], + "score": 1.0, + "content": "otherwise.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 213, + 507, + 282 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 295, + 356, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 357, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 357, + 307 + ], + "score": 1.0, + "content": "2.1 GRAPH NEURAL NETWORKS VIA MESSAGE PASSING", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 504, + 329 + ], + "score": 1.0, + "content": "There are two primary deep learning methods on graphs (Bronstein et al.); those are, spectral meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "ods and spatial methods. The spectral method in Bruna et al. (2014) extends convolutional neural", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "networks (LeCun et al., 1989) to the graph domain based on the spectrum of the graph Laplacian.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "The main limitation of spectral methods is the high complexity. ChebNet (Defferrard et al., 2016)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "and GCN (Kipf & Welling, 2016) simplify the spectral methods and can be understood from the spa-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "tial perspective. In this work, we focus on the analysis of the current mainstream spatial methods.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "Generally, most existing spatial methods, such as ChebNet (Defferrard et al., 2016), GCN (Kipf", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "& Welling, 2016), GG-NN (Li et al., 2016), GAT (Velickovi ˇ c et al., 2018), and GIN (Xu et al., ´", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "2019), can be understood from the message passing perspective (Gilmer et al., 2017; Battaglia et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "2018). Specifically, we iteratively update node representations by aggregating representations from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "its immediate neighbors. These message passing methods have been shown to be effective in many", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "fields. However, they have inherent difficulties when applied on large graphs due to their excessive", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 380, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 380, + 460 + ], + "score": 1.0, + "content": "computation and memory requirements, as described in Section 2.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 317, + 506, + 460 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 340, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 341, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 341, + 487 + ], + "score": 1.0, + "content": "2.2 GRAPH NEURAL NETWORKS ON LARGE GRAPHS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "The above message passing methods are often trained in full batch. This requires the whole graph,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "i.e., all the node representations and edge connections, to be in memory to allow recursive message", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "passing on the whole graph. Usually, the number of neighbors grows very rapidly with the increase", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "of receptive field. Hence, these methods cannot be applied directly on large-scale graphs due to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "prohibitive requirements on computation and memory. To enable deep learning on large graphs, two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 501, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 501, + 564 + ], + "score": 1.0, + "content": "families of methods have been proposed; those are methods based on sampling and precomputing.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 495, + 505, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "To circumvent the recursive expansion of neighbors across layers, sampling methods apply GNNs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "on a sampled subset of nodes with mini-batch training. Sampling methods can be further divided", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "into three categories. First, node-wise sampling methods perform message passing for each node in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "its sampled neighborhood. This strategy is first proposed in GraphSAGE (Hamilton et al., 2017),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "where neighbors are randomly sampled. This is extended by PinSAGE (Ying et al., 2018), which", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "selects neighbors based on random walks. VR-GCN (Chen et al., 2018a) further proposes to use", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "variance reduction techniques to obtain a convergence guarantee. Although these node-wise sam-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 504, + 657 + ], + "score": 1.0, + "content": "pling methods can reduce computation, the remaining computation is still very expensive and some", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "redundancy might have been introduced, as described in Huang et al. (2018). Second, layer-wise", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "sampling methods sample a fixed number of nodes for each layer. In particular, FastGCN (Chen", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "et al., 2018b) samples a fixed number of nodes for each layer independently based on the degree of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "each node. AS-GCN (Huang et al., 2018) and LADIES (Zou et al., 2019) introduce between-layer", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "dependencies during sampling, thus alleviating the loss of information. Layer-wise sampling meth-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ods can avoid the redundancy introduced by node-wise sampling methods. However, the expensive", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "sampling algorithms that aim to ensure performance may themselves incur high computational cost,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "as pointed out in Zeng et al. (2020). Third, graph-wise sampling methods build mini-batches on", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "sampled subgraphs. Specifically, LGCN (Gao et al., 2018) proposes to leverage mini-batch training", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "on subgraphs selected by Breadth-First-Search algorithms. ClusterGCN (Chiang et al., 2019) con-", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "ducts mini-batch training on sampled subgraphs that are obtained by a graph clustering algorithm.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "GraphSAINT (Zeng et al., 2020) proposes to derive subgraphs by importance-sampling and intro-", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "duces normalization techniques to eliminate biases. These graph-wise sampling methods usually", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "have high efficiency. The main limitation is that the nodes in a sampled subgraph are usually clus-", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "tered together. This implies that two distant nodes in the original graph usually cannot be feeded into", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 502, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 502, + 183 + ], + "score": 1.0, + "content": "the GNNs in the same mini-batch during training, potentially leading to bias in the trained models.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 567, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "as pointed out in Zeng et al. (2020). Third, graph-wise sampling methods build mini-batches on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "sampled subgraphs. Specifically, LGCN (Gao et al., 2018) proposes to leverage mini-batch training", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "on subgraphs selected by Breadth-First-Search algorithms. ClusterGCN (Chiang et al., 2019) con-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "ducts mini-batch training on sampled subgraphs that are obtained by a graph clustering algorithm.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "GraphSAINT (Zeng et al., 2020) proposes to derive subgraphs by importance-sampling and intro-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "duces normalization techniques to eliminate biases. These graph-wise sampling methods usually", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "have high efficiency. The main limitation is that the nodes in a sampled subgraph are usually clus-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "tered together. This implies that two distant nodes in the original graph usually cannot be feeded into", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 502, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 502, + 183 + ], + "score": 1.0, + "content": "the GNNs in the same mini-batch during training, potentially leading to bias in the trained models.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "The second family of methods for enabling GNNs training on large graphs are based on procomput-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "ing. Specifically, SGC (Wu et al., 2019) removes the non-linearity between GCN layers, resulting", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 192, + 225 + ], + "score": 1.0, + "content": "in a simplification as", + "type": "text" + }, + { + "bbox": [ + 193, + 209, + 294, + 223 + ], + "score": 0.9, + "content": "Y = \\mathrm { s o f t m a x } ( \\hat { A } ^ { L } X W )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 208, + 380, + 225 + ], + "score": 1.0, + "content": ". In this formulation,", + "type": "text" + }, + { + "bbox": [ + 380, + 209, + 456, + 221 + ], + "score": 0.91, + "content": "\\hat { \\pmb { A } } = \\tilde { \\pmb { D } } ^ { - \\frac 1 2 } \\tilde { \\pmb { A } } \\tilde { \\pmb { D } } ^ { - \\frac 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 208, + 505, + 225 + ], + "score": 1.0, + "content": "is the sym-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 271, + 237 + ], + "score": 1.0, + "content": "metrically normalized adjacency matrix,", + "type": "text" + }, + { + "bbox": [ + 271, + 222, + 322, + 234 + ], + "score": 0.9, + "content": "{ \\tilde { A } } = A + I", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 223, + 483, + 237 + ], + "score": 1.0, + "content": "is the adjacency matrix with self-loops,", + "type": "text" + }, + { + "bbox": [ + 484, + 222, + 495, + 234 + ], + "score": 0.85, + "content": "\\tilde { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 319, + 250 + ], + "score": 1.0, + "content": "the corresponding diagonal node degree matrix with", + "type": "text" + }, + { + "bbox": [ + 319, + 234, + 399, + 250 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\tilde { D } _ { ( i , i ) } = \\mathrm { \\bar { \\sum } } _ { j } \\tilde { A } _ { ( i , j ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 235, + 403, + 250 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 403, + 236, + 411, + 246 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 235, + 506, + 250 + ], + "score": 1.0, + "content": "is the size of receptive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 424, + 261 + ], + "score": 1.0, + "content": "field (i.e., the number of considered neighboring hops), which is the same as a", + "type": "text" + }, + { + "bbox": [ + 425, + 248, + 433, + 258 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 247, + 483, + 261 + ], + "score": 1.0, + "content": "-layer GCN,", + "type": "text" + }, + { + "bbox": [ + 484, + 248, + 495, + 258 + ], + "score": 0.75, + "content": "\\mathbf { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 258, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 420, + 275 + ], + "score": 1.0, + "content": "the output of the softmax classifier. Since there is no learnable parameters in", + "type": "text" + }, + { + "bbox": [ + 420, + 259, + 446, + 271 + ], + "score": 0.91, + "content": "\\hat { A } ^ { L } X", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 258, + 506, + 275 + ], + "score": 1.0, + "content": ", this term can", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "be precomputed as a feature pre-processing step. Similarly, SIGN (Rossi et al., 2020) applies an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 307, + 297 + ], + "score": 1.0, + "content": "inception-like model to the precomputed features", + "type": "text" + }, + { + "bbox": [ + 308, + 282, + 331, + 295 + ], + "score": 0.9, + "content": "{ \\hat { A } } ^ { \\ell } X", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 283, + 347, + 297 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 347, + 284, + 411, + 296 + ], + "score": 0.93, + "content": "\\ell \\in \\{ 1 , \\cdots , L \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 283, + 443, + 297 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 443, + 284, + 451, + 294 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 283, + 505, + 297 + ], + "score": 1.0, + "content": "is the prede-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "fined size of receptive field. Instead of precomputing the smoothing features as in SGC and SIGN,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 305, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 319 + ], + "score": 1.0, + "content": "PPRGo (Bojchevski et al., 2020) extends the idea of PPNP (Klicpera et al., 2018) by approximately", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "precomputing the personalized PageRank (Page et al., 1999) matrix, thereby enabling model training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "on large graphs using mini-batches. Generally, the precomputing methods can scale to larger graphs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 353 + ], + "score": 1.0, + "content": "than sampling methods because the latter still needs to perform the recursive message passing during", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "training. Differing from these precomputing methods, we consider the hierarchical neighborhood of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "each node as an ordered sequence, thus retaining the useful information about hops between nodes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 343, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 343, + 385 + ], + "score": 1.0, + "content": "and enabling subsequent powerful and efficient operations.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 423, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 424, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 424, + 416 + ], + "score": 1.0, + "content": "3 THE PROPOSED NEIGHBOR2SEQ METHOD AND ANALYSIS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "In this section, we describe our proposed method, known as Neighbor2Seq, which transforms the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "hierarchical neighborhood of each node into an ordered sequence, thus enabling the subsequent use", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "of general deep learning operations. We analyze the scalability of our method (See Section 3.5) and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "score": 1.0, + "content": "describe how our method can alleviate the over-squashing issue suffered by current message passing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 471, + 216, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 216, + 483 + ], + "score": 1.0, + "content": "methods (See Section 3.6).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 498, + 178, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 180, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 180, + 510 + ], + "score": 1.0, + "content": "3.1 OVERVIEW", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "As described in Section 2.1, existing message passing methods recursively update each node’s rep-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "resentation by aggregating information from its immediate neighbors. Hence, what these methods", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "aim at capturing for each node is essentially its corresponding hierarchical neighborhood, i.e., the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "neighborhood tree rooted at current node, as illustrated in Figure 1 (b). In this work, we attempt", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "to go beyond the message passing scheme to overcome the limitations mentioned in Section 2. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "propose to capture the information of this hierarchical neighborhood by transforming it into an or-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "dered sequence, instead of recursively squashing it into a fixed-length vector. Our proposed method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "is composed of three steps. First, we transform a neighborhood to a sequence for each node. Second,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "we apply a normalization technique to the derived sequence features. Third, we use general deep", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "learning operations, i.e., convolution and attention, to learn from these sequence features and then", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 629, + 444, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 444, + 641 + ], + "score": 1.0, + "content": "make predictions for nodes. 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Differing from these precomputing methods, we consider the hierarchical neighborhood of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "each node as an ordered sequence, thus retaining the useful information about hops between nodes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 343, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 343, + 385 + ], + "score": 1.0, + "content": "and enabling subsequent powerful and efficient operations.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 186, + 506, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 423, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 424, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 424, + 416 + ], + "score": 1.0, + "content": "3 THE PROPOSED NEIGHBOR2SEQ METHOD AND ANALYSIS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "In this section, we describe our proposed method, known as Neighbor2Seq, which transforms the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "hierarchical neighborhood of each node into an ordered sequence, thus enabling the subsequent use", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "of general deep learning operations. 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We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "propose to capture the information of this hierarchical neighborhood by transforming it into an or-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "dered sequence, instead of recursively squashing it into a fixed-length vector. Our proposed method", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "is composed of three steps. First, we transform a neighborhood to a sequence for each node. 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Fol-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 306, + 713 + ], + "score": 1.0, + "content": "lowing the notations defined in Section 2, we let", + "type": "text" + }, + { + "bbox": [ + 307, + 699, + 366, + 711 + ], + "score": 0.91, + "content": "z _ { 0 } ^ { i } , z _ { 1 } ^ { i } , \\cdots , z _ { L } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "denote the resulting sequence for", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 128, + 722 + ], + "score": 1.0, + "content": "node", + "type": "text" + }, + { + "bbox": [ + 128, + 711, + 133, + 720 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 710, + 163, + 722 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 163, + 711, + 172, + 720 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "is the height (i.e., the number of hops we consider) of the neighborhood tree rooted", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 138, + 734 + ], + "score": 1.0, + "content": "at node", + "type": "text" + }, + { + "bbox": [ + 138, + 721, + 142, + 730 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 719, + 146, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 147, + 720, + 181, + 733 + ], + "score": 0.9, + "content": "z _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 719, + 230, + 734 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 230, + 721, + 235, + 730 + ], + "score": 0.76, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "-th feature of the sequence. 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We define", + "type": "text" + }, + { + "bbox": [ + 226, + 439, + 262, + 451 + ], + "score": 0.93, + "content": "w ( i , j , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 439, + 295, + 451 + ], + "score": 1.0, + "content": "as 1 for", + "type": "text" + }, + { + "bbox": [ + 295, + 440, + 317, + 451 + ], + "score": 0.91, + "content": "j = i", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 439, + 415, + 451 + ], + "score": 1.0, + "content": "and 0 otherwise. 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Note that the derived sequence has meaningful", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "order information, indicating the hops between nodes. 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The activation function between", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 349, + 272, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 272, + 361 + ], + "score": 1.0, + "content": "layers is ReLU (Krizhevsky et al., 2012).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 304, + 505, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "Incorporating attention is another natural idea to learn from sequences. 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Formally, for each node", + "type": "text" + }, + { + "bbox": [ + 351, + 421, + 356, + 430 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 421, + 505, + 433 + ], + "score": 1.0, + "content": ", we add positional encoding for each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 431, + 211, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 211, + 445 + ], + "score": 1.0, + "content": "feature in the sequence as", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 366, + 506, + 445 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 448, + 436, + 505 + ], + "lines": [ + { + "bbox": [ + 173, + 448, + 436, + 505 + ], + "spans": [ + { + "bbox": [ + 173, + 448, + 436, + 505 + ], + "score": 0.95, + "content": "k _ { \\ell } ^ { i } = o _ { \\ell } ^ { i } + p _ { \\ell } ^ { i } , \\quad p _ { \\ell } ^ { i } [ m ] = \\left\\{ \\begin{array} { l l } { { \\sin \\left( \\displaystyle { \\frac { \\ell } { 1 0 0 0 0 ^ { \\frac { 2 n } { d ^ { \\prime } } } } } \\right) } } & { { m = 2 n } } \\\\ { { \\cos \\left( \\displaystyle { \\frac { \\ell } { 1 0 0 0 0 ^ { \\frac { 2 n } { d ^ { \\prime } } } } } \\right) } } & { { m = 2 n + 1 } } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "a8d764b29b380ea5970f6076db24a83373f9976d1fba2cc51aa5a129e43d705f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 173, + 448, + 436, + 467.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 173, + 467.0, + 436, + 486.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 173, + 486.0, + 436, + 505.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 506, + 566 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 222, + 524 + ], + "score": 1.0, + "content": "The positional encoding for", + "type": "text" + }, + { + "bbox": [ + 222, + 511, + 227, + 521 + ], + "score": 0.69, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 509, + 304, + 524 + ], + "score": 1.0, + "content": "-th feature of node", + "type": "text" + }, + { + "bbox": [ + 305, + 512, + 309, + 521 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 509, + 367, + 524 + ], + "score": 1.0, + "content": "is denoted as", + "type": "text" + }, + { + "bbox": [ + 367, + 509, + 407, + 523 + ], + "score": 0.88, + "content": "\\pmb { p } _ { \\ell } ^ { i } \\in \\mathbb { R } ^ { d ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 509, + 413, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 414, + 510, + 494, + 523 + ], + "score": 0.9, + "content": "m \\in \\{ 1 , 2 , \\cdots , d ^ { \\prime } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "the dimensional index. Intuitively, a position-dependent vector is added to each feature such that the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "order information can be captured. Then we use the attention mechanism with learnable query (Yang", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 460, + 556 + ], + "score": 1.0, + "content": "et al., 2016) to combine these sequential feature vectors to obtain the final representations", + "type": "text" + }, + { + "bbox": [ + 460, + 543, + 469, + 554 + ], + "score": 0.86, + "content": "r ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 178, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 128, + 568 + ], + "score": 1.0, + "content": "node", + "type": "text" + }, + { + "bbox": [ + 129, + 556, + 133, + 564 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 553, + 178, + 568 + ], + "score": 1.0, + "content": ". Formally,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 509, + 506, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 564, + 393, + 599 + ], + "lines": [ + { + "bbox": [ + 218, + 564, + 393, + 599 + ], + "spans": [ + { + "bbox": [ + 218, + 564, + 393, + 599 + ], + "score": 0.94, + "content": "\\pmb { r } ^ { i } = \\sum _ { \\ell = 0 } ^ { L } \\alpha _ { \\ell } ^ { i } \\pmb { k } _ { \\ell } ^ { i } , \\quad \\alpha _ { \\ell } ^ { i } = \\frac { \\exp ( \\pmb { k } _ { \\ell } ^ { i ^ { T } } \\pmb { q } ) } { \\sum _ { \\ell = 0 } ^ { L } \\exp ( \\pmb { k } _ { \\ell } ^ { i ^ { T } } \\pmb { q } ) } .", + "type": "interline_equation", + "image_path": "2f7d6802d2122f35927e768b610a02f08b246cc689cbd8a7b1a6f6abf9ce9a03.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 564, + 393, + 581.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 218, + 581.5, + 393, + 599.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 107, + 601, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 107, + 602, + 139, + 615 + ], + "score": 0.91, + "content": "\\boldsymbol { q } \\in \\mathbb { R } ^ { d ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 601, + 506, + 617 + ], + "score": 1.0, + "content": "is the learnable query vector that is trained along with other model parameters. The derived", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 613, + 482, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 165, + 627 + ], + "score": 1.0, + "content": "representation", + "type": "text" + }, + { + "bbox": [ + 166, + 614, + 175, + 624 + ], + "score": 0.85, + "content": "r ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 613, + 472, + 627 + ], + "score": 1.0, + "content": "will be taken as the input to a linear classifier to make prediction for node", + "type": "text" + }, + { + "bbox": [ + 473, + 615, + 477, + 624 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 613, + 482, + 627 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 601, + 506, + 627 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 639, + 250, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 252, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 252, + 651 + ], + "score": 1.0, + "content": "3.5 ANALYSIS OF SCALABILITY", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 659, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 401, + 672 + ], + "score": 1.0, + "content": "Precomputing Neighbor2Seq. A well-known fact is that the value of", + "type": "text" + }, + { + "bbox": [ + 401, + 660, + 437, + 672 + ], + "score": 0.93, + "content": "w ( i , j , \\ell )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "in Equation (1)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 416, + 683 + ], + "score": 1.0, + "content": "can be obtained by computing the power of the original adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 417, + 672, + 426, + 681 + ], + "score": 0.76, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 670, + 506, + 683 + ], + "score": 1.0, + "content": ". Following GCN,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 681, + 507, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 392, + 697 + ], + "score": 1.0, + "content": "we add self-loops to make each node connected to itself. Concretely,", + "type": "text" + }, + { + "bbox": [ + 392, + 681, + 468, + 697 + ], + "score": 0.93, + "content": "w ( i , j , \\ell ) = \\tilde { A } _ { ( i , j ) } ^ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 681, + 507, + 697 + ], + "score": 1.0, + "content": "Hence,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 439, + 711 + ], + "score": 1.0, + "content": "the Neighbor2Seq can be implemented by computing the matrix multiplications", + "type": "text" + }, + { + "bbox": [ + 439, + 697, + 463, + 709 + ], + "score": 0.88, + "content": "\\tilde { A } ^ { \\ell } X", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 698, + 480, + 711 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 481, + 698, + 505, + 710 + ], + "score": 0.85, + "content": "\\forall \\ell \\in", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 172, + 722 + ], + "score": 0.91, + "content": "\\{ 0 , 1 , 2 , \\bar { \\cdot } \\cdot \\cdot , L \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 709, + 504, + 721 + ], + "score": 1.0, + "content": ". Since there is no learnable parameters in the Neighbor2Seq step, these matrix mul-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "tiplications can be precomputed sequentially for large graphs on CPU platforms with large memory.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 660, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 311, + 96 + ], + "score": 1.0, + "content": "This can be easily precomputed because the matrix", + "type": "text" + }, + { + "bbox": [ + 312, + 81, + 322, + 93 + ], + "score": 0.84, + "content": "\\tilde { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "is usually sparse. 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Our mini-batches can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "be randomly extracted over all nodes, opening the possibility that any pair of nodes can be sampled", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "in the same mini-batch. In contrast, mini-batches in sampling methods are usually restricted by the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "fixed sampling strategies. This advantage opens the door for subsequent model training on extremely", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 440, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 440, + 211 + ], + "score": 1.0, + "content": "large graphs, as long as the corresponding Neighbor2Seq step can be precomputed.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 315, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "score": 1.0, + "content": "Computational complexity comparison. We com-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 317, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 317, + 239 + ], + "score": 1.0, + "content": "pare our methods with several existing sampling and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 237, + 317, + 248 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 317, + 248 + ], + "score": 1.0, + "content": "precomputing methods in terms of computational", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 317, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 191, + 260 + ], + "score": 1.0, + "content": "complexity. 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For sam-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 316, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 171, + 292 + ], + "score": 1.0, + "content": "pling methods,", + "type": "text" + }, + { + "bbox": [ + 171, + 283, + 177, + 291 + ], + "score": 0.53, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 281, + 316, + 292 + ], + "score": 1.0, + "content": "is the number of sampled neigh-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 316, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 316, + 303 + ], + "score": 1.0, + "content": "bors for each node. 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MethodPrecomputingForward Pass
GCNO(Ldm + Ld²n)
GraphSAGEO(sLn)O(sLd²n)
ClusterGCNO(m)O(Ldm + Ld²n)
GraphSAINTO(sn)O(Ldm +Ld²n)
SGCO(Ldm)O(d²n)
SIGNO(Ldm)O(Ld²n)
Neighbor2Seq+ConvO(Ldm)O(Ld²+Lkd²)n)
Neighbor2Seq+AttnO(Ldm)O(Ld²+Ld)n)
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Hence, the computational complexity for the forward pass of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 143, + 360 + ], + "score": 1.0, + "content": "Neighbo", + "type": "text" + }, + { + "bbox": [ + 143, + 347, + 192, + 357 + ], + "score": 0.4, + "content": "_ { 1 2 \\mathrm { S e q + C o n v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 345, + 203, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 204, + 346, + 285, + 358 + ], + "score": 0.91, + "content": "\\mathcal { O } ( ( L d ^ { 2 } + L k d ^ { 2 } ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 345, + 505, + 360 + ], + "score": 1.0, + "content": ". Neighbor2Seq+Attn has a computational complexity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 117, + 370 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 185, + 370 + ], + "score": 0.92, + "content": "\\mathcal { \\bar { O } } ( ( L d ^ { 2 } + \\bar { L } d ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "because the attention mechanism is more efficient than 1-D convolutional neural", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "networks. As shown in Table 1, the forward pass complexities of precomputing methods, including", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "our Neighbor2Seq+Conv and Neighbor2Seq+Attn, are all linear with respect to the number of nodes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 114, + 400 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 390, + 293, + 403 + ], + "score": 1.0, + "content": "and do not depend on the number of edges", + "type": "text" + }, + { + "bbox": [ + 293, + 392, + 303, + 400 + ], + "score": 0.52, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ". Hence, the training processes of our models are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 402, + 211, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 211, + 414 + ], + "score": 1.0, + "content": "computationally efficient.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 425, + 321, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 322, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 322, + 438 + ], + "score": 1.0, + "content": "3.6 ALLEVIATING THE OVER-SQUASHING ISSUE", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "An inherent problem in message passing methods is known as the over-squashing (Alon & Yahav,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "score": 1.0, + "content": "2020). In particular, recursively propagating information between neighbors creates a bottleneck", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "because the number of nodes in the receptive field grows exponentially with the number of layers.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "This bottleneck causes the over-squashing issue; that is, information from the exponentially-growing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "receptive field is compressed into fixed-length vectors. Consequently, message passing methods fail", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "to capture the message flowing from distant nodes and performs poorly when long-range information", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "is essential for the prediction tasks. Note that the over-squashing issue is not identical to the over-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "smoothing issue. Over-smoothing is related to the phenomenon that node representations converge", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "to indistinguishable limits when the number of layers increases (Li et al., 2018; Wu et al., 2019; NT", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "& Maehara, 2019; Liu et al., 2020a; Oono & Suzuki, 2020; Cai & Wang, 2020; Chen et al., 2020).", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "score": 1.0, + "content": "The virtual edges added in Gilmer et al. (2017) and recent non-local aggregations (Pei et al., 2020;", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "Liu et al., 2020b) can be viewed as attempts to alleviate the over-squashing issue by incorporating", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "distant nodes. Another study (Ma et al., 2019) considers message passing along all possible paths", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 589, + 408, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 408, + 600 + ], + "score": 1.0, + "content": "between two nodes, instead of propagating information between neighbors.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 108, + 605, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Our Neighbor2Seq can alleviate the over-squashing issue because we transform the exponentially-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "growing nodes in hierarchical neighborhoods into an ordered sequence, instead of recursively", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "squashing them into a fixed-size vector. With our Neighbor2Seq, capturing long-range informa-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "score": 1.0, + "content": "tion on graphs becomes similar to achieving this on sequence data, such as texts.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54.5 + }, + { + "type": "title", + "bbox": [ + 108, + 665, + 196, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 198, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 198, + 681 + ], + "score": 1.0, + "content": "4 DISCUSSIONS", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 302, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 303, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 303, + 703 + ], + "score": 1.0, + "content": "4.1 INFORMATION LOSS IN NEIGHBOR2SEQ", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "As shown in Figure 1 (c), Neighbor2Seq obtains the sequence by integrating features of nodes", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "in each layer of the neighborhood tree. This transformation may lose the cross-layer dependency", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 59.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 311, + 96 + ], + "score": 1.0, + "content": "This can be easily precomputed because the matrix", + "type": "text" + }, + { + "bbox": [ + 312, + 81, + 322, + 93 + ], + "score": 0.84, + "content": "\\tilde { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "is usually sparse. For extremely large graphs,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 375, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 375, + 106 + ], + "score": 1.0, + "content": "this precomputation can even be performed on distributed systems.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "Enabling mini-batch training. After we obtain the precomputed sequence features, each node in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "the graph corresponds to a sequence of feature vectors. Therefore, each node can be viewed as an", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "score": 1.0, + "content": "independent sample. That is, we are no longer restricted by the original graph connectivity anymore.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "Then, we can randomly sample from all the training nodes to conduct mini-batch training. This is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "more flexible and unbiased than sampling methods as reviewed in Section 2.2. Our mini-batches can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "be randomly extracted over all nodes, opening the possibility that any pair of nodes can be sampled", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "in the same mini-batch. In contrast, mini-batches in sampling methods are usually restricted by the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "fixed sampling strategies. This advantage opens the door for subsequent model training on extremely", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 440, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 440, + 211 + ], + "score": 1.0, + "content": "large graphs, as long as the corresponding Neighbor2Seq step can be precomputed.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 110, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 315, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 317, + 227 + ], + "score": 1.0, + "content": "Computational complexity comparison. We com-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 317, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 317, + 239 + ], + "score": 1.0, + "content": "pare our methods with several existing sampling and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 237, + 317, + 248 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 317, + 248 + ], + "score": 1.0, + "content": "precomputing methods in terms of computational", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 317, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 191, + 260 + ], + "score": 1.0, + "content": "complexity. We let", + "type": "text" + }, + { + "bbox": [ + 191, + 248, + 200, + 258 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 248, + 317, + 260 + ], + "score": 1.0, + "content": "denote the number of hops", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 317, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 317, + 271 + ], + "score": 1.0, + "content": "we consider. For simplicity, we assume the fea-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 317, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 171, + 281 + ], + "score": 1.0, + "content": "ture dimension", + "type": "text" + }, + { + "bbox": [ + 171, + 270, + 178, + 280 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 270, + 317, + 281 + ], + "score": 1.0, + "content": "is fixed for all layers. For sam-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 316, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 171, + 292 + ], + "score": 1.0, + "content": "pling methods,", + "type": "text" + }, + { + "bbox": [ + 171, + 283, + 177, + 291 + ], + "score": 0.53, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 281, + 316, + 292 + ], + "score": 1.0, + "content": "is the number of sampled neigh-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 316, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 316, + 303 + ], + "score": 1.0, + "content": "bors for each node. 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MethodPrecomputingForward Pass
GCNO(Ldm + Ld²n)
GraphSAGEO(sLn)O(sLd²n)
ClusterGCNO(m)O(Ldm + Ld²n)
GraphSAINTO(sn)O(Ldm +Ld²n)
SGCO(Ldm)O(d²n)
SIGNO(Ldm)O(Ld²n)
Neighbor2Seq+ConvO(Ldm)O(Ld²+Lkd²)n)
Neighbor2Seq+AttnO(Ldm)O(Ld²+Ld)n)
", + "type": "table", + "image_path": "54349962c508f86d4ef5fdbdbdc1d389ed999225fc0dfc0a04923d637b69ad13.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 325, + 249, + 503, + 262.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 325, + 262.6666666666667, + 503, + 276.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 325, + 276.33333333333337, + 503, + 290.00000000000006 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 325, + 290.00000000000006, + 503, + 303.66666666666674 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 325, + 303.66666666666674, + 503, + 317.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 325, + 317.3333333333334, + 503, + 331.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "index": 25.25 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 107, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 335, + 151, + 347 + ], + "score": 0.92, + "content": "O ( L k d ^ { 2 } n )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 334, + 181, + 349 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 181, + 336, + 188, + 345 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 334, + 506, + 349 + ], + "score": 1.0, + "content": "is the kernel size). Hence, the computational complexity for the forward pass of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 345, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 143, + 360 + ], + "score": 1.0, + "content": "Neighbo", + "type": "text" + }, + { + "bbox": [ + 143, + 347, + 192, + 357 + ], + "score": 0.4, + "content": "_ { 1 2 \\mathrm { S e q + C o n v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 345, + 203, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 204, + 346, + 285, + 358 + ], + "score": 0.91, + "content": "\\mathcal { O } ( ( L d ^ { 2 } + L k d ^ { 2 } ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 345, + 505, + 360 + ], + "score": 1.0, + "content": ". Neighbor2Seq+Attn has a computational complexity", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 117, + 370 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 185, + 370 + ], + "score": 0.92, + "content": "\\mathcal { \\bar { O } } ( ( L d ^ { 2 } + \\bar { L } d ) n )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "because the attention mechanism is more efficient than 1-D convolutional neural", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "networks. As shown in Table 1, the forward pass complexities of precomputing methods, including", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "our Neighbor2Seq+Conv and Neighbor2Seq+Attn, are all linear with respect to the number of nodes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 114, + 400 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 390, + 293, + 403 + ], + "score": 1.0, + "content": "and do not depend on the number of edges", + "type": "text" + }, + { + "bbox": [ + 293, + 392, + 303, + 400 + ], + "score": 0.52, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ". Hence, the training processes of our models are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 402, + 211, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 211, + 414 + ], + "score": 1.0, + "content": "computationally efficient.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 334, + 506, + 414 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 425, + 321, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 322, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 322, + 438 + ], + "score": 1.0, + "content": "3.6 ALLEVIATING THE OVER-SQUASHING ISSUE", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 446, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "An inherent problem in message passing methods is known as the over-squashing (Alon & Yahav,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 506, + 471 + ], + "score": 1.0, + "content": "2020). In particular, recursively propagating information between neighbors creates a bottleneck", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "because the number of nodes in the receptive field grows exponentially with the number of layers.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 493 + ], + "score": 1.0, + "content": "This bottleneck causes the over-squashing issue; that is, information from the exponentially-growing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "receptive field is compressed into fixed-length vectors. Consequently, message passing methods fail", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "to capture the message flowing from distant nodes and performs poorly when long-range information", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "is essential for the prediction tasks. Note that the over-squashing issue is not identical to the over-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "smoothing issue. Over-smoothing is related to the phenomenon that node representations converge", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "to indistinguishable limits when the number of layers increases (Li et al., 2018; Wu et al., 2019; NT", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "& Maehara, 2019; Liu et al., 2020a; Oono & Suzuki, 2020; Cai & Wang, 2020; Chen et al., 2020).", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 504, + 568 + ], + "score": 1.0, + "content": "The virtual edges added in Gilmer et al. (2017) and recent non-local aggregations (Pei et al., 2020;", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 505, + 581 + ], + "score": 1.0, + "content": "Liu et al., 2020b) can be viewed as attempts to alleviate the over-squashing issue by incorporating", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "distant nodes. Another study (Ma et al., 2019) considers message passing along all possible paths", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 589, + 408, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 408, + 600 + ], + "score": 1.0, + "content": "between two nodes, instead of propagating information between neighbors.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 446, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 605, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Our Neighbor2Seq can alleviate the over-squashing issue because we transform the exponentially-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "growing nodes in hierarchical neighborhoods into an ordered sequence, instead of recursively", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "squashing them into a fixed-size vector. With our Neighbor2Seq, capturing long-range informa-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "score": 1.0, + "content": "tion on graphs becomes similar to achieving this on sequence data, such as texts.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 605, + 506, + 651 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 665, + 196, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 198, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 198, + 681 + ], + "score": 1.0, + "content": "4 DISCUSSIONS", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 57 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 302, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 303, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 303, + 703 + ], + "score": 1.0, + "content": "4.1 INFORMATION LOSS IN NEIGHBOR2SEQ", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "As shown in Figure 1 (c), Neighbor2Seq obtains the sequence by integrating features of nodes", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "in each layer of the neighborhood tree. This transformation may lose the cross-layer dependency", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "information in the tree. Specifically, the Neighbor2Seq ignores the identities of nodes that each", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "walk passes through and only considers what are the nodes in each layer of the neighborhood tree.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "Nevertheless, this information can neither be captured by message passing methods because the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "aggregation is usually permutation-invariant. This implies that messages from different neighbors", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "cannot be distinguished, as pointed in Pei et al. (2020). According to our experimental results in", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "Table 5, our models without this information can outperform message passing methods, such as", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "GCN. It is intriguing to have an in-depth exploration of whether such information is useful and how", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 182, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 182, + 171 + ], + "score": 1.0, + "content": "it can be captured.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 59.5, + "bbox_fs": [ + 104, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "information in the tree. Specifically, the Neighbor2Seq ignores the identities of nodes that each", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "walk passes through and only considers what are the nodes in each layer of the neighborhood tree.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "Nevertheless, this information can neither be captured by message passing methods because the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "aggregation is usually permutation-invariant. This implies that messages from different neighbors", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "cannot be distinguished, as pointed in Pei et al. (2020). According to our experimental results in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "Table 5, our models without this information can outperform message passing methods, such as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "GCN. It is intriguing to have an in-depth exploration of whether such information is useful and how", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 182, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 182, + 171 + ], + "score": 1.0, + "content": "it can be captured.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 184, + 378, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 379, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 379, + 196 + ], + "score": 1.0, + "content": "4.2 RELATIONS WITH THE WEISFEILER-LEHMAN HIERARCHY", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "As shown in Xu et al. (2019), most of current GNNs are at most as powerful as the Weisfeiler-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "Lehman (WL) graph isomorphism test (Weisfeiler & Lehman, 1968) in distinguishing graph struc-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "tures. Our Neighbor2Seq is still under the WL hierarchy because the neighborhood tree used to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "obtain the sequence is indeed the one that the WL test uses to distinguish different graphs. We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "would be interested in exploring how Neighbor2Seq can be extended to go beyond the WL hierar-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 260, + 208, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 208, + 272 + ], + "score": 1.0, + "content": "chy as a future direction.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 284, + 392, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 394, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 394, + 298 + ], + "score": 1.0, + "content": "4.3 BRIDGING THE GAP BETWEEN GRAPH AND GRID-LIKE DATA", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "The main difference between graph and grid-like data lies in the notion and properties of locality.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "Specifically, the numbers of neighbors differ for different nodes, and there is no order information", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 328, + 504, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 504, + 340 + ], + "score": 1.0, + "content": "among the neighbors of a node in graphs. These are the main obstacles preventing the use of generic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "deep learning operations on graphs. Our Neighbor2Seq is an attempt to bridge the gap between", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "graph and grid-like data. Base on our Neighbor2Seq, many effective strategies for grid-like data can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 359, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 374 + ], + "score": 1.0, + "content": "be naturally transferred to graph data. These include self-supervised learning and pre-training on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "graphs (Hu et al., 2019; Velickovic et al., 2019; Sun et al., 2019; Hassani & Khasahmadi, 2020; You", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 359, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 359, + 394 + ], + "score": 1.0, + "content": "et al., 2020; Hu et al., 2020b; Qiu et al., 2020; Jin et al., 2020).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "We notice an existing work AWE Ivanov & Burnaev (2018) which also embed the information in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "graph as a sequence. 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DatasetScale#Nodes#EdgesAvg. Deg.#Features#ClassesTrain/Val/Test
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ogbn-productsMedium2,449,02961,859,14051100470.08/0.02/0.90
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DatasetScale#Nodes#EdgesAvg. Deg.#Features#ClassesTrain/Val/Test
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For our proposed methods, we conduct the precomputation on", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "index": 43 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "the CPU, after which we train our models on a GeForce RTX 2080 Ti GPU. We perform a grid search", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 316, + 106 + ], + "score": 1.0, + "content": "for the following hyperparameters: number of hops", + "type": "text" + }, + { + "bbox": [ + 316, + 94, + 324, + 104 + ], + "score": 0.4, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ", batch size, learning rate, hidden dimension", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 115, + 115 + ], + "score": 0.79, + "content": "d ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 104, + 348, + 118 + ], + "score": 1.0, + "content": ", dropout rate, weight decay, and convolutional kernel size", + "type": "text" + }, + { + "bbox": [ + 348, + 105, + 355, + 115 + ], + "score": 0.51, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 104, + 506, + 118 + ], + "score": 1.0, + "content": ". The chosen hyperparameters for our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Neighbor2Seq+Conv and Neighbor2Seq+Attn are summarized in Appendix A.2 for reproducibility.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 109, + 140, + 298, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 300, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 300, + 153 + ], + "score": 1.0, + "content": "5.2 RESULTS ON MASSIVE-SCALE GRAPHS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 259, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 260, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 260, + 172 + ], + "score": 1.0, + "content": "Since ogbn-papers100M is a mas-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 260, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 221, + 183 + ], + "score": 1.0, + "content": "sive graph with more than", + "type": "text" + }, + { + "bbox": [ + 221, + 172, + 255, + 182 + ], + "score": 0.32, + "content": "1 1 1 \\ \\mathrm { m i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 172, + 260, + 183 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 260, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 260, + 194 + ], + "score": 1.0, + "content": "lion nodes and 1.6 billion edges, most", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 259, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 259, + 205 + ], + "score": 1.0, + "content": "existing methods have difficulty han-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 260, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 260, + 216 + ], + "score": 1.0, + "content": "dling such a graph. We consider three", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 260, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 260, + 226 + ], + "score": 1.0, + "content": "baselines that have available results", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 226, + 260, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 260, + 237 + ], + "score": 1.0, + "content": "evaluated by OGB: Multilayer Per-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 260, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 260, + 248 + ], + "score": 1.0, + "content": "ceptron (MLP), Node2Vec (Grover &", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 260, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 260, + 260 + ], + "score": 1.0, + "content": "Leskovec, 2016), and SGC (Wu et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 260, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 260, + 270 + ], + "score": 1.0, + "content": "2019). The results under transductive", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 260, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 260, + 282 + ], + "score": 1.0, + "content": "setting is reported in Table 3. Fol-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 260, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 260, + 293 + ], + "score": 1.0, + "content": "lowing OGB, we report accuracies for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 292, + 260, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 260, + 304 + ], + "score": 1.0, + "content": "all models on training, validation, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 303, + 259, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 259, + 314 + ], + "score": 1.0, + "content": "test sets. The previous state-of-the-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 313, + 259, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 259, + 325 + ], + "score": 1.0, + "content": "art result on ogbn-papers100M is ob-", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 268, + 162, + 504, + 250 + ], + "lines": [ + { + "bbox": [ + 266, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 266, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "Table 3: Results on ogbn-papers100M in terms of classi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 266, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 266, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "fication accuracy (in percent). The reported accuracy is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 266, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 266, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "averaged over 10 random runs. Note that existing sam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 266, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 266, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "pling methods cannot scale to this massive graph. Dur-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 266, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 266, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "ing precomputation, both SGC and our models have to ran-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 266, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 266, + 216, + 328, + 228 + ], + "score": 1.0, + "content": "domly remove", + "type": "text" + }, + { + "bbox": [ + 328, + 216, + 348, + 227 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "edges to avoid a memory overflow on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 266, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 266, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "CPU. This implies that the performance could be further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 267, + 239, + 501, + 251 + ], + "spans": [ + { + "bbox": [ + 267, + 239, + 501, + 251 + ], + "score": 1.0, + "content": "improved if more advanced precomuting platform is used.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "table", + "bbox": [ + 269, + 250, + 505, + 324 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 269, + 250, + 505, + 324 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 269, + 250, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 269, + 250, + 505, + 324 + ], + "score": 0.977, + "html": "
MethodTrainingValidationTest
MLP54.84±0.4349.60±0.2947.24±0.31
Node2vec158.07±0.2855.60±0.23
SGC67.54±0.4366.48±0.2063.29±0.19
Neighbor2Seq+Conv69.87±0.8167.46±0.1664.04±0.22
Neighbor2Seq+Attn68.83±0.3066.90±0.1063.59±0.17
", + "type": "table", + "image_path": "8231c469d959eeb37180f0fd515f142db613000891510b0e7b65999a6aa05a52.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 269, + 250, + 505, + 262.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 269, + 262.3333333333333, + 505, + 274.66666666666663 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 269, + 274.66666666666663, + 505, + 286.99999999999994 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 269, + 286.99999999999994, + 505, + 299.33333333333326 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 269, + 299.33333333333326, + 505, + 311.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 269, + 311.6666666666666, + 505, + 323.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 108, + 325, + 505, + 358 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "tained by the precomputing method SGC. Our models outperform the baselines consistently in terms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "of training, validation, and test, which demonstrates the expressive power and the generalization", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 346, + 272, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 272, + 360 + ], + "score": 1.0, + "content": "ability of our method on massive graphs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "index": 32.75 + }, + { + "type": "title", + "bbox": [ + 108, + 371, + 296, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 298, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 298, + 384 + ], + "score": 1.0, + "content": "5.3 RESULTS ON MEDIUM-SCALE GRAPHS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 406 + ], + "score": 1.0, + "content": "We also evaluate our models on medium-scale graphs, thus enabling comparison with more existing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "works. We conduct transductive learning on ogbn-products, a medium-scale graph from OGB. We", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "also conduct inductive learning on Reddit, Yelp, and Flickr, which are frequently used for inductive", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "learning by the community. The following baselines are considered: MLP, Node2Vec (Grover &", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Leskovec, 2016), GCN (Kipf & Welling, 2016), SGC Wu et al. (2019), GraphSAGE (Hamilton", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 444, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 506, + 460 + ], + "score": 1.0, + "content": "et al., 2017), FastGCN (Chen et al., 2018b), VR-GCN (Chen et al., 2018a), AS-GCN (Huang et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "2018), ClusterGCN (Chiang et al., 2019), GraphSAINT (Zeng et al., 2020), and SIGN (Rossi et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 466, + 136, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 136, + 482 + ], + "score": 1.0, + "content": "2020).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "table", + "bbox": [ + 108, + 540, + 298, + 639 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 485, + 299, + 540 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 484, + 299, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 299, + 497 + ], + "score": 1.0, + "content": "Table 4: Results on ogbn-products in terms of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 496, + 299, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 299, + 508 + ], + "score": 1.0, + "content": "classification accuracy (in percent). The re-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 507, + 299, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 299, + 519 + ], + "score": 1.0, + "content": "ported accuracy is averaged over 10 random", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 516, + 300, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 300, + 531 + ], + "score": 1.0, + "content": "runs. Obtaining the results of GCN requires a", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 528, + 224, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 224, + 542 + ], + "score": 1.0, + "content": "GPU with 33GB of memory.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + }, + { + "type": "table_body", + "bbox": [ + 108, + 540, + 298, + 639 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 540, + 298, + 639 + ], + "spans": [ + { + "bbox": [ + 108, + 540, + 298, + 639 + ], + "score": 0.978, + "html": "
MethodTrainingValidationTest
MLP84.03±0.9375.54±0.1461.06±0.08
Node2vec93.39±0.1090.32±0.0672.49±0.10
GCN93.56±0.0992.00±0.0375.64±0.21
GraphSAGE92.96±0.0791.70±0.0978.70±0.36
ClusterGCN93.75±0.1392.12±0.0978.97±0.33
GraphSAINT92.71±0.1491.62±0.0879.08±0.24
SGC92.60±0.1091.19±0.0672.46±0.27
SIGN96.92±0.4693.10±0.0877.60±0.13
Neighbor2Seq+Conv95.32±0.1092.92±0.0579.67±0.16
Neighbor2Seq+Attn92.82±0.1492.20±0.0279.35±0.17
", + "type": "table", + "image_path": "13e6ab87cd4ea0a977ccc7d5ab16c1388f68a89d6bdfa052b236c1f0c4dc45b1.jpg" + } + ] + } + ], + "index": 54.5, + "virtual_lines": [ + { + "bbox": [ + 108, + 540, + 298, + 552.375 + ], + "spans": [], + "index": 51 + }, + { + "bbox": [ + 108, + 552.375, + 298, + 564.75 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 108, + 564.75, + 298, + 577.125 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 108, + 577.125, + 298, + 589.5 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 108, + 589.5, + 298, + 601.875 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 108, + 601.875, + 298, + 614.25 + ], + "spans": [], + "index": 56 + }, + { + "bbox": [ + 108, + 614.25, + 298, + 626.625 + ], + "spans": [], + "index": 57 + }, + { + "bbox": [ + 108, + 626.625, + 298, + 639.0 + ], + "spans": [], + "index": 58 + } + ] + } + ], + "index": 51.25 + }, + { + "type": "table", + "bbox": [ + 316, + 538, + 505, + 640 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 314, + 483, + 506, + 538 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 314, + 482, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 314, + 482, + 507, + 496 + ], + "score": 1.0, + "content": "Table 5: Results for inductive learning on three", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 315, + 495, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 315, + 495, + 507, + 505 + ], + "score": 1.0, + "content": "datasets in terms of F1-micro score. The re-", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 313, + 505, + 507, + 517 + ], + "spans": [ + { + "bbox": [ + 313, + 505, + 507, + 517 + ], + "score": 1.0, + "content": "ported score is averaged over 10 random runs.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 313, + 515, + 507, + 528 + ], + "spans": [ + { + "bbox": [ + 313, + 515, + 507, + 528 + ], + "score": 1.0, + "content": "The results of baselines are partially obtained", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 314, + 526, + 491, + 539 + ], + "spans": [ + { + "bbox": [ + 314, + 526, + 491, + 539 + ], + "score": 1.0, + "content": "from Zeng et al. (2020); Rossi et al. (2020).", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 61 + }, + { + "type": "table_body", + "bbox": [ + 316, + 538, + 505, + 640 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 316, + 538, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 316, + 538, + 505, + 640 + ], + "score": 0.976, + "html": "
MethodRedditFlickrYelp
GCN0.933±0.0000.492±0.0030.378±0.001
FastGCN0.924±0.0010.504±0.0010.265±0.053
VR-GCN0.964±0.0010.482±0.0030.640±0.002
AS-GCN0.958±0.0010.504±0.002-
GraphSAGE0.953±0.0010.501±0.0130.634±0.006
ClusterGCN0.954±0.0010.481±0.0050.609±0.005
GraphSAINT0.966±0.0010.511±0.0010.653±0.003
SGC0.949±0.0000.502±0.0010.358±0.006
SIGN0.968±0.0000.514±0.0010.631±0.003
Neighbor2Seq+Conv0.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn0.967±0.0000.523±0.0020.647±0.001
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The chosen hyperparameters for our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Neighbor2Seq+Conv and Neighbor2Seq+Attn are summarized in Appendix A.2 for reproducibility.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 140, + 298, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 300, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 300, + 153 + ], + "score": 1.0, + "content": "5.2 RESULTS ON MASSIVE-SCALE GRAPHS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 259, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 260, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 260, + 172 + ], + "score": 1.0, + "content": "Since ogbn-papers100M is a mas-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 260, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 221, + 183 + ], + "score": 1.0, + "content": "sive graph with more than", + "type": "text" + }, + { + "bbox": [ + 221, + 172, + 255, + 182 + ], + "score": 0.32, + "content": "1 1 1 \\ \\mathrm { m i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 172, + 260, + 183 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 260, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 260, + 194 + ], + "score": 1.0, + "content": "lion nodes and 1.6 billion edges, most", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 259, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 259, + 205 + ], + "score": 1.0, + "content": "existing methods have difficulty han-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 260, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 260, + 216 + ], + "score": 1.0, + "content": "dling such a graph. 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The results under transductive", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 260, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 260, + 282 + ], + "score": 1.0, + "content": "setting is reported in Table 3. Fol-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 260, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 260, + 293 + ], + "score": 1.0, + "content": "lowing OGB, we report accuracies for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 292, + 260, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 260, + 304 + ], + "score": 1.0, + "content": "all models on training, validation, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 303, + 259, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 259, + 314 + ], + "score": 1.0, + "content": "test sets. 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Dur-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 266, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 266, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "ing precomputation, both SGC and our models have to ran-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 266, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 266, + 216, + 328, + 228 + ], + "score": 1.0, + "content": "domly remove", + "type": "text" + }, + { + "bbox": [ + 328, + 216, + 348, + 227 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "edges to avoid a memory overflow on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 266, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 266, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "CPU. 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MethodTrainingValidationTest
MLP54.84±0.4349.60±0.2947.24±0.31
Node2vec158.07±0.2855.60±0.23
SGC67.54±0.4366.48±0.2063.29±0.19
Neighbor2Seq+Conv69.87±0.8167.46±0.1664.04±0.22
Neighbor2Seq+Attn68.83±0.3066.90±0.1063.59±0.17
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Our models outperform the baselines consistently in terms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "of training, validation, and test, which demonstrates the expressive power and the generalization", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 346, + 272, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 272, + 360 + ], + "score": 1.0, + "content": "ability of our method on massive graphs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "index": 32.75 + }, + { + "type": "title", + "bbox": [ + 108, + 371, + 296, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 298, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 298, + 384 + ], + "score": 1.0, + "content": "5.3 RESULTS ON MEDIUM-SCALE GRAPHS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 406 + ], + "score": 1.0, + "content": "We also evaluate our models on medium-scale graphs, thus enabling comparison with more existing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "works. We conduct transductive learning on ogbn-products, a medium-scale graph from OGB. We", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "also conduct inductive learning on Reddit, Yelp, and Flickr, which are frequently used for inductive", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "learning by the community. The following baselines are considered: MLP, Node2Vec (Grover &", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Leskovec, 2016), GCN (Kipf & Welling, 2016), SGC Wu et al. (2019), GraphSAGE (Hamilton", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 444, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 506, + 460 + ], + "score": 1.0, + "content": "et al., 2017), FastGCN (Chen et al., 2018b), VR-GCN (Chen et al., 2018a), AS-GCN (Huang et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "2018), ClusterGCN (Chiang et al., 2019), GraphSAINT (Zeng et al., 2020), and SIGN (Rossi et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 466, + 136, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 136, + 482 + ], + "score": 1.0, + "content": "2020).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 389, + 506, + 482 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 540, + 298, + 639 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 485, + 299, + 540 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 484, + 299, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 299, + 497 + ], + "score": 1.0, + "content": "Table 4: Results on ogbn-products in terms of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 496, + 299, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 299, + 508 + ], + "score": 1.0, + "content": "classification accuracy (in percent). The re-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 507, + 299, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 299, + 519 + ], + "score": 1.0, + "content": "ported accuracy is averaged over 10 random", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 516, + 300, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 300, + 531 + ], + "score": 1.0, + "content": "runs. Obtaining the results of GCN requires a", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 528, + 224, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 224, + 542 + ], + "score": 1.0, + "content": "GPU with 33GB of memory.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + }, + { + "type": "table_body", + "bbox": [ + 108, + 540, + 298, + 639 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 540, + 298, + 639 + ], + "spans": [ + { + "bbox": [ + 108, + 540, + 298, + 639 + ], + "score": 0.978, + "html": "
MethodTrainingValidationTest
MLP84.03±0.9375.54±0.1461.06±0.08
Node2vec93.39±0.1090.32±0.0672.49±0.10
GCN93.56±0.0992.00±0.0375.64±0.21
GraphSAGE92.96±0.0791.70±0.0978.70±0.36
ClusterGCN93.75±0.1392.12±0.0978.97±0.33
GraphSAINT92.71±0.1491.62±0.0879.08±0.24
SGC92.60±0.1091.19±0.0672.46±0.27
SIGN96.92±0.4693.10±0.0877.60±0.13
Neighbor2Seq+Conv95.32±0.1092.92±0.0579.67±0.16
Neighbor2Seq+Attn92.82±0.1492.20±0.0279.35±0.17
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MethodRedditFlickrYelp
GCN0.933±0.0000.492±0.0030.378±0.001
FastGCN0.924±0.0010.504±0.0010.265±0.053
VR-GCN0.964±0.0010.482±0.0030.640±0.002
AS-GCN0.958±0.0010.504±0.002-
GraphSAGE0.953±0.0010.501±0.0130.634±0.006
ClusterGCN0.954±0.0010.481±0.0050.609±0.005
GraphSAINT0.966±0.0010.511±0.0010.653±0.003
SGC0.949±0.0000.502±0.0010.358±0.006
SIGN0.968±0.0000.514±0.0010.631±0.003
Neighbor2Seq+Conv0.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn0.967±0.0000.523±0.0020.647±0.001
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The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 224, + 206, + 383, + 218 + ], + "spans": [ + { + "bbox": [ + 224, + 206, + 383, + 218 + ], + "score": 1.0, + "content": "reported time is averaged over 10 runs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 225, + 221, + 503, + 289 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 221, + 503, + 289 + ], + "spans": [ + { + "bbox": [ + 225, + 221, + 503, + 289 + ], + "score": 0.979, + "html": "
MethodPreprocessing (↓)Training (↓)Inferenece (↓)Test Accuracy (↑)
ClusterGCN44.15±0.7711.87±0.8487.03±0.2478.97±0.33
GraphSAINT80.78±3.54.29±0.48107.26±0.9479.08±0.24
SGC153.36±3.60.15±0.011.08±0.0172.46±0.27
SIGN151.47±3.51.22±0.022.93±0.0677.60±0.13
Neighbor2Seq+Conv153.42±3.24.09±0.1231.52±1.4479.67±0.16
Neighbor2Seq+Attn153.42±3.22.67±0.0831.24±0.6179.35±0.17
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ModelOrderogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Conv：64.04±0.2279.67±0.160.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn63.59±0.1779.35±0.170.967±0.0000.523±0.0020.647±0.001
Neighbor2Seq+Attn w/o PEX63.61±0.0978.54±0.250.965±0.0000.521±0.0030.646±0.001
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Hence, we conduct an ablation study to verify the significance of this order in-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 366, + 501 + ], + "score": 1.0, + "content": "formation. We remove the positional encoding in Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 366, + 489, + 372, + 497 + ], + "score": 0.43, + "content": "+ .", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "Attn, leading to a model without", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "the ability to capture the order information. The comparison is demonstrated in Table 7. Note", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 183, + 523 + ], + "score": 1.0, + "content": "that Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 183, + 511, + 190, + 520 + ], + "score": 0.32, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 509, + 286, + 523 + ], + "score": 1.0, + "content": "Attn and Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 286, + 511, + 293, + 519 + ], + "score": 0.48, + "content": "+ .", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "Attn w/o PE have the same number of parameters.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "Hence, Comparing the results of these two models, we can conclude that the order information is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 345, + 544 + ], + "score": 1.0, + "content": "usually necessary. Neighbor2Seq+Conv and Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 346, + 533, + 352, + 541 + ], + "score": 0.39, + "content": "+ .", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "Attn both can capture the order infor-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 350, + 555 + ], + "score": 1.0, + "content": "mation. There are two possible reasons why Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 351, + 544, + 357, + 552 + ], + "score": 0.29, + "content": "\\Vdash", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "Conv performs better. First, Neigh-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "bor2Seq+Conv has more learnable parameters than Neighbor2Se+Attn, which only has a learnable", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 376, + 577 + ], + "score": 1.0, + "content": "query. Second, the convolutional neural network in Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 376, + 566, + 383, + 574 + ], + "score": 0.25, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "Conv can additionally investi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "gate the dependencies between feature dimensions because each feature dimension of the output", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 587, + 303, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 303, + 599 + ], + "score": 1.0, + "content": "depends on every feature dimension of the input.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 614, + 281, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 284, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 284, + 629 + ], + "score": 1.0, + "content": "6 CONCLUSIONS AND OUTLOOK", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "In this work, we propose Neighbor2Seq, for transforming the heirarchical neighborhoods to ordered", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "sequences. Neighbor2Seq enables the subsequent use of powerful general deep learning operations,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 258, + 673 + ], + "score": 1.0, + "content": "leading to the proposed Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 259, + 662, + 265, + 671 + ], + "score": 0.38, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Conv and Neighbor2Seq+Attn. Our models can be deployed", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "on massive graphs and trained efficiently. The extensive expriments demonstrate the scalability and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 681, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 697 + ], + "score": 1.0, + "content": "the promising performance of our method. 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On Reddit, our models perform better", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "than all sampling methods and achieve the competitive result as SIGN. On Flickr, our models obtain", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 339, + 117 + ], + "score": 1.0, + "content": "significantly better results. 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We consider the fol-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 217, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 217, + 283 + ], + "score": 1.0, + "content": "lowing representative sam-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 217, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 217, + 294 + ], + "score": 1.0, + "content": "pling methods and precom-", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 182, + 218, + 294 + ] + }, + { + "type": "table", + "bbox": [ + 225, + 221, + 503, + 289 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 224, + 184, + 504, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 182, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 223, + 182, + 505, + 198 + ], + "score": 1.0, + "content": "Table 6: Computational efficiency in terms of preprocessing, training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 224, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 224, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "(per epoch), and inference times (in seconds) on ogbn-products. The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 224, + 206, + 383, + 218 + ], + "spans": [ + { + "bbox": [ + 224, + 206, + 383, + 218 + ], + "score": 1.0, + "content": "reported time is averaged over 10 runs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 225, + 221, + 503, + 289 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 221, + 503, + 289 + ], + "spans": [ + { + "bbox": [ + 225, + 221, + 503, + 289 + ], + "score": 0.979, + "html": "
MethodPreprocessing (↓)Training (↓)Inferenece (↓)Test Accuracy (↑)
ClusterGCN44.15±0.7711.87±0.8487.03±0.2478.97±0.33
GraphSAINT80.78±3.54.29±0.48107.26±0.9479.08±0.24
SGC153.36±3.60.15±0.011.08±0.0172.46±0.27
SIGN151.47±3.51.22±0.022.93±0.0677.60±0.13
Neighbor2Seq+Conv153.42±3.24.09±0.1231.52±1.4479.67±0.16
Neighbor2Seq+Attn153.42±3.22.67±0.0831.24±0.6179.35±0.17
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ModelOrderogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Conv：64.04±0.2279.67±0.160.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn63.59±0.1779.35±0.170.967±0.0000.523±0.0020.647±0.001
Neighbor2Seq+Attn w/o PEX63.61±0.0978.54±0.250.965±0.0000.521±0.0030.646±0.001
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Note", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 183, + 523 + ], + "score": 1.0, + "content": "that Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 183, + 511, + 190, + 520 + ], + "score": 0.32, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 509, + 286, + 523 + ], + "score": 1.0, + "content": "Attn and Neighbor2Seq", + "type": "text" + }, + { + "bbox": [ + 286, + 511, + 293, + 519 + ], + "score": 0.48, + "content": "+ .", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "Attn w/o PE have the same number of parameters.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "Hence, Comparing the results of these two models, we can conclude that the order information is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 345, + 544 + ], + "score": 1.0, + "content": "usually necessary. 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MethodPrecomputingForward Pass
GCNO(Ldm + Ld²n)
GraphSAGEO(sLn)O(sLd²n)
ClusterGCNO(m)O(Ldm + Ld²n)
GraphSAINTO(sn)O(Ldm +Ld²n)
SGCO(Ldm)O(d²n)
SIGNO(Ldm)O(Ld²n)
Neighbor2Seq+ConvO(Ldm)O(Ld²+Lkd²)n)
Neighbor2Seq+AttnO(Ldm)O(Ld²+Ld)n)
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DatasetScale#Nodes#EdgesAvg. Deg.#Features#ClassesTrain/Val/Test
ogbn-papers100MMassive111,059,9561,615,685,872291281720.78/0.08/0.14
ogbn-productsMedium2,449,02961,859,14051100470.08/0.02/0.90
RedditMedium232,96511,606,91950602410.66/0.10/0.24
YelpMedium716,8576,997,41010300100(m)0.75/0.10/0.15
FlickrMedium89,250899,7561050070.50/0.25/0.25
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MethodTrainingValidationTest
MLP84.03±0.9375.54±0.1461.06±0.08
Node2vec93.39±0.1090.32±0.0672.49±0.10
GCN93.56±0.0992.00±0.0375.64±0.21
GraphSAGE92.96±0.0791.70±0.0978.70±0.36
ClusterGCN93.75±0.1392.12±0.0978.97±0.33
GraphSAINT92.71±0.1491.62±0.0879.08±0.24
SGC92.60±0.1091.19±0.0672.46±0.27
SIGN96.92±0.4693.10±0.0877.60±0.13
Neighbor2Seq+Conv95.32±0.1092.92±0.0579.67±0.16
Neighbor2Seq+Attn92.82±0.1492.20±0.0279.35±0.17
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MethodTrainingValidationTest
MLP54.84±0.4349.60±0.2947.24±0.31
Node2vec158.07±0.2855.60±0.23
SGC67.54±0.4366.48±0.2063.29±0.19
Neighbor2Seq+Conv69.87±0.8167.46±0.1664.04±0.22
Neighbor2Seq+Attn68.83±0.3066.90±0.1063.59±0.17
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MethodRedditFlickrYelp
GCN0.933±0.0000.492±0.0030.378±0.001
FastGCN0.924±0.0010.504±0.0010.265±0.053
VR-GCN0.964±0.0010.482±0.0030.640±0.002
AS-GCN0.958±0.0010.504±0.002-
GraphSAGE0.953±0.0010.501±0.0130.634±0.006
ClusterGCN0.954±0.0010.481±0.0050.609±0.005
GraphSAINT0.966±0.0010.511±0.0010.653±0.003
SGC0.949±0.0000.502±0.0010.358±0.006
SIGN0.968±0.0000.514±0.0010.631±0.003
Neighbor2Seq+Conv0.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn0.967±0.0000.523±0.0020.647±0.001
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MethodPreprocessing (↓)Training (↓)Inferenece (↓)Test Accuracy (↑)
ClusterGCN44.15±0.7711.87±0.8487.03±0.2478.97±0.33
GraphSAINT80.78±3.54.29±0.48107.26±0.9479.08±0.24
SGC153.36±3.60.15±0.011.08±0.0172.46±0.27
SIGN151.47±3.51.22±0.022.93±0.0677.60±0.13
Neighbor2Seq+Conv153.42±3.24.09±0.1231.52±1.4479.67±0.16
Neighbor2Seq+Attn153.42±3.22.67±0.0831.24±0.6179.35±0.17
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ModelOrderogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Conv：64.04±0.2279.67±0.160.967±0.0000.527±0.0030.647±0.003
Neighbor2Seq+Attn63.59±0.1779.35±0.170.967±0.0000.523±0.0020.647±0.001
Neighbor2Seq+Attn w/o PEX63.61±0.0978.54±0.250.965±0.0000.521±0.0030.646±0.001
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ModelHyperparameterogbn-papers100Mogbn-productsRedditFlickrYelp
Neighbor2Seq+Convnumber of hopsL373102
hidden dimension d'512512256256512
convolutional kernel size k57573
learning rate5e-42e-58e-58e-45e-4
batch size122886432768245768192
weight decay5e-55e-505e-50
dropout rate0.250.50.50.50
Neighbor2Seq+Attnnumber of hopsL373102
hidden dimension d'512512256256512
learning rate5e-41e-32e-32e-35e-4
batch size122883072327682568192
weight decay5e-65e-505e-50
dropout rate0.250.50.50.50
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Snell +University of Toronto +Vector Institute +jsnell@cs.toronto.edu +Richard Zemel +University of Toronto +Vector Institute +Canadian Institute for Advanced Research +zemel@cs.toronto.edu + +# ABSTRACT + +Few-shot classification (FSC), the task of adapting a classifier to unseen classes given a small labeled dataset, is an important step on the path toward human-like machine learning. Bayesian methods are well-suited to tackling the fundamental issue of overfitting in the few-shot scenario because they allow practitioners to specify prior beliefs and update those beliefs in light of observed data. Contemporary approaches to Bayesian few-shot classification maintain a posterior distribution over model parameters, which is slow and requires storage that scales with model size. Instead, we propose a Gaussian process classifier based on a novel combination of Polya-Gamma augmentation and the one-vs-each softmax ´ approximation (Titsias, 2016) that allows us to efficiently marginalize over functions rather than model parameters. We demonstrate improved accuracy and uncertainty quantification on both standard few-shot classification benchmarks and few-shot domain transfer tasks. + +# 1 INTRODUCTION + +Few-shot classification (FSC) is a rapidly growing area of machine learning that seeks to build classifiers able to adapt to novel classes given only a few labeled examples. It is an important step towards machine learning systems that can successfully handle challenging situations such as personalization, rare classes, and time-varying distribution shift. The shortage of labeled data in FSC leads to uncertainty over the parameters of the model, known as model uncertainty or epistemic uncertainty. If model uncertainty is not handled properly in the few-shot setting, there is a significant risk of overfitting. In addition, FSC is increasingly being used for risk-averse applications such as medical diagnosis (Prabhu, 2019) and human-computer interfaces (Wang et al., 2019) where it is important for a few-shot classifier to know when it is uncertain. + +Bayesian methods maintain a distribution over model parameters and thus provide a natural framework for capturing this inherent model uncertainty. In a Bayesian approach, a prior distribution is first placed over the parameters of a model. After data is observed, the posterior distribution over parameters is computed using Bayesian inference. This elegant treatment of model uncertainty has led to a surge of interest in Bayesian approaches to FSC that infer a posterior distribution over the weights of a neural network (Finn et al., 2018; Yoon et al., 2018; Ravi & Beatson, 2019). + +Although conceptually appealing, there are several practical obstacles to applying Bayesian inference directly to the weights of a neural network. Bayesian neural networks (BNNs) are expensive from both a computational and memory perspective. Moreover, specifying meaningful priors in parameter space is known to be difficult due to the complex relationship between weights and network outputs (Sun et al., 2019). + +Gaussian processes (GPs) instead maintain a distribution over functions rather than model parameters. The prior is directly specified by a mean and covariance function, which may be parameterized by deep neural networks. When used with Gaussian likelihoods, GPs admit closed form expressions for the posterior and predictive distributions. They exchange the computational drawbacks of BNNs for cubic scaling with the number of examples. In FSC, where the number of examples is small, this is often an acceptable trade-off. + +When applying GPs to classification with a softmax likelihood, the non-conjugacy of the GP prior renders posterior inference intractable. Many approximate inference methods have been proposed to circumvent this, including variational inference and expectation propagation. In this paper we investigate a particularly promising class of approaches that augment the GP model with a set of auxiliary random variables, such that when they are marginalized out the original model is recovered (Albert & Chib, 1993; Girolami & Rogers, 2006; Linderman et al., 2015). Such augmentation-based approaches typically admit efficient Gibbs sampling procedures for generating posterior samples which when combined with Fisher’s identity (Douc et al., 2014) can be used to optimize the parameters of the mean and covariance functions. + +In particular, augmentation with Polya-Gamma random variables (Polson et al., 2013) makes in- ´ ference tractable in logistic models. Naively, this is useful for handling binary classification, but in this paper we show how to extend Polya-Gamma augmentation to multiple classes by using the ´ one-vs-each softmax approximation (Titsias, 2016), which can be expressed as a product of logistic sigmoids. We further show that the one-vs-each approximation can be interpreted as a composite likelihood (Lindsay, 1988; Varin et al., 2011), a connection which to our knowledge has not been made in the literature. + +In this work, we make several contributions: + +• We show how the one-vs-each softmax approximation (Titsias, 2016) can be interpreted as a composite likelihood consisting of pairwise conditional terms. +• We propose a novel GP classification method that combines the one-vs-each softmax approximation with Polya-Gamma augmentation for tractable inference. ´ We demonstrate competitive classification accuracy of our method on standard FSC benchmarks and challenging domain transfer settings. We propose several new benchmarks for uncertainty quantification in FSC, including calibration, robustness to input noise, and out-of-episode detection. +• We demonstrate improved uncertainty quantification of our method on the proposed benchmarks relative to standard few-shot baselines. + +# 2 RELATED WORK + +Our work is related to both GP methods for handling non-conjugate classification likelihoods and Bayesian approaches to few-shot classification. We summarize relevant work here. + +# 2.1 GP CLASSIFICATION + +Non-augmentation approaches. There are several classes of approaches for applying Gaussian processes to classification. The most straightforward method, known as least squares classification (Rifkin & Klautau, 2004), treats class labels as real-valued observations and performs inference with a Gaussian likelihood. The Laplace approximation (Williams & Barber, 1998) constructs a Gaussian approximate posterior centered at the posterior mode. Variational approaches (Titsias, 2009; Matthews et al., 2016) maximize a lower bound on the log marginal likelihood. In expectation propagation (Minka, 2001; Kim & Ghahramani, 2006; Hernandez-Lobato & Hernandez-Lobato, 2016), local Gaussian approximations to the likelihood are fitted iteratively to minimize KL divergence from the true posterior. + +Augmentation approaches. Augmentation-based approaches introduce auxiliary random variables such that the original model is recovered when marginalized out. Girolami & Rogers (2006) propose a Gaussian augmentation for multinomial probit regression. Linderman et al. (2015) utilize Polya-Gamma augmentation (Polson et al., 2013) and a stick-breaking construction to decom- ´ pose a multinomial distribution into a product of binomials. Galy-Fajou et al. (2020) propose a logistic-softmax likelihood for classification and uses Gamma and Poisson augmentation in addition to Polya-Gamma augmentation in order to perform inference. ´ + +# 2.2 FEW-SHOT CLASSIFICATION + +Meta-learning. A common approach to FSC is meta-learning, which seeks to learn a strategy to update neural network parameters when faced with a novel learning task. The Meta-learner LSTM (Ravi & Larochelle, 2017) learns a meta-level LSTM to recurrently output a new set of parameters for a base learner. MAML (Finn et al., 2017) learns initializations of deep neural networks that perform well on task-specific losses after one or a few steps of gradient descent by backpropagating through the gradient descent procedure itself. LEO (Rusu et al., 2019) performs meta-learning in a learned low-dimensional latent space from which the parameters of a classifier are generated. + +Metric learning. Metric learning approaches learn distances such that input examples can be meaningfully compared. Siamese Networks (Koch, 2015) learn a shared embedding network along with a distance layer for computing the probability that two examples belong to the same class. Matching Networks (Vinyals et al., 2016) uses a nonparametric classification in the form of attention over nearby examples, which can be interpreted as a form of soft $k$ -nearest neighbors in the embedding space. Prototypical Networks (Snell et al., 2017) make predictions based on distances to nearest class centroids. Relation Networks (Sung et al., 2018) instead learn a more complex neural network distance function on top of the embedding layer. + +Bayesian Few-shot Classification. More recently, Bayesian FSC approaches that attempt to infer a posterior over task-specific parameters have appeared. Grant et al. (2018) reinterpret MAML as an approximate empirical Bayes algorithm and propose LLAMA, which optimizes the Laplace approximation to the marginal likelihood. Bayesian MAML (Yoon et al., 2018) instead uses Stein Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to approximate the posterior distribution over model parameters. VERSA (Gordon et al., 2019) uses amortized inference networks to obtain an approximate posterior distribution over task-specific parameters. ABML (Ravi & Beatson, 2019) uses a few steps of Bayes by Backprop (Blundell et al., 2015) on the support set to produce an approximate posterior over network parameters. CNAPs (Requeima et al., 2019) modulate taskspecific Feature-wise Linear Modulation (FiLM) (Perez et al., 2018) layer parameters as the output of an adaptation network that takes the support set as input. + +GPs for Few-shot Learning. There have been relatively few works applying GPs to few-shot learning. Tossou et al. (2020) consider Gaussian processes in the context of few-shot regression with Gaussian likelihoods. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) uses Gaussian processes with least squares classification to perform few-shot classification and learns covariance functions parameterized by deep neural networks. More recently, Titsias et al. (2020) applies GPs to meta-learning by maximizing the mutual information between the query set and a latent representation of the support set. + +# 3 BACKGROUND + +In this section we first review Polya-Gamma augmentation for binary classification and the one-vs- ´ each approximation before we introduce our method in Section 4. + +# 3.1 POLYA ´ -GAMMA AUGMENTATION + +The Polya-Gamma augmentation scheme was originally introduced to address Bayesian inference ´ in logistic models (Polson et al., 2013). Suppose we have a vector of logits $\boldsymbol { \psi } \in \mathbf { \mathbb { R } } ^ { N }$ with corresponding binary labels $\mathbf { y } \in \{ 0 , 1 \} ^ { N }$ . The logistic likelihood is + +$$ +p ( \mathbf { y } \vert \psi ) = \prod _ { i = 1 } ^ { N } \sigma ( \psi _ { i } ) ^ { y _ { i } } ( 1 - \sigma ( \psi _ { i } ) ) ^ { 1 - y _ { i } } = \prod _ { i = 1 } ^ { N } \frac { ( e ^ { \psi _ { i } } ) ^ { y _ { i } } } { 1 + e ^ { \psi _ { i } } } , +$$ + +where $\sigma ( \cdot )$ is the logistic sigmoid function. Let the prior over $\psi$ be Gaussian: $p ( \psi ) = \mathcal { N } ( \psi | \mu , \Sigma )$ . In Bayesian inference, we are interested in the posterior $p ( \psi | \mathbf { y } ) \propto p ( \mathbf { y } | \psi ) p ( \psi )$ but the form of (1) does not admit analytic computation of the posterior due to non-conjugacy. The main idea of Polya-Gamma augmentation is to introduce auxiliary random variables ´ $\omega$ to the likelihood such that the original model is recovered when $\omega$ is marginalized out: $\begin{array} { r } { p ( \mathbf { y } \vert \psi ) = \int p ( \omega ) p ( \mathbf { y } \vert \psi , \omega ) d \omega } \end{array}$ . + +Conditioned on $\omega \sim \mathrm { P G } ( 1 , 0 )$ , the batch likelihood is proportional to a diagonal Gaussian (see Section A for a full derivation): + +$$ +p ( \mathbf { y } \vert \psi , \boldsymbol { \omega } ) \propto \prod _ { i = 1 } ^ { N } e ^ { - \omega _ { i } \psi _ { i } ^ { 2 } / 2 } e ^ { \kappa _ { i } \psi _ { i } } \propto \mathcal { N } ( \Omega ^ { - 1 } \kappa \vert \psi , \Omega ^ { - 1 } ) , +$$ + +where $\kappa _ { i } = y _ { i } - 1 / 2$ and $\pmb { \Omega } = \mathrm { d i a g } ( \omega )$ . The conditional distribution over $\psi$ given $\mathbf { y }$ and $\omega$ is now tractable: + +$$ +p ( \psi | \mathbf { y } , \boldsymbol { \omega } ) \propto p ( \mathbf { y } | \psi , \boldsymbol { \omega } ) p ( \psi ) \propto \mathcal { N } ( \psi | \tilde { \boldsymbol { \Sigma } } ( \Sigma ^ { - 1 } \mu + \kappa ) , \tilde { \boldsymbol { \Sigma } } ) , +$$ + +where $\tilde { \Sigma } = ( \Sigma ^ { - 1 } + \Omega ) ^ { - 1 }$ . The conditional distribution of $\omega$ given $\psi$ and $\mathbf { y }$ can also be easily computed: + +$$ +p ( \omega _ { i } | y _ { i } , \psi _ { i } ) \propto \mathrm { P G } ( \omega _ { i } | 1 , 0 ) e ^ { - \omega _ { i } \psi _ { i } ^ { 2 } / 2 } \propto \mathrm { P G } ( \omega _ { i } | 1 , \psi _ { i } ) , +$$ + +where the last expression follows from the exponential tilting property of Polya-Gamma random ´ variables. This suggest a Gibbs sampling procedure in which iterates ${ \bf \dot { \boldsymbol { \omega } } } ^ { ( t ) } \sim \dot { p } ( \boldsymbol { \omega } | { \bf y } , \psi ^ { ( t - 1 ) } )$ and $\psi ^ { ( t ) } \sim p ( \psi | \mathbf { X } , \mathbf { y } , \omega ^ { ( t ) } )$ are drawn sequentially until the Markov chain reaches its stationary distribution, which is the joint posterior $p ( \boldsymbol { \psi } , \boldsymbol { \omega } | \mathbf { y } )$ . Fortunately, efficient samplers for the Polya-Gamma ´ distribution have been developed (Windle et al., 2014) to facilitate this. + +# 3.2 ONE-VS-EACH APPROXIMATION TO SOFTMAX + +The one-vs-each (OVE) approximation (Titsias, 2016) was formulated as a lower bound to the softmax likelihood in order to handle classification over a large number of output classes, where computation of the normalizing constant is prohibitive. We employ the OVE approximation not to deal with extreme classification, but rather due to its compatibility with Polya-Gamma augmentation, as ´ we shall soon see. The one-vs-each approximation can be derived by first rewriting the softmax likelihood as follows: + +$$ +p ( y = i \mid { \bf f } ) \triangleq { \frac { e ^ { f _ { i } } } { \sum _ { j } e ^ { f _ { j } } } } = { \frac { 1 } { 1 + \sum _ { j \neq i } e ^ { - ( f _ { i } - f _ { j } ) } } } , +$$ + +where $\mathbf { f } \triangleq ( f _ { 1 } , \ldots , f _ { C } ) ^ { \intercal }$ are the logits. Since in general $\begin{array} { r } { \prod _ { k } ( 1 + \alpha _ { k } ) \ge \left( 1 + \sum _ { k } \alpha _ { k } \right) } \end{array}$ for $\alpha _ { k } \geq 0$ , the softmax likelihood (5) can be bounded as follows: + +$$ +p ( y = i \mid \mathbf { f } ) \geq \prod _ { j \neq i } { \frac { 1 } { 1 + e ^ { - ( f _ { i } - f _ { j } ) } } } = \prod _ { j \neq i } \sigma ( f _ { i } - f _ { j } ) , +$$ + +which is the OVE lower bound. This expression avoids the normalizing constant and factorizes into a product of pairwise sigmoids, which is amenable to Polya-Gamma augmentation for tractable ´ inference. + +# 4 ONE-VS-EACH POLYA ´ -GAMMA GPS + +In this section, we first show how the one-vs-each (OVE) approximation can be interpreted as a pairwise composite likelihood. We then we introduce our method for GP-based Bayesian few-shot classification, which brings together OVE and Polya-Gamma augmentation in a novel combination. ´ + +# 4.1 OVE AS A COMPOSITE LIKELIHOOD + +Titsias (2016) showed that the OVE approximation shares the same global optimum as the softmax maximum likelihood, suggesting a close relationship between the two. We show here that in fact OVE can be interpreted as a pairwise composite likelihood version of the softmax. Composite likelihoods (Lindsay, 1988; Varin et al., 2011) are a type of approximate likelihood often employed when the exact likelihood is intractable or otherwise difficult to compute. Given a collection of marginal or conditional events $\{ E _ { 1 } , \ldots , E _ { K } \}$ and parameters $\mathbf { f }$ , a composite likelihood is defined as: + +$$ +\mathcal { L } _ { \mathrm { C L } } ( \mathbf { f } \mid y ) \triangleq \prod _ { k = 1 } ^ { K } \mathcal { L } _ { k } ( \mathbf { f } \mid y ) ^ { w _ { k } } , +$$ + +where $\mathcal { L } _ { k } ( \mathbf { f } \mid y ) \propto p ( y \in E _ { k } \mid \mathbf { f } )$ and $w _ { k } \geq 0$ are arbitrary weights. + +In order to make the connection to OVE, it will be useful to let the one-hot encoding of the label $y$ be denoted as $\mathbf { y } \in \{ 0 , 1 \} ^ { C }$ . Define a set of $C ( C - 1 ) / 2$ pairwise conditional events $E _ { i j }$ , one each for all pairs of classes $i \neq j$ , indicating the event that the model’s output matches the target label for classes $i$ and $j$ conditioned on all the other classes: + +$$ +p ( \mathbf { y } \in E _ { i j } \mid \mathbf { f } ) \triangleq p ( y _ { i } , y _ { j } \mid \mathbf { y } _ { \lnot i j } , \mathbf { f } ) , +$$ + +where $\neg i j$ denotes the set of classes not equal to either $i$ or $j$ . This expression resembles the pseudolikelihood (Besag, 1975), but instead of a single conditional event per output site, the expression in (8) considers all pairs of sites. Stoehr $\&$ Friel (2015) explored similar composite likelihood generalizations of the pseudolikelihood in the context of random fields. + +Now suppose that $y _ { c } = 1$ for some class $c \notin \{ i , j \}$ . Then $p ( y _ { i } , y _ { j } \mid \mathbf { y } _ { \lnot i j } , \mathbf { f } ) = 1$ due to the one-hot constraint. Otherwise either $y _ { i } = 1$ or $y _ { j } = 1$ . In this case, assume without loss of generality that $y _ { i } = 1$ and $y _ { j } = 0$ and thus + +$$ +p ( y _ { i } , y _ { j } \mid \mathbf { y } _ { - i j } , \mathbf { f } ) = \frac { e ^ { f _ { i } } } { e ^ { f _ { i } } + e ^ { f _ { j } } } = \sigma ( f _ { i } - f _ { j } ) . +$$ + +The composite likelihood defined in this way with unit component weights is therefore + +$$ +\mathcal { L } _ { \mathrm { O V E } } ( \mathbf { f } \mid \mathbf { y } ) = \prod _ { i } \prod _ { j \neq i } p ( y _ { i } , y _ { j } \mid \mathbf { y } _ { - i j } , \mathbf { f } ) = \prod _ { i } \prod _ { j \neq i } \sigma ( f _ { i } - f _ { j } ) ^ { y _ { i } } . +$$ + +Alternatively, we may simply write $\begin{array} { r } { \mathcal { L } _ { \mathrm { O V E } } ( \mathbf { f } \mid y = i ) = \prod _ { j \neq i } \sigma ( f _ { i } - f _ { j } ) } \end{array}$ , which is identical to the OVE bound (6). + +# 4.2 GP CLASSIFICATION WITH THE OVE LIKELIHOOD + +We now turn our attention to GP classification. Suppose we have access to examples $\mathbf { X } \in \mathbb { R } ^ { N \times D }$ with corresponding one-hot labels $\mathbf { Y } \in \{ 0 , 1 \} ^ { N \times C }$ , where $C$ is the number of classes. We consider the logits jointly as a single vector + +$$ +\mathbf { f } \triangleq ( f _ { 1 } ^ { 1 } , \hdots , f _ { N } ^ { 1 } , f _ { 1 } ^ { 2 } , \hdots , f _ { N } ^ { 2 } , \hdots , f _ { 1 } ^ { C } , \hdots , f _ { N } ^ { C } ) ^ { \intercal } +$$ + +and place an independent GP prior on the logits for each class: $\mathbf { f } ^ { c } ( \mathbf { x } ) \sim \mathcal { G P } ( m ( \mathbf { x } ) , k ( \mathbf { x } , \mathbf { x } ^ { \prime } ) )$ . Therefore we have $p ( \mathbf { f } | \mathbf { X } ) = \bar { \mathcal { N } } ( \mathbf { f } | \pmb { \mu } , \mathbf { K } )$ , where $\mu _ { i } ^ { c } = m ( \mathbf { x } _ { i } )$ and $\mathbf { K }$ is block diagonal with $K _ { i j } ^ { c } =$ $k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } )$ for each block ${ \bf K } ^ { c }$ . + +The Polya-Gamma integral identity used to derive (2) does not have a multi-class analogue and thus ´ a direct application of the augmentation scheme to the softmax likelihood is nontrivial. Instead, we propose to directly replace the softmax with the OVE-based composite likelihood function from (10) with unit weights. The posterior over f when using OVE as the likelihood function can be expressed as: + +$$ +p ( \mathbf { f } | \mathbf { X } , \mathbf { y } ) \propto p ( \mathbf { f } | \mathbf { X } ) \prod _ { i = 1 } ^ { N } \prod _ { c ^ { \prime } \neq y _ { i } } \sigma ( f _ { i } ^ { y _ { i } } - f _ { i } ^ { c ^ { \prime } } ) , +$$ + +to which Polya-Gamma augmentation can be applied as we show in the next section. Our motiva- ´ tion for using a composite likelihood therefore differs from the traditional motivation, which is to avoid the use of a likelihood function which is intractable to evaluate. Instead, we employ a composite likelihood because it makes posterior inference tractable when coupled with Polya-Gamma ´ augmentation. + +Prior work on Bayesian inference with composite likelihoods has shown that the composite posterior is consistent under fairly general conditions for correctly specified models (Miller, 2019) but can produce overly concentrated posteriors (Pauli et al., 2011; Ribatet et al., 2012) since each component likelihood event is treated as independent when in reality there may be significant dependencies. Nevertheless, we show in Section 5 that in practice our method exhibits competitive accuracy and strong calibration relative to baseline few-shot learning algorithms. We leave further theoretical analysis of the OVE composite posterior and its properties for future work. + +Compared to choices of likelihoods used by previous approaches, there are several reasons to prefer OVE. Relative to the Gaussian augmentation approach of Girolami & Rogers (2006), Polya-Gamma ´ augmentation has the benefit of fast mixing and the ability of a single value of $\omega$ to capture much of the marginal distribution over function values1. The stick-breaking construction of Linderman et al. (2015) induces a dependence on the ordering of classes, which leads to undesirable asymmetry. Finally, the logistic-softmax likelihood of Galy-Fajou et al. (2020) requires three augmentations and careful learning of the mean function to avoid a priori underconfidence (see Section F.1 for more details). + +# 4.3 POSTERIOR INFERENCE VIA GIBBS SAMPLING + +We now describe how we perform tractable posterior inference in our model with Gibbs sampling. Define the matrix $\mathbf { A } \ { \overset { \Delta } { = } } \ \operatorname { O V E - M A T R I X } ( \mathbf { Y } )$ to be a $C N \times C N$ sparse block matrix with $C$ row partitions and $C$ column partitions. Each block $\mathbf { A } _ { c c ^ { \prime } }$ is a diagonal $N \times N$ matrix defined as follows: + +$$ +\mathbf { A } _ { c c ^ { \prime } } \triangleq \mathrm { d i a g } ( \mathbf { Y } _ { \cdot c ^ { \prime } } ) - \mathbb { 1 } [ c = c ^ { \prime } ] \mathbf { I } _ { n } , +$$ + +where $\mathbf { Y } _ { \cdot c ^ { \prime } }$ denotes the $c ^ { \prime }$ th column of $\mathbf { Y }$ . Now the binary logit vector $\pmb { \psi } \triangleq \mathbf { A } \mathbf { f } \in \mathbb { R } ^ { C N }$ will have entries equal to $f _ { i } ^ { y _ { i } } - f _ { i } ^ { c }$ for each unique combination of $c$ and $i$ , of which there are $C N$ in total. The OVE composite likelihood can now be written as $\begin{array} { r } { \mathcal { L } ( \psi | \mathbf { Y } ) = 2 ^ { N } \prod _ { j = 1 } ^ { N C } \sigma ( \psi _ { j } ) } \end{array}$ , where the $2 ^ { N }$ term arises from the $N$ cases in which $\psi _ { j } = 0$ due to comparing the ground truth logit with itself. + +Analogous to (2), the likelihood of $\psi$ conditioned on $\omega$ and $\mathbf { Y }$ is proportional to a diagonal Gaussian: + +$$ +\mathcal { L } ( \psi | \mathbf { Y } , \omega ) \propto \prod _ { j = 1 } ^ { N C } e ^ { - \omega _ { j } \psi _ { j } ^ { 2 } / 2 } e ^ { \kappa _ { j } \psi _ { j } } \propto \mathcal { N } ( \Omega ^ { - 1 } \kappa | \psi , \Omega ^ { - 1 } ) , +$$ + +where $\kappa _ { j } ~ = ~ 1 / 2$ and $\Omega = \mathrm { d i a g } ( \omega )$ . By exploiting the fact that $\psi = \mathbf { A } \mathbf { f }$ , we can express the likelihood in terms of $\mathbf { f }$ and write down the conditional composite posterior as follows: + +$$ +p ( \mathbf { f } | \mathbf { X } , \mathbf { Y } , \omega ) \propto \mathcal { N } ( \Omega ^ { - 1 } \kappa | \mathbf { A } \mathbf { f } , \Omega ^ { - 1 } ) \mathcal { N } ( \mathbf { f } | \boldsymbol { \mu } , \mathbf { K } ) \propto \mathcal { N } ( \mathbf { f } | \tilde { \boldsymbol { \Sigma } } ( \mathbf { K } ^ { - 1 } \boldsymbol { \mu } + \mathbf { A } ^ { \top } \boldsymbol { \kappa } ) , \tilde { \boldsymbol { \Sigma } } ) , +$$ + +where $\tilde { \boldsymbol { \Sigma } } = ( \boldsymbol { \mathbf { K } } ^ { - 1 } + \boldsymbol { \mathbf { A } } ^ { \top } \boldsymbol { \Omega } \boldsymbol { \mathbf { A } } ) ^ { - 1 }$ , which is an expression remarkably similar to (3). Analogous to (4), the conditional distribution over $\omega$ given f and the data becomes $p ( \omega | \mathbf { y } , \mathbf { f } ) = \mathrm { P G } ( \omega | \mathbf { 1 } , \mathbf { A } \mathbf { f } )$ . + +The primary computational bottleneck of posterior inference lies in sampling $\mathbf { f }$ from (15). Since $\tilde { \Sigma }$ is a $C N \times C N$ matrix, a naive implementation has complexity $\mathcal { O } ( C ^ { 3 } N ^ { 3 } )$ . By utilizing the matrix inversion lemma and Gaussian sampling techniques summarized in (Doucet, 2010), this can be brought down to $\mathcal { O } ( C N ^ { 3 } )$ . Details may be found in Section B. + +# 4.4 LEARNING COVARIANCE HYPERPARAMETERS FOR FEW-SHOT CLASSIFICATION + +We now describe how we apply OVE Polya-Gamma augmented GPs to few-shot classification. ´ We assume the standard episodic few-shot setup in which one observes a labeled support set ${ \boldsymbol { s } } =$ $( \mathbf { X } , \mathbf { Y } )$ . Predictions must then be made for a query example $\left( \mathbf { x } _ { * } , \mathbf { y } _ { * } \right)$ . We consider a zero-mean GP prior over the class logits $\mathbf { f } ^ { c } ( \mathbf { x } ) \sim \mathcal { G P } ( \mathbf { 0 } , k _ { \theta } ( \mathbf { x } , \mathbf { x } ^ { \prime } ) )$ , where $\pmb \theta$ are learnable parameters of our covariance function. These could include traditional hyperparameters such as lengthscales or the weights of a deep neural network as in deep kernel learning (Wilson et al., 2016). + +We consider two objectives for learning hyperparameters of the covariance function: the marginal likelihood (ML) and the predictive likelihood (PL). Marginal likelihood measures the likelihood of the hyperparameters given the observed data and is intuitively appealing from a Bayesian perspective. On the other hand, many standard FSC methods optimize for predictive likelihood on the query set (Vinyals et al., 2016; Finn et al., 2017; Snell et al., 2017). Both objectives marginalize over latent functions, thereby making full use of our Bayesian formulation. + +The details of these objectives and how we compute gradients can be found in Section C. Our learning algorithm for both marginal and predictive likelihood may be found in Section D. Details of computing the posterior predictive distribution $p ( \mathbf { y } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } , \omega )$ may be found in Section E. Finally, details of our chosen “cosine” kernel may be found in Section $_ \mathrm { H }$ . + +# 5 EXPERIMENTS + +In this section, we present our results on few-shot classification both in terms of accuracy and uncertainty quantification. Additional results comparing the one-vs-each composite likelihood to the softmax, logistic softmax, and Gaussian likelihoods may be found in Section F. + +One of our aims is to compare methods based on uncertainty quantification. We therefore developed new benchmark evaluations and tasks: few-shot calibration, robustness, and out-of-episode detection. In order to empirically compare methods, we could not simply borrow the accuracy results from other papers, but instead needed to train each of these baselines ourselves. For all baselines except Bayesian MAML, ABML, and Logistic Softmax GP, we ran the code from (Patacchiola et al., 2020) and verified that the accuracies matched closely to their reported results. We have made PyTorch code for our experiments publicly available2. + +# 5.1 FEW-SHOT CLASSIFICATION + +For our few-shot classification experiments, we follow the training and evaluation protocol of Patacchiola et al. (2020). We train both 1-shot and 5-shot versions of our model in four different settings: Caltech-UCSD Birds (CUB) (Wah et al., 2011), mini-Imagenet with the split proposed by Ravi & Larochelle (2017), as well as two cross-domain transfer tasks. The first transfer task entails training on mini-ImageNet and testing on CUB, and the second measures transfer from Omniglot (Lake et al., 2011) to EMNIST (Cohen et al., 2017). Experimental details and an overview of the baselines we used can be found in Section G. Classification results are shown in Table 1 and 2. We find that our proposed Polya-Gamma OVE GPs yield strong classification results, outperforming the baselines in ´ five of the eight scenarios. + +Table 1: Average accuracy and standard deviation (percentage) on 5-way FSC. Baseline results (through DKT) are from Patacchiola et al. (2020). Evaluation is performed on 3,000 randomly generated test episodes. Standard deviation for the remaining methods are computed by averaging over 5 batches of 600 episodes with different random seeds. The best results are highlighted in bold. + +
MethodCUBmini-ImageNet
1-shot5-shot1-shot5-shot
FeatureTransfer46.19± 0.6468.40± 0.7939.51 ± 0.2360.51 ± 0.55
Baseline++61.75 ± 0.9578.51 ± 0.5947.15 ± 0.4966.18 ± 0.18
MatchingNet60.19 ± 1.0275.11 ± 0.3548.25 ± 0.6562.71 ± 0.44
ProtoNet52.52 ± 1.9075.93 ± 0.4644.19 ± 1.3064.07 ± 0.65
RelationNet62.52 ± 0.3478.22 ± 0.0748.76 ± 0.1764.20 ± 0.28
MAML56.11 ± 0.6974.84 ± 0.6245.39 ± 0.4961.58 ± 0.53
DKT + Cosine63.37 ± 0.1977.73 ± 0.2648.64 ± 0.4562.85 ± 0.37
BayesianMAML55.93 ± 0.7172.87 ± 0.2644.46 ± 0.3062.60 ± 0.25
Bayesian MAML(Chaser)53.93 ± 0.7271.16 ± 0.3243.74 ± 0.4659.23 ± 0.34
ABML48.80 ± 0.4070.91 ± 0.3240.88 ± 0.2558.19 ± 0.17
Logistic Softmax GP + Cosine (ML)60.23 ± 0.5474.58 ± 0.2546.75 ± 0.2059.93 ± 0.31
Logistic Softmax GP + Cosine (PL)60.07 ±0.2978.14 ± 0.0747.05 ± 0.2066.01 ± 0.25
OVE PG GP + Cosine (ML) [ours]63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
OVE PG GP +Cosine (PL) [ours]60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
+ +# 5.2 UNCERTAINTY QUANTIFICATION THROUGH CALIBRATION + +We next turn to uncertainty quantification, an important concern for few-shot classifiers. When used in safety-critical applications such as medical diagnosis, it is important for a machine learning system to defer when there is not enough evidence to make a decision. Even in non-critical applications, precise uncertainty quantification helps practitioners in the few-shot setting determine when a class has an adequate amount of labeled data or when more labels are required, and can facilitate active learning. + +Table 2: Average accuracy and standard deviation (percentage) on 5-way cross-domain FSC, with the same experimental setup as in Table 1. Baseline results (through DKT) are from (Patacchiola et al., 2020). + +
MethodOmniglot→EMNISTmini-ImageNet→CUB
1-shot5-shot1-shot5-shot
Feature Transfer64.22 ±1.2486.10± 0.8432.77 ± 0.3550.34± 0.27
Baseline++56.84 ± 0.9180.01 ± 0.9239.19 ± 0.1257.31 ± 0.11
MatchingNet75.01 ± 2.0987.41 ± 1.7936.98 ± 0.0650.72 ± 0.36
ProtoNet72.04 ± 0.8287.22 ± 1.0133.27 ± 1.0952.16 ± 0.17
RelationNet75.62 ± 1.0087.84 ± 0.2737.13 ± 0.2051.76 ± 1.48
MAML72.68 ± 1.8583.54 ± 1.7934.01 ±1.2548.83 ± 0.62
DKT + Cosine73.06 ± 2.3688.10 ± 0.7840.22 ± 0.5455.65 ± 0.05
Bayesian MAML63.94 ± 0.4765.26 ± 0.3033.52 ± 0.3651.35 ± 0.16
Bayesian MAML (Chaser)55.04 ± 0.3454.19 ± 0.3236.22 ± 0.5051.53 ± 0.43
ABML73.89 ± 0.2487.28 ± 0.4031.51 ± 0.3247.80 ± 0.51
Logistic Softmax GP + Cosine (ML)62.91 ± 0.4983.80 ± 0.1336.41 ± 0.1850.33 ± 0.13
Logistic Softmax GP + Cosine (PL)70.70 ±0.3686.59 ± 0.1536.73 ± 0.2656.70 ± 0.31
OVE PG GP + Cosine (ML) [ours]68.43± 0.6786.22 ±0.2039.66± 0.1855.71 ± 0.31
OVE PG GP + Cosine (PL) [ours]77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
+ +We chose several commonly used metrics for calibration. Expected calibration error (ECE) (Guo et al., 2017) measures the expected binned difference between confidence and accuracy. Maximum calibration error (MCE) is similar to ECE but measures maximum difference instead of expected difference. Brier score (BRI) (Brier, 1950) is a proper scoring rule computed as the squared error between the output probabilities and the one-hot label. For a recent perspective on metrics for uncertainty evaluation, please refer to Ovadia et al. (2019). The results for representative approaches on 5-shot, 5-way CUB can be found in Figure 1. Our OVE PG GPs are the best calibrated overall across the metrics. + +![](images/1830b095a5b4a6ce7ed6b21a00496d1191e5c709cd10ae31e6c6e7a1cada7456.jpg) +Figure 1: Reliability diagrams, expected calibration error (ECE), maximum calibration error (MCE), and Brier Score (BRI) for 5-shot 5-way tasks on CUB (additional calibration results can be found in Appendix I). Metrics are computed on 3,000 random tasks from the test set. The last two plots are our proposed method. + +# 5.3 ROBUSTNESS TO INPUT NOISE + +Input examples for novel classes in FSC may have been collected under conditions that do not match those observed at training time. For example, labeled support images in a medical diagnosis application may come from a different hospital than the training set. To mimic a simplified version of this scenario, we investigate robustness to input noise. We used the Imagecorruptions package (Michaelis et al., 2019) to apply Gaussian noise, impulse noise, and defocus blur to both the support set and query sets of episodes at test time and evaluated both accuracy and calibration. We used corruption severity of 5 (severe) and evaluated across 1,000 randomly generated tasks on the three datasets involving natural images. The robustness results for Gaussian noise are shown in Figure 2. Full quantitative results tables for each noise type may be found in Section J. We find that in general Bayesian approaches tend to be robust due to their ability to marginalize over hypotheses consistent with the support labels. Our approach is one of the top performing methods across all settings. + +![](images/010f14a398ac09e9735aa1817cc4ec54d2d9cb8c053ad8077145d91ee24e15af.jpg) +Figure 2: Accuracy (↑) and Brier Score (↓) when corrupting both support and query with Gaussian noise on 5-way 5-shot tasks. Quantitative results may be found in Appendix J. + +# 5.4 OUT-OF-EPISODE DETECTION + +Finally, we measure performance on out-of-episode detection, another application in which uncertainty quantification is important. In this experiment, we used 5-way, 5-shot support sets at test time but incorporated out-of-episode examples into the query set. Each episode had 150 query examples: 15 from each of 5 randomly chosen in-episode classes and 15 from each of 5 randomly chosen outof-episode classes. We then computed the AUROC of binary outlier detection using the negative of the maximum logit as the score. Intuitively, if none of the support classes assign a high logit to the example, it can be classified as an outlier. The results are shown in Figure 3. Our approach generally performs the best across the datasets. + +![](images/790f796e9c00d9b88a81ad2a638b51abb81455e1bffbd2051471191cde9d6417.jpg) +Figure 3: Average AUROC (↑) for out-of-episode detection. The AUC is computed separately for each episode and averaged across 1,000 episodes. Bars indicate a $9 5 \%$ bootstrapped confidence interval. + +# 6 CONCLUSION + +In this work, we have proposed a Bayesian few-shot classification approach based on Gaussian processes. Our method replaces the ordinary softmax likelihood with a one-vs-each pairwise composite likelihood and applies Polya-Gamma augmentation to perform inference. This allows us to model ´ class logits directly as function values and efficiently marginalize over uncertainty in each few-shot episode. Modeling functions directly enables our approach to avoid the dependence on model size that posterior inference in weight-space based models inherently have. Our approach compares favorably to baseline FSC methods under a variety of dataset and shot configurations, including dataset transfer. We also demonstrate strong uncertainty quantification, robustness to input noise, and out-of-episode detection. We believe that Bayesian modeling is a powerful tool for handling uncertainty and hope that our work will lead to broader adoption of efficient Bayesian inference in the few-shot scenario. + +# ACKNOWLEDGMENTS + +We would like to thank Ryan Adams, Ethan Fetaya, Mike Mozer, Eleni Triantafillou, Kuan-Chieh Wang, and Max Welling for helpful discussions. JS also thanks SK T-Brain for supporting him on an internship that led to precursors of some ideas in this paper. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute (https://www.vectorinstitute. ai/partners). This project is supported by NSERC and the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (DoI/IBC) contract number D16PC00003. 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First, recall the logistic likelihood: + +$$ +p ( \mathbf { y } \vert \psi ) = \prod _ { i = 1 } ^ { N } \sigma ( \psi _ { i } ) ^ { y _ { i } } ( 1 - \sigma ( \psi _ { i } ) ) ^ { 1 - y _ { i } } = \prod _ { i = 1 } ^ { N } \frac { ( e ^ { \psi _ { i } } ) ^ { y _ { i } } } { 1 + e ^ { \psi _ { i } } } , +$$ + +where $\sigma ( \cdot )$ is the logistic sigmoid function. We have a Gaussian prior $p ( \psi ) = \mathcal { N } ( \psi | \mu , \Sigma )$ and introduce Polya-Gamma auxiliary random variables ´ $\omega$ to the likelihood such that the original model is recovered when $\omega$ is marginalized out: $\begin{array} { r } { p ( \mathbf { y } \vert \psi ) = \int p ( \omega ) p ( \mathbf { y } \vert \psi , \omega ) d \omega } \end{array}$ . + +The Polya-Gamma distribution ´ $\omega \sim \mathrm { P G } ( b , c )$ can be written as an infinite convolution of Gamma distributions: + +$$ +\omega \stackrel { \cal D } { = } \frac { 1 } { 2 \pi ^ { 2 } } \sum _ { k = 1 } ^ { \infty } \frac { \mathrm { G a } ( b , 1 ) } { ( k - 1 / 2 ) ^ { 2 } + c ^ { 2 } / ( 4 \pi ^ { 2 } ) } . +$$ + +The following integral identity holds for $b > 0$ : + +$$ +\frac { ( e ^ { \psi } ) ^ { a } } { ( 1 + e ^ { \psi } ) ^ { b } } = 2 ^ { - b } e ^ { \kappa \psi } \int _ { 0 } ^ { \infty } e ^ { - \omega \psi ^ { 2 } / 2 } p ( \omega ) d \omega , +$$ + +where $\kappa = a - b / 2$ and $\omega \sim \mathrm { P G } ( b , 0 )$ . Specifically, when $a = y$ and $b = 1$ , we recover an individual term of the logistic likelihood (16): + +$$ +p ( y | \psi ) = \frac { ( e ^ { \psi } ) ^ { y } } { 1 + e ^ { \psi } } = \frac { 1 } { 2 } e ^ { \kappa \psi } \int _ { 0 } ^ { \infty } e ^ { - \omega \psi ^ { 2 } / 2 } p ( \omega ) d \omega , +$$ + +where $\kappa = y - 1 / 2$ and $\omega \sim P G ( 1 , 0 )$ . Conditioned on $\omega$ , the batch likelihood is proportional to a diagonal Gaussian: + +$$ +p ( \mathbf { y } \vert \psi , \boldsymbol { \omega } ) \propto \prod _ { i = 1 } ^ { N } e ^ { - \omega _ { i } \psi _ { i } ^ { 2 } / 2 } e ^ { \kappa _ { i } \psi _ { i } } \propto \mathcal { N } ( \Omega ^ { - 1 } \kappa \vert \psi , \Omega ^ { - 1 } ) , +$$ + +where $\kappa _ { i } = y _ { i } - 1 / 2$ and $\pmb { \Omega } = \mathrm { d i a g } ( \omega )$ . The conditional distribution over $\psi$ given $\mathbf { y }$ and $\omega$ is now tractable: + +$$ +p ( \psi | \mathbf { y } , \boldsymbol { \omega } ) \propto p ( \mathbf { y } | \psi , \boldsymbol { \omega } ) p ( \psi ) \propto \mathcal { N } ( \psi | \tilde { \boldsymbol { \Sigma } } ( \Sigma ^ { - 1 } \mu + \kappa ) , \tilde { \boldsymbol { \Sigma } } ) , +$$ + +where $\tilde { \Sigma } = ( \Sigma ^ { - 1 } + \Omega ) ^ { - 1 }$ . + +# B EFFICIENT GIBBS SAMPLING + +The Gibbs conditional distribution over f is given by: + +$$ +p ( \mathbf { f } | \mathbf { X } , \mathbf { y } , \omega ) = \mathcal { N } ( \mathbf { f } | \tilde { \boldsymbol { \Sigma } } ( \mathbf { K } ^ { - 1 } \boldsymbol { \mu } + \mathbf { A } ^ { \top } \boldsymbol { \kappa } ) , \tilde { \boldsymbol { \Sigma } } ) , +$$ + +where $\tilde { \mathbf { \Sigma } } = ( \mathbf { K } ^ { - 1 } { + } \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { A } ) ^ { - 1 }$ . Naively sampling from this distribution requires $\mathcal { O } ( C ^ { 3 } N ^ { 3 } )$ computation since $\tilde { \Sigma }$ is a $C N \times C N$ matrix. Here we describe a method for sampling from this distribution that requires $\mathcal { O } ( C N ^ { 3 } )$ computation instead. + +First, we note that (22) can be interpreted as the conditional distribution $p ( \mathbf { f } | \mathbf { z } = \Omega ^ { - 1 } \kappa )$ ) resulting from the following marginal distribution $p ( \mathbf { f } )$ and conditional $p ( \mathbf { z } | \mathbf { f } )$ : + +$$ +\begin{array} { r } { p ( \mathbf { f } ) = \mathcal { N } ( \mathbf { f } \vert \boldsymbol { \mu } , \mathbf { K } ) \qquad } \\ { p ( \mathbf { z } \vert \mathbf { f } ) = \mathcal { N } ( \mathbf { z } \vert \mathbf { A } \mathbf { f } , \boldsymbol { \Omega } ^ { - 1 } ) , } \end{array} +$$ + +where we have made implicit the dependence on $\mathbf { X } , \mathbf { Y }$ , and $\omega$ for brevity of notation. Equivalently, the distribution over f and $\mathbf { z }$ can be represented by the partitioned Gaussian + +$$ +\left[ \begin{array} { c } { \mathbf { f } } \\ { \mathbf { z } } \end{array} \right] \sim \mathcal { N } \left( \left[ \begin{array} { c } { \ \mu } \\ { \mathbf { A } \mu } \end{array} \right] , \left[ \begin{array} { c c } { \mathbf { K } } & { \mathbf { K } \mathbf { A } ^ { \top } } \\ { \mathbf { A } \mathbf { K } } & { \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \boldsymbol { \Omega } ^ { - 1 } } \end{array} \right] \right) . +$$ + +The conditional distribution $p ( \mathbf { f } | \mathbf { z } )$ is given as: + +$$ +p ( \mathbf { f } \vert \mathbf { z } ) = \mathcal { N } ( \mathbf { f } \vert \tilde { \boldsymbol { \Sigma } } ( \mathbf { K } ^ { - 1 } \boldsymbol { \mu } + \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { z } ) , \tilde { \boldsymbol { \Sigma } } ) , +$$ + +where $\tilde { \boldsymbol { \Sigma } } = ( \mathbf { K } ^ { - 1 } + \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { A } ) ^ { - 1 }$ . Note that $p ( \mathbf { f } | \mathbf { z } = \Omega ^ { - 1 } \boldsymbol { \kappa } )$ recovers our desired Gibbs conditional distribution from (22). + +An efficient approach to conditional Gaussian sampling is due to Hoffman & Ribak (1991) and described in greater clarity by Doucet (2010). The procedure is as follows: + +1. Sample $\mathbf { f } _ { 0 } \sim p ( \mathbf { f } )$ and $\mathbf { z } _ { 0 } \sim p ( \mathbf { z } | \mathbf { f } )$ . +2. Return $\bar { \mathbf { f } } = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { A } ^ { \top } ( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 } ( \Omega ^ { - 1 } \pmb { \kappa } - \mathbf { z } _ { 0 } )$ as the sample from $p ( \mathbf { f } | \mathbf { z } )$ + +$\mathbf { K }$ is block diagonal and thus sampling from $p ( \mathbf { f } )$ requires $\mathcal { O } ( C N ^ { 3 } )$ time. Af can be computed in $\mathcal { O } ( C N )$ time, since each entry is the difference between $f _ { i } ^ { y _ { i } }$ and $f _ { i } ^ { c }$ for some $i$ and $c$ . Overall, step 1 requires $\mathcal { O } ( C N ^ { 3 } )$ time. + +We now show how to compute ¯f from step 2 in $\mathcal { O } ( C N ^ { 3 } )$ time. We first expand $( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 }$ + +$$ +( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 } = \Omega - \Omega \mathbf { A } ( \mathbf { K } ^ { - 1 } + \mathbf { A } ^ { \top } \Omega \mathbf { A } ) ^ { - 1 } \mathbf { A } ^ { \top } \Omega +$$ + +We substitute into the expression for $\bar { \bf f }$ : + +$$ +\begin{array} { r l } & { \bar { \mathbf { f } } = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { A } ^ { \top } ( \Omega - \Omega \mathbf { A } ( \mathbf { K } ^ { - 1 } + \mathbf { A } ^ { \top } \Omega \mathbf { A } ) ^ { - 1 } \mathbf { A } ^ { \top } \Omega ) ( \Omega ^ { - 1 } \kappa - \mathbf { z } _ { 0 } ) } \\ & { \quad = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { A } ^ { \top } \Omega ( \Omega ^ { - 1 } \kappa - \mathbf { z } _ { 0 } ) - \mathbf { K } \mathbf { A } ^ { \top } \Omega \mathbf { A } ( \mathbf { K } ^ { - 1 } + \mathbf { A } ^ { \top } \Omega \mathbf { A } ) ^ { - 1 } \mathbf { A } ^ { \top } \Omega ( \Omega ^ { - 1 } \kappa - \mathbf { z } _ { 0 } ) } \\ & { \quad = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { v } - \mathbf { K } \mathbf { A } ^ { \top } \Omega \mathbf { A } ( \mathbf { K } ^ { - 1 } + \mathbf { A } ^ { \top } \Omega \mathbf { A } ) ^ { - 1 } \mathbf { v } , } \end{array} +$$ + +where we have defined $\mathbf { v } \triangleq \mathbf { A } ^ { \top } \pmb { \Omega } ( \Omega ^ { - 1 } \pmb { \kappa } - \mathbf { z } _ { 0 } )$ + +Now let $\mathbf { d } \triangleq ( d _ { 1 } ^ { 1 } , \hdots , d _ { N } ^ { 1 } , d _ { 1 } ^ { 2 } , \hdots , d _ { N } ^ { 2 } , \hdots , d _ { 1 } ^ { C } , \hdots , d _ { N } ^ { C } ) ^ { \intercal }$ , where $\begin{array} { r } { d _ { i } ^ { c } = Y _ { i c } \sum _ { c ^ { \prime } } \omega _ { i } ^ { c ^ { \prime } } } \end{array}$ . Define $\mathbf { Y } ^ { \dagger }$ to be the $C N \times N$ matrix produced by vertically stacking $\mathrm { d i a g } ( Y _ { \cdot c } )$ , and let $\mathbf { W } ^ { \dagger }$ be the $C N \times N$ matrix produced by vertically stacking $\mathrm { d i a g } ( ( \omega _ { 1 } ^ { c } , \dots , \omega _ { N } ^ { c } ) ^ { \top } ,$ ). $\mathbf { A } ^ { \top } \mathbf { \widehat {Omega } } \mathbf { \widehat { \Omega } } \mathbf { A }$ may then be written as follows: + +$$ +\begin{array} { r l r } & { } & { \boldsymbol { \Omega } \mathbf { A } = \mathbf { D } - \mathbf { S } \mathbf { P S } ^ { \mathrm { ~ I ~ } } , \mathrm { w i } } \\ & { } & { \mathbf { D } = \boldsymbol { \Omega } + \mathrm { d i a g } ( \mathbf { d } ) , } \\ & { } & { \mathbf { S } = \left[ \begin{array} { l l } { \mathbf { Y } ^ { \dagger } } & { \mathbf { W } ^ { \dagger } } \end{array} \right] , } \\ & { } & { \mathbf { P } = \left[ \begin{array} { l l } { \mathbf { 0 } _ { N } } & { \mathbf { I } _ { N } } \\ { \mathbf { I } _ { N } } & { \mathbf { 0 } _ { N } } \end{array} \right] . } \end{array} +$$ + +Substituting (31) into (30): + +Now we expand + +$$ +\begin{array} { r l } & { \bar { \mathbf { f } } = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { v } - \mathbf { K } \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { A } ( \mathbf { K } ^ { - 1 } + \mathbf { D } - \mathbf { S } \mathbf { P } \mathbf { S } ^ { \top } ) ^ { - 1 } \mathbf { v } . } \\ & { . . . } \\ & { . . . + \mathbf { D } - \mathbf { S } \mathbf { P } \mathbf { S } ^ { \top } ) ^ { - 1 } : } \\ & { \mathbf { K } ^ { - 1 } + \mathbf { D } - \mathbf { S } \mathbf { P } \mathbf { S } ^ { \top } ) ^ { - 1 } = \mathbf { E } - \mathbf { E } \mathbf { S } ( \mathbf { S } ^ { \top } \mathbf { E } \mathbf { S } - \mathbf { P } ^ { - 1 } ) ^ { - 1 } \mathbf { S } ^ { \top } \mathbf { F } } \end{array} +$$ + +where $\mathbf { E } = ( \mathbf { K } ^ { - 1 } + \mathbf { D } ) ^ { - 1 } = \mathbf { K } ( \mathbf { K } + \mathbf { D } ^ { - 1 } ) ^ { - 1 } \mathbf { D } ^ { - 1 }$ is a block-diagonal matrix that can be computed in $\mathcal { O } ( C N ^ { 3 } )$ time, since $\mathbf { D }$ is diagonal and $\mathbf { K }$ is block diagonal. Now, substituting (36) back into (35), + +$$ +\bar { \mathbf { f } } = \mathbf { f } _ { 0 } + \mathbf { K } \mathbf { v } - \mathbf { K } \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { A } \mathbf { E } \mathbf { v } + \mathbf { K } \mathbf { A } ^ { \top } \boldsymbol { \Omega } \mathbf { A } \mathbf { E } \mathbf { S } ( \mathbf { S } ^ { \top } \mathbf { E } \mathbf { S } - \mathbf { P } ^ { - 1 } ) ^ { - 1 } \mathbf { S } ^ { \top } \mathbf { E } \mathbf { v } . +$$ + +Note that $( \mathbf { S } ^ { \top } \mathbf { E } \mathbf { S } - \mathbf { P } ^ { - 1 } ) ^ { - 1 }$ is a $2 N \times 2 N$ matrix and thus can be inverted in $\mathcal { O } ( N ^ { 3 } )$ time. The overall complexity is therefore $\mathcal { O } ( C N ^ { 3 } )$ . + +# C MARGINAL LIKELIHOOD AND PREDICTIVE LIKELIHOOD OBJECTIVES + +Marginal Likelihood (ML). The log marginal likelihood can be written as follows: + +$$ +\begin{array} { c } { { { \cal L } _ { \mathrm { M L } } ( \theta ; { \bf X } , { \bf Y } ) \triangleq \log p _ { \theta } ( { \bf Y } | { \bf X } ) = \log \displaystyle \int p ( \omega ) p _ { \theta } ( { \bf Y } | \omega , { \bf X } ) d \omega } } \\ { { \displaystyle = \log \displaystyle \int p ( \omega ) \int \mathcal { L } ( { \bf f } | { \bf Y } , \omega ) p _ { \theta } ( { \bf f } | { \bf X } ) d { \bf f } d \omega } } \end{array} +$$ + +The gradient of the log marginal likelihood can be estimated by posterior samples $\boldsymbol { \omega } \sim p _ { \boldsymbol { \theta } } ( \boldsymbol { \omega } | { \bf X } , { \bf Y } )$ . In practice, we use a stochastic training objective based on samples of $\omega$ from Gibbs chains. We use Fisher’s identity (Douc et al., 2014) to derive the following gradient estimator: + +$$ +\nabla _ { \theta } L _ { \mathrm { M L } } = \int p _ { \theta } ( \boldsymbol { \omega } | \mathbf { X } , \mathbf { Y } ) \nabla _ { \theta } \log p _ { \theta } ( \mathbf { Y } | \boldsymbol { \omega } , \mathbf { X } ) d \boldsymbol { \omega } \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \nabla _ { \theta } \log p _ { \theta } ( \mathbf { Y } | \mathbf { X } , \boldsymbol { \omega } ^ { ( m ) } ) , +$$ + +where $\omega ^ { ( 1 ) } , \ldots , \omega ^ { ( M ) }$ are samples from the posterior Gibbs chain. As suggested by Patacchiola et al. (2020), who applied GPs to FSC via least-squares classification, we merge the support and query sets during learning to take full advantage of the available data within each episode. + +Predictive Likelihood (PL). The log predictive likelihood for a query example $\mathbf { x } _ { * }$ is: + +$$ +L _ { \mathrm { P L } } ( \theta ; \mathbf { X } , \mathbf { Y } , \mathbf { x } _ { * } , \mathbf { y } _ { * } ) \triangleq \log p \varrho ( \mathbf { y } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } ) = \log \int p ( \omega ) p \varrho ( \mathbf { y } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } , \omega ) d \omega . +$$ + +We use an approximate gradient estimator again based on posterior samples of $\omega$ : + +$$ +\nabla _ { \theta } L _ { \mathrm { P L } } \approx \int p _ { \theta } ( \omega | \mathbf { X } , \mathbf { Y } ) \nabla _ { \theta } \log p _ { \theta } ( \mathbf { y } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } ) d \omega \approx \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \nabla _ { \theta } \log p _ { \theta } ( \mathbf { y } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } , \omega ^ { ( m ) } ) . +$$ + +We note that this is not an unbiased estimator of the gradient, but find it works well in practice. + +# D LEARNING ALGORITHM + +Our learning algorithm for both marginal and predictive likelihood is summarized in Algorithm 1. + +# Algorithm 1 One-vs-Each Polya-Gamma GP Learning ´ + +
Input: Objective L ∈ {LML,LPL}, Task distribution T, number of parallel Gibbs chains M, number of steps T,learning rate n. Initialize hyperparameters 0 randomly.
repeat Sample S= (X,Y),Q= (X*,Y*)~ T
if L= LML then X←XUX*,Y←YUY*
end if
A ← OVE-MATRIX(Y) for m = 1 to M do
~ pe(f|X)
fort=1 toTdo
（m）~PG(1b（m))
f（m） ~pe(f|x,Y,ω(m）)
end for end for
if L= LML then
0←0+∑m-1 Velogpe(Y|x,(m)）
else 0←0+∑m=1∑,∀θlogpe(y*j｜x*j,S,ω(m）)
end if until convergence
+ +# E POSTERIOR PREDICTIVE DISTRIBUTION + +The posterior predictive distribution for a query example $\mathbf { x } _ { * }$ conditioned on $\omega$ is: + +$$ +p ( { \bf y } _ { * } | { \bf x } _ { * } , { \bf X } , { \bf Y } , \omega ) = \int p ( { \bf y } _ { * } | { \bf f } _ { * } ) p ( { \bf f } _ { * } | { \bf x } _ { * } , { \bf X } , { \bf Y } , \omega ) d { \bf f } _ { * } , +$$ + +where $\mathbf { f } _ { * }$ are the query example’s logits. The predictive distribution over $\mathbf { f } _ { * }$ can be obtained by noting that $\psi$ and the query logits are jointly Gaussian: + +$$ +\left[ \begin{array} { c } { \psi } \\ { \mathbf { f } _ { * } } \end{array} \right] \sim { \mathcal { N } } \left( 0 , \left[ \begin{array} { c c } { \mathbf { A K A ^ { \top } } + \boldsymbol { \Omega } ^ { - 1 } } & { \mathbf { A K } _ { * } } \\ { \left( \mathbf { A K } _ { * } \right) ^ { \top } } & { \mathbf { K } _ { * * } } \end{array} \right] \right) , +$$ + +where $\mathbf { K } _ { * }$ is the $N C \times C$ block diagonal matrix with blocks $K _ { \theta } ( \mathbf { X } , \mathbf { x } _ { * } )$ and $\mathbf { K } _ { * * }$ is the $C \times C$ diagonal matrix with diagonal entries $k _ { \theta } ( \mathbf { x } _ { * } , \mathbf { x } _ { * } )$ . The predictive distribution becomes: + +$p ( \mathbf { f } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } , \omega ) = \mathcal { N } ( \mathbf { f } _ { * } | \mu _ { * } , \Sigma _ { * } )$ , where + +$$ +\begin{array} { r l } & { \pmb { \jmath } ^ { \top } ( \mathbf { A } \mathbf { x } ) ^ { \top } \mathbf { \Phi } ^ { \mathbf { a } \mathbf { x } } ) ^ { \top } ( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 } \Omega ^ { - 1 } \mathbf { \Lambda } ^ { \mathrm { a n d } } } \\ & { \pmb { \mu } _ { * } = ( \mathbf { A } \mathbf { K } _ { * } ) ^ { \top } ( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 } \Omega ^ { - 1 } \mathbf { \Lambda } ^ { \mathrm { a n d } } } \\ & { \pmb { \Sigma } _ { * } = \mathbf { K } _ { * * } - ( \mathbf { A } \mathbf { K } _ { * } ) ^ { \top } ( \mathbf { A } \mathbf { K } \mathbf { A } ^ { \top } + \Omega ^ { - 1 } ) ^ { - 1 } \mathbf { A } \mathbf { K } _ { * } . } \end{array} +$$ + +With $p ( \mathbf { f } _ { * } | \mathbf { x } _ { * } , \mathbf { X } , \mathbf { Y } , \omega )$ in hand, the integral in (42) can easily be computed numerically for each class $c$ by forming the corresponding OVE linear transformation matrix $\mathbf { A } ^ { c }$ and then performing 1D Gaussian-Hermite quadrature on each dimension of $\mathcal { N } ( \psi _ { * } ^ { c } | \mathbf { A } ^ { c } \pmb { \mu } ^ { * } , \mathbf { A } ^ { c } \pmb { \Sigma } _ { * } \mathbf { A } ^ { c \top } )$ . + +# F DETAILED COMPARISON OF LIKELIHOODS + +In this section we seek to better understand the behaviors of the softmax, OVE, logistic softmax, and Gaussian likelihoods for classification. For convenience, we summarize the forms of these likelihoods in Table 3. + +Table 3: Likelihoods used in Section F. + +
LikelihoodC(f|y =c)
Softmaxexp(fe) ∑c exp(fe)
GaussianIIN(2·1[c' =c]-1|μ= fe,σ² =1) c'
Logistic Softmax (LSM)g(fc) ∑eσ(fe)
One-vs-Each (OVE)IIσ(fe- fe) c≠c
+ +# F.1 HISTOGRAM OF CONFIDENCES + +We sampled logits from $f _ { c } \sim \mathcal { N } ( 0 , 1 )$ and plotted a histogram and kernel density estimate of the maximum output probability m $\arg ( y = c | \mathbf { f } )$ for each of the likelihoods shown in Table 3, where $C = 5$ . The results are shown in Figure 4. Logistic softmax is a priori underconfident: it puts little probability mass on confidence above 0.4. This may be due to the use of the sigmoid function which squashes large values of $f$ . Gaussian likelihood and OVE are a priori overconfident in that they put a large amount of probability mass on confident outputs. Note that this is not a complete explanation, because GP hyperparameters such as the prior mean or Gaussian likelihood variance may be able to compensate for these imperfections to some degree. Indeed, we found it helpful to learn a constant mean for the logistic softmax likelihood, as mentioned in Section G.2. + +# F.2 LIKELIHOOD VISUALIZATION + +In order to visualize the various likelihoods under consideration, we consider a trivial classification task with a single observed example. We assume that there are three classes $C = 3$ ) and the single example belongs to the first class $y = 1 \mathrm { \cdot }$ ). We place the following prior on $\mathbf { f } = \left( f _ { 1 } , f _ { 2 } , f _ { 3 } \right) ^ { \top }$ : + +$$ +\begin{array} { r } { p ( \mathbf { f } ) = \mathcal { N } \left( \mathbf { f } \bigg | \mu = \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 0 } \end{array} \right] , \Sigma = \left[ \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right] \right) . } \end{array} +$$ + +![](images/90f74595999ffa71f93112e5993b98c03df1ad7c6c88dcbaea08535a25ad34fe.jpg) +Figure 4: Histogram and kernel density estimate of confidence for randomly generated function samples $f _ { c } \sim \mathcal { N } ( 0 , 1 )$ . Normalized output probabilities were computed for $C = 5$ and a histogram of $\operatorname* { m a x } _ { c } p ( y = c | \mathbf { f } )$ was computed for 50,000 randomly generated simulations. + +In other words, the prior for $f _ { 1 }$ and $f _ { 2 }$ is a standard normal and $f _ { 3 }$ is clamped at zero (for ease of visualization). The likelihoods are plotted in Figure 5 and the corresponding posteriors are plotted in Figure 6. + +![](images/c475772f592b02bc778a43d7039536becbe66815b4c507f1cb35ac1a177e623e.jpg) +Figure 5: Plot of $\mathcal { L } ( \mathbf { f } \mid y = 1 )$ , where $f _ { 3 }$ is clamped to 0. The Gaussian likelihood penalizes configurations far away from $( f _ { 1 } , f _ { 2 } ) = ( 1 , - 1 ) .$ . Logistic softmax is much flatter compared to softmax and has visibly different contours. One-vs-Each is visually similar to the softmax but penalizes $( f _ { 1 } , f _ { 2 } )$ near the origin slightly more. + +![](images/c80f9c03f1647cfa1a69f4086877bb5121d476ef745a0037b2f946a73d185a49.jpg) +Figure 6: Plot of posterior $p ( \mathbf { f } \mid y = 1 )$ , where $f _ { 3 }$ is clamped to 0. The mode of each posterior distribution is similar, but each differs slightly in shape. Gaussian is more peaked about its mode, while logistic softmax is more spread out. One-vs-Each is similar to softmax, but is slightly more elliptical. + +We also conducted experiments on a 2D version of the Iris dataset (Fisher, 1936), which contains 150 examples across 3 classes. The first two features of the dataset were retained (sepal length and width). We used a zero-mean GP prior and an RBF kernel $k ( \mathbf { x } , \mathbf { x } ^ { \prime } ) = \exp \left( - \frac { 1 } { 2 } d ( \mathbf { x } , \mathbf { \bar { x } } ^ { \prime } ) ^ { 2 } \right)$ , where $d ( \cdot , \cdot )$ is Euclidean distance. We considered training set sizes with 1, 2, 3, 4, 5, 10, 15, 20, 25, and 30 examples per class. For each training set size, we performed GP inference on 200 randomly generated train/test splits and compared the predictions across Gaussian, logistic softmax, and onevs-each likelihoods. + +Predictions at a test point $\mathbf { x } _ { * }$ were made by applying the (normalized) likelihood to the posterior predictive mean $\bar { \mathbf { f } } _ { * }$ . The predictive probabilities for each likelihood is shown in Figure 7 for a randomly generated train/test split with 30 examples per class. Test predictive accuracy, Brier score, expected calibration error, and evidence lower bound (ELBO) results across various training set sizes are shown in Figure 8. + +The ELBO is computed by treating each likelihood’s posterior $q ( \mathbf { f } | \mathbf { X } , \mathbf { Y } )$ as an approximation to the softmax posterior $p ( \mathbf { f } | \mathbf { X } , \mathbf { Y } )$ . + +$$ +\begin{array} { r l } & { \mathrm { E L B O } ( q ) = \mathbb { E } _ { q } [ \log p ( \mathbf { f } | \mathbf { X } ) ] + \mathbb { E } _ { q } [ \log p ( \mathbf { Y } | \mathbf { f } ) ] - \mathbb { E } _ { q } [ \log q ( \mathbf { f } | \mathbf { X } , \mathbf { Y } ) ] } \\ & { \quad \quad \quad = \log p ( \mathbf { x } ) - \mathrm { K L } ( q ( \mathbf { f } | \mathbf { X } , \mathbf { Y } ) | | p ( \mathbf { f } | \mathbf { X } , \mathbf { Y } ) ) . } \end{array} +$$ + +Even though direct computation of the softmax posterior $p ( \mathbf { f } | \mathbf { X } , \mathbf { y } )$ is intractable, computing the ELBO is tractable. A larger ELBO indicates a lower KL divergence to the softmax posterior. + +One-vs-Each performs well for accuracy, Brier score, and ELBO across the training set sizes. Gaussian performs best on expected calibration error through 15 examples per class, beyond which onevs-each is better. + +![](images/c313125f2f323d8dd5fc6d7b997993186867c8fd3f117136626d8758be027a88.jpg) +Figure 7: Training points (colored points) and maximum predictive probability for various likelihoods on the Iris dataset. The Gaussian likelihood produces more warped decision boundaries than the others. Logistic softmax tends to produce lower confidence predictions, while one-vs-each produces larger regions of greater confidence than the others. + +![](images/b1ac3ed75011a27c35369b9894f46e6eb1989c84e9f1d028b3593dfc778ba366.jpg) +Figure 8: Comparison across likelihoods in terms of test predictive accuracy, Brier score, expected calibration error (computed with 10 bins), and ELBO. Results are averaged over 200 randomly generated splits for each training set size (1, 2, 3, 4, 5, 10, 15, 20, 25, and 30 examples per class). Error bars indicate $9 5 \%$ confidence intervals. + +# G FEW-SHOT EXPERIMENTAL DETAILS + +Here we provide more details about our experimental setup for our few-shot classification experiments, which are based on the protocol of (Patacchiola et al., 2020). + +# G.1 DATASETS + +We used the four dataset scenarios described below. The first three are the same used by Chen et al. +(2019) and the final was proposed by Patacchiola et al. (2020). +• CUB. Caltech-UCSD Birds (CUB) (Wah et al., 2011) consists of 200 classes and 11,788 images. A split of 100 training, 50 validation, and 50 test classes was used (Hilliard et al., 2018; Chen et al., 2019). mini-Imagenet. The mini-Imagenet dataset (Vinyals et al., 2016) consists of 100 classes with 600 images per class. We used the split proposed by Ravi & Larochelle (2017), which has 64 classes for training, 16 for validation, and 20 for test. +mini-Imagenet $\scriptstyle \mathbf { C U B }$ . This cross-domain transfer scenario takes the training split of mini-Imagenet and the validation & test splits of CUB. +Omniglot EMNIST. We use the same setup as proposed by Patacchiola et al. (2020). Omniglot (Lake et al., 2011) consists of 1,623 classes, each with 20 examples, and is augmented by rotations of 90 degrees to create 6,492 classes, of which 4,114 are used for training. The EMNIST dataset (Cohen et al., 2017), consisting of 62 classes, is split into 31 training and 31 test classes. + +# G.2 FEW-SHOT CLASSIFICATION BASELINES + +Here we explain the few-shot baselines in greater detail. + +• Feature Transfer (Chen et al., 2019) involves first training an off-line classifier on the training classes and then training a new classification layer on the episode. Baseline $^ { + + }$ (Chen et al., 2019) is similar to Feature Transfer except it uses a cosine distance module prior to the softmax during fine-tuning. Matching Networks (Vinyals et al., 2016) can be viewed as a soft form of $k$ -nearest neighbors that computes attention and sums over the support examples to form a predictive distribution over classes. +• Prototypical Networks (Snell et al., 2017) computes class means (prototypes) and forms a predictive distribution based on Euclidean distance to the prototypes. It can be viewed as a Gaussian classifier operating in an embedding space. MAML (Finn et al., 2017) performs one or a few steps of gradient descent on the support set and then makes predictions on the query set, backpropagating through the gradient descent procedure. For this baseline, we simply quote the classification accuracy reported by (Patacchiola et al., 2020). RelationNet (Sung et al., 2018) rather than using a predefined distance metric as in Matching Networks or Prototypical Networks instead learns a deep distance metric as the output of a neural network that accepts as input the latent representation of both examples. It is trained to minimize squared error of output predictions. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) relies on least squares classification (Rifkin & Klautau, 2004) to maintain tractability of Gaussian process posterior inference. In DKT, a separate binary classification task is formed for each class in one-vs-rest fashion by treating labels in $\{ - \dot { 1 } , + 1 \}$ as continuous targets. We include the results of DKT with the cosine kernel as implemented by Patacchiola et al. (2020), which is parameterized slightly differently from the version we used in (47): + +$$ +k _ { \mathrm { d k t } } ^ { \mathrm { c o s } } ( \mathbf { x } , \mathbf { x } ^ { \prime } ; \pmb { \theta } , \alpha , \nu ) = \mathrm { s o f t p l u s } ( \alpha ) \cdot \mathrm { s o f t p l u s } ( \nu ) \cdot \frac { g _ { \pmb { \theta } } ( \mathbf { x } ) ^ { \top } g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) } { \| g _ { \pmb { \theta } } ( \mathbf { x } ) \| \| g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) \| } . +$$ + +• Bayesian MAML (Yoon et al., 2018) relies on Stein Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to get an approximate posterior distribution in weight-space. We compare to both the non-chaser version, which optimizes cross-entropy of query predictions, and the chaser version, which optimizes mean squared error between the approximate posterior on the support set and the approximate posterior on the merged support & query set. The non-chaser version is therefore related to predictive likelihood methods and the chaser version is more analogous to the marginal likelihood methods. For the non-chaser version, we used 20 particles and 1 step of adaptation at both train and test time. For the chaser version, we also used 20 particles. At train time, the chaser took 1 step and the leader 1 additional step. At test time, we used 5 steps of adaptation. Due to the slow performance of this method, we followed the advice of Yoon et al. (2018) and only performed adaptation on the final layer of weights, which may help explain the drop in performance relative to MAML. The authors released Tensorflow code for regression only, so we reimplemented this baseline for classification in PyTorch. + +• Amortized Bayesian Meta-Learning (ABML) (Ravi & Beatson, 2019) performs a few steps of Bayes-by-backprop (Blundell et al., 2015) in order to infer a fully-factorized approximate posterior over the weights. The authors did not release code and so we implemented our own version of ABML in PyTorch. We found the weighting on the inner and outer KL divergences to be important for achieving good performance. We took the negative log likelihood to be mean cross entropy and used an inner KL weight of 0.01 and an outer KL weight of 0.001. These values were arrived upon by doing a small amount of hyperparameter tuning on the Omniglot EMNIST dataset. We used $\alpha = 1 . 0$ and $\beta = 0 . 0 1$ for the Gamma prior over the weights. We only applied ABML to the weights of the network; the biases were learned as point estimates. We used 4 steps of adaptation and took 5 samples when computing expectations (using any more than this did not fit into GPU memory). We used the local reparameterization trick (Kingma et al., 2015) and flipout (Wen et al., 2018) when computing expectations in order to reduce variance. In order to match the architecture used by Ravi & Beatson (2019), we trained this baseline with 32 filters throughout the classification network. We trained each 1-shot ABML model for 800 epochs and each 5-shot ABML model for 600 epochs as the learning had not converged within the epoch limits specified in Section G.3. + +• Logistic Softmax GP (Galy-Fajou et al., 2020) is the multi-class Gaussian process classification method that relies on the logistic softmax likelihood. Galy-Fajou et al. (2020) did not consider few-shot, but we use the same objectives described in Section 4.4 to adapt this method to FSC. In addition, we used the cosine kernel (see Section H for a description) that we found to work best with our OVE PG GPs. For this method, we found it important to learn a constant mean function (rather than a zero mean) in order to improve calibration. + +# G.3 TRAINING DETAILS + +All methods employed the commonly-used Conv4 architecture (Vinyals et al., 2016) (see Table 4 for a detailed specification), except ABML which used 32 filters throughout. All of our experiments used the Adam (Kingma & Ba, 2015) optimizer with learning rate $1 0 ^ { - 3 }$ . During training, all models used epochs consisting of 100 randomly sampled episodes. A single gradient descent step on the encoder network and relevant hyperparameters is made per episode. All 1-shot models are trained for 600 epochs and 5-shot models are trained for 400 epochs, except for ABML which was trained for an extra 200 epochs. Each episode contained 5 classes (5-way) and 16 query examples. At test time, 15 query examples are used for each episode. Early stopping was performed by monitoring accuracy on the validation set. The validation set was not used for retraining. + +We train both marginal likelihood and predictive likelihood versions of our models. For Polya- ´ Gamma sampling we use the PyPolyaGamma package ´ 3. During training, we use a single step of Gibbs $( T { = } 1 )$ ). For evaluation, we run until $T = 5 0$ . In both training and evaluation, we use $M = 2 0$ parallel Gibbs chains to reduce variance. + +Table 4: Specification of Conv4 architecture. Conv2d layers are $3 \times 3$ with stride 1 and same padding. MaxPool2d layers are $2 \times 2$ with stride 2 and valid padding. + +
Output SizeLayers
1×28×28Input image
64 × 14 ×14Conv2d BatchNorm2d
ReLU
MaxPool2d
64×7×7Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×3×3Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×1×1Conv2d BatchNorm2d
ReLU
MaxPool2d
64Flatten
+ +(a) Omniglot EMNIST dataset. + +
Output SizeLayers
3× 84×84Input image
64 × 42 × 42Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 × 21 × 21Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 ×10×10Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×5×5Conv2d BatchNorm2d
ReLU
MaxPool2d
1600Flatten
+ +(b) All other datasets. + +# H EFFECT OF KERNEL CHOICE ON CLASSIFICATION ACCURACY + +In this section, we examine the effect of kernel choice on classification accuracy for our proposed One-vs-Each Polya-Gamma OVE GPs. ´ + +Cosine Kernel. In the main paper, we showed results for the following kernel, which we refer to as the “cosine” kernel due to its resemblance to cosine similarity: + +$$ +k ^ { \mathrm { { c o s } } } ( \mathbf { x } , \mathbf { x } ^ { \prime } ; \pmb { \theta } , \alpha ) = \mathrm { { e x p } } ( \alpha ) \frac { g _ { \pmb { \theta } } ( \mathbf { x } ) ^ { \top } g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) } { \| g _ { \pmb { \theta } } ( \mathbf { x } ) \| \| g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) \| } , +$$ + +where $g _ { \pmb { \theta } } ( \cdot )$ is a deep neural network that outputs a fixed-dimensional encoded representation of the input and $\alpha$ is the scalar log output scale. Both $\pmb { \theta }$ and $\alpha$ are considered hyperparameters and learned simultaneously as shown in Algorithm 1. We found that this kernel works well for a range of datasets and shot settings. We note that the use of cosine similarity is reminiscent of the approach taken by Baseline $^ { + + }$ method of (Chen et al., 2019), which computes the softmax over cosine similarity to class weights. + +Here we consider three additional kernels: linear, RBF, and normalized RBF. + +Linear Kernel. The linear kernel is defined as follows: + +$$ +k ^ { \mathrm { l i n } } ( \mathbf { x } , \mathbf { x } ^ { \prime } ; \pmb { \theta } , \alpha ) = \frac { 1 } { D } \exp ( \alpha ) g _ { \pmb { \theta } } ( \mathbf { x } ) ^ { \top } g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) , +$$ + +where $D$ is the output dimensionality of $g _ { \pmb { \theta } } ( \mathbf { x } )$ . We apply this dimensionality scaling because the dot product between $g _ { \pmb { \theta } } ( \mathbf { x } )$ and $g _ { \boldsymbol { \theta } } ( \mathbf { x } ^ { \prime } )$ may be large depending on $D$ . + +RBF Kernel. The RBF (also known as squared exponential) kernel can be defined as follows: + +$$ +k ^ { \mathrm { t b f } } ( \mathbf { x } , \mathbf { x } ^ { \prime } ; \pmb { \theta } , \alpha , \ell ) = \mathrm { e x p } ( \alpha ) \mathrm { e x p } \left( - \frac { 1 } { 2 D \mathrm { e x p } ( \ell ) ^ { 2 } } \| g _ { \pmb { \theta } } ( \mathbf { x } ) - g _ { \pmb { \theta } } ( \mathbf { x } ^ { \prime } ) \| ^ { 2 } \right) , +$$ + +where $\ell$ is the log lengthscale parameter (as with $\alpha$ , we learn the $\ell$ alongside $\pmb { \theta }$ ). + +Normalized RBF Kernel. Finally, we consider a normalized RBF kernel similar in spirit to the cosine kernel: + +$$ +k ^ { \mathrm { \scriptsize { t h f - n o r m } } } ( \mathbf { x } , \mathbf { x } ^ { \prime } ; \theta , \alpha , \ell ) = \exp ( \alpha ) \exp \left( - \frac { 1 } { 2 \exp ( \ell ) ^ { 2 } } \left\| \frac { g _ { \theta } ( \mathbf { x } ) } { \| g _ { \theta } ( \mathbf { x } ) \| } - \frac { g _ { \theta } ( \mathbf { x } ^ { \prime } ) } { \| g _ { \theta } ( \mathbf { x } ^ { \prime } ) \| } \right\| ^ { 2 } \right) . +$$ + +The results of our Polya-Gamma OVE GPs with different kernels can be found in Tables 5 ´ and 6. In general, we find that the cosine kernel works best overall, with the exception of Omniglot EMNIST, where RBF does best. + +Table 5: Classification accuracy for Polya-Gamma OVE GPs (our method) using different kernels. ´ Cosine is overall the best, followed closely by linear. RBF-based kernels perform worse, except for the Omniglot EMNIST dataset. Evaluation is performed on 5 randomly generated sets of 600 test episodes. Standard deviation of the mean accuracy is also shown. $\mathbf { M L } = \mathbf { I }$ Marginal Likelihood, $\mathrm { P L } =$ Predictive Likelihood. + +
CUBmini-ImageNet
KernelObjective1-shot5-shot1-shot5-shot
CosineML63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
LinearML62.48 ± 0.2677.94 ± 0.2150.81 ± 0.3066.66 ± 0.45
RBFML58.49 ± 0.4075.50 ± 0.1850.33 ± 0.2664.62 ± 0.37
RBF (normalized)ML62.75 ± 0.3278.71 ± 0.0850.26 ± 0.3164.84 ± 0.39
CosinePL60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
LinearPL60.44 ± 0.3978.54 ± 0.1947.29 ± 0.3166.66 ± 0.36
RBFPL56.18 ± 0.6977.96 ± 0.1948.06 ± 0.2866.66 ± 0.39
RBF (normalized)PL59.78 ± 0.3478.42 ± 0.1347.51 ± 0.2066.42 ± 0.36
+ +Table 6: Cross-domain classification accuracy for Polya-Gamma OVE GPs (our method) using dif- ´ ferent kernels. The experimental setup is the same as Table 5. + +
Omniglot→EMNISTmini-ImageNet-→CUB
KernelObjective1-shot5-shot1-shot5-shot
CosineML68.43 ± 0.6786.22 ± 0.2039.66± 0.1855.71 ± 0.31
LinearML72.42 ± 0.4988.27 ± 0.2039.61 ± 0.1955.07 ± 0.29
RBFML78.05 ± 0.3888.98 ± 0.1636.99 ± 0.0751.75 ± 0.27
RBF (normalized)ML75.51 ± 0.4788.86 ± 0.1638.42 ± 0.1654.20 ± 0.13
CosinePL77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
LinearPL75.87 ± 0.4388.77 ± 0.1036.83 ± 0.2756.46 ± 0.22
RBFPL74.62 ± 0.3589.87 ± 0.1335.06 ± 0.2555.12 ± 0.21
RBF (normalized)PL76.01 ± 0.3189.42 ± 0.1637.50 ± 0.2856.80 ± 0.39
+ +# I ADDITIONAL CALIBRATION RESULTS + +In Figure 9, we include calibration results for mini-Imagenet and Omniglot EMNIST. They follow similar trends to the results presented in Section 5.2. + +![](images/3ce955918e7771637ee77a82ff033e887c6e839cbfde137d0515cbb078a00a51.jpg) +Figure 9: Reliability diagrams, expected calibration error, maximum calibration error, and Brier scores for 5-shot 5-way tasks on mini-Imagenet, Omniglot EMNIST, and mini-Imagenet CUB. Metrics are computed on 3,000 random tasks from the test set. + +# J QUANTITATIVE ROBUSTNESS TO INPUT NOISE RESULTS + +In this section we include quantitative results for the robustness to input noise results presented in Figure 2. Results for Gaussian noise are shown in Table 7, impulse noise in Table 8, and defocus blur in Table 9. + +Table 7: Accuracy $( \% )$ and Brier Score when applying Gaussian noise corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. + +
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.450.77522.580.79922.750.799
Baseline++22.600.79823.820.79724.130.797
MatchingNet26.720.80324.800.79723.590.804
ProtoNet32.280.77829.970.78132.300.779
RelationNet25.230.79923.690.80020.000.800
DKT+Cosine29.540.77927.780.79231.940.782
Bayesian MAML22.790.90520.520.96320.460.949
Bayesian MAML (Chaser)20.201.13320.411.11821.391.039
LSM GP + Cosine (ML)27.920.78722.430.79822.360.799
LSM GP + Cosine (PL)31.210.77231.770.76834.740.754
OVE PG GP + Cosine (ML) [ours]32.270.77429.990.77629.970.784
OVE PG GP + Cosine (PL) [ours]33.010.77133.290.76031.410.764
+ +Table 8: Accuracy $( \% )$ and Brier Score when applying impulse noise corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. + +
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.200.77623.540.79822.870.799
Baseline++28.050.79023.720.79825.580.795
MatchingNet28.250.79023.800.80323.210.811
ProtoNet32.120.77428.810.78332.700.775
RelationNet25.230.79923.130.80020.000.800
DKT+Cosine29.740.77829.110.78932.260.781
Bayesian MAML22.760.90320.500.97020.560.950
Bayesian MAML (Chaser)20.251.17220.511.11621.451.022
LSM GP + Cosine (ML)28.180.78721.820.79923.640.797
LSM GP + Cosine (PL)32.100.76930.220.77635.090.751
OVE PG GP + Cosine (ML) [ours]31.410.77829.660.77830.280.783
OVE PG GP + Cosine (PL) [ours]33.360.77233.230.76132.060.762
+ +Table 9: Accuracy $( \% )$ and Brier Score when applying defocus blur corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. + +
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer38.030.73433.060.79133.470.792
Baseline++42.550.71035.890.76139.880.740
MatchingNet44.430.68234.430.75435.950.741
ProtoNet46.780.67636.920.73741.450.714
RelationNet40.810.75930.110.79025.690.794
DKT + Cosine45.340.69538.290.73745.170.703
Bayesian MAML42.650.69730.630.80837.320.736
Bayesian MAML (Chaser)40.660.88129.931.12131.331.125
LSM GP + Cosine (ML)45.370.70634.100.76939.660.753
LSM GP + Cosine (PL)48.550.69039.460.73743.150.714
OVE PG GP + Cosine (ML) [ours]46.460.70137.650.77543.480.723
OVE PG GP + Cosine (PL) [ours]49.440.69538.950.78043.660.720
\ No newline at end of file diff --git a/parse/train/lgNx56yZh8a/lgNx56yZh8a_content_list.json b/parse/train/lgNx56yZh8a/lgNx56yZh8a_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d38c2601f5b40194527ad5b2708df1b0c517fe7b --- /dev/null +++ b/parse/train/lgNx56yZh8a/lgNx56yZh8a_content_list.json @@ -0,0 +1,3431 @@ +[ + { + "type": "text", + "text": "BAYESIAN FEW-SHOT CLASSIFICATION WITH ONE-VS-EACH POLYA ´ -GAMMA AUGMENTED GAUSSIAN PROCESSES ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jake C. Snell \nUniversity of Toronto \nVector Institute \njsnell@cs.toronto.edu \nRichard Zemel \nUniversity of Toronto \nVector Institute \nCanadian Institute for Advanced Research \nzemel@cs.toronto.edu ", + "bbox": [ + 184, + 195, + 392, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 472, + 195, + 751, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 301, + 544, + 316 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Few-shot classification (FSC), the task of adapting a classifier to unseen classes given a small labeled dataset, is an important step on the path toward human-like machine learning. Bayesian methods are well-suited to tackling the fundamental issue of overfitting in the few-shot scenario because they allow practitioners to specify prior beliefs and update those beliefs in light of observed data. Contemporary approaches to Bayesian few-shot classification maintain a posterior distribution over model parameters, which is slow and requires storage that scales with model size. Instead, we propose a Gaussian process classifier based on a novel combination of Polya-Gamma augmentation and the one-vs-each softmax ´ approximation (Titsias, 2016) that allows us to efficiently marginalize over functions rather than model parameters. We demonstrate improved accuracy and uncertainty quantification on both standard few-shot classification benchmarks and few-shot domain transfer tasks. ", + "bbox": [ + 233, + 332, + 764, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 537, + 336, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Few-shot classification (FSC) is a rapidly growing area of machine learning that seeks to build classifiers able to adapt to novel classes given only a few labeled examples. It is an important step towards machine learning systems that can successfully handle challenging situations such as personalization, rare classes, and time-varying distribution shift. The shortage of labeled data in FSC leads to uncertainty over the parameters of the model, known as model uncertainty or epistemic uncertainty. If model uncertainty is not handled properly in the few-shot setting, there is a significant risk of overfitting. In addition, FSC is increasingly being used for risk-averse applications such as medical diagnosis (Prabhu, 2019) and human-computer interfaces (Wang et al., 2019) where it is important for a few-shot classifier to know when it is uncertain. ", + "bbox": [ + 173, + 568, + 825, + 693 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bayesian methods maintain a distribution over model parameters and thus provide a natural framework for capturing this inherent model uncertainty. In a Bayesian approach, a prior distribution is first placed over the parameters of a model. After data is observed, the posterior distribution over parameters is computed using Bayesian inference. This elegant treatment of model uncertainty has led to a surge of interest in Bayesian approaches to FSC that infer a posterior distribution over the weights of a neural network (Finn et al., 2018; Yoon et al., 2018; Ravi & Beatson, 2019). ", + "bbox": [ + 174, + 700, + 823, + 784 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Although conceptually appealing, there are several practical obstacles to applying Bayesian inference directly to the weights of a neural network. Bayesian neural networks (BNNs) are expensive from both a computational and memory perspective. Moreover, specifying meaningful priors in parameter space is known to be difficult due to the complex relationship between weights and network outputs (Sun et al., 2019). ", + "bbox": [ + 174, + 791, + 823, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gaussian processes (GPs) instead maintain a distribution over functions rather than model parameters. The prior is directly specified by a mean and covariance function, which may be parameterized by deep neural networks. When used with Gaussian likelihoods, GPs admit closed form expressions for the posterior and predictive distributions. They exchange the computational drawbacks of BNNs for cubic scaling with the number of examples. In FSC, where the number of examples is small, this is often an acceptable trade-off. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "When applying GPs to classification with a softmax likelihood, the non-conjugacy of the GP prior renders posterior inference intractable. Many approximate inference methods have been proposed to circumvent this, including variational inference and expectation propagation. In this paper we investigate a particularly promising class of approaches that augment the GP model with a set of auxiliary random variables, such that when they are marginalized out the original model is recovered (Albert & Chib, 1993; Girolami & Rogers, 2006; Linderman et al., 2015). Such augmentation-based approaches typically admit efficient Gibbs sampling procedures for generating posterior samples which when combined with Fisher’s identity (Douc et al., 2014) can be used to optimize the parameters of the mean and covariance functions. ", + "bbox": [ + 174, + 138, + 825, + 263 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In particular, augmentation with Polya-Gamma random variables (Polson et al., 2013) makes in- ´ ference tractable in logistic models. Naively, this is useful for handling binary classification, but in this paper we show how to extend Polya-Gamma augmentation to multiple classes by using the ´ one-vs-each softmax approximation (Titsias, 2016), which can be expressed as a product of logistic sigmoids. We further show that the one-vs-each approximation can be interpreted as a composite likelihood (Lindsay, 1988; Varin et al., 2011), a connection which to our knowledge has not been made in the literature. ", + "bbox": [ + 174, + 270, + 825, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we make several contributions: ", + "bbox": [ + 174, + 376, + 464, + 390 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We show how the one-vs-each softmax approximation (Titsias, 2016) can be interpreted as a composite likelihood consisting of pairwise conditional terms. \n• We propose a novel GP classification method that combines the one-vs-each softmax approximation with Polya-Gamma augmentation for tractable inference. ´ We demonstrate competitive classification accuracy of our method on standard FSC benchmarks and challenging domain transfer settings. We propose several new benchmarks for uncertainty quantification in FSC, including calibration, robustness to input noise, and out-of-episode detection. \n• We demonstrate improved uncertainty quantification of our method on the proposed benchmarks relative to standard few-shot baselines. ", + "bbox": [ + 215, + 401, + 825, + 561 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 582, + 344, + 598 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work is related to both GP methods for handling non-conjugate classification likelihoods and Bayesian approaches to few-shot classification. We summarize relevant work here. ", + "bbox": [ + 174, + 614, + 823, + 642 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GP CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 659, + 357, + 674 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Non-augmentation approaches. There are several classes of approaches for applying Gaussian processes to classification. The most straightforward method, known as least squares classification (Rifkin & Klautau, 2004), treats class labels as real-valued observations and performs inference with a Gaussian likelihood. The Laplace approximation (Williams & Barber, 1998) constructs a Gaussian approximate posterior centered at the posterior mode. Variational approaches (Titsias, 2009; Matthews et al., 2016) maximize a lower bound on the log marginal likelihood. In expectation propagation (Minka, 2001; Kim & Ghahramani, 2006; Hernandez-Lobato & Hernandez-Lobato, 2016), local Gaussian approximations to the likelihood are fitted iteratively to minimize KL divergence from the true posterior. ", + "bbox": [ + 174, + 685, + 825, + 810 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Augmentation approaches. Augmentation-based approaches introduce auxiliary random variables such that the original model is recovered when marginalized out. Girolami & Rogers (2006) propose a Gaussian augmentation for multinomial probit regression. Linderman et al. (2015) utilize Polya-Gamma augmentation (Polson et al., 2013) and a stick-breaking construction to decom- ´ pose a multinomial distribution into a product of binomials. Galy-Fajou et al. (2020) propose a logistic-softmax likelihood for classification and uses Gamma and Poisson augmentation in addition to Polya-Gamma augmentation in order to perform inference. ´ ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 FEW-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 410, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Meta-learning. A common approach to FSC is meta-learning, which seeks to learn a strategy to update neural network parameters when faced with a novel learning task. The Meta-learner LSTM (Ravi & Larochelle, 2017) learns a meta-level LSTM to recurrently output a new set of parameters for a base learner. MAML (Finn et al., 2017) learns initializations of deep neural networks that perform well on task-specific losses after one or a few steps of gradient descent by backpropagating through the gradient descent procedure itself. LEO (Rusu et al., 2019) performs meta-learning in a learned low-dimensional latent space from which the parameters of a classifier are generated. ", + "bbox": [ + 173, + 128, + 825, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Metric learning. Metric learning approaches learn distances such that input examples can be meaningfully compared. Siamese Networks (Koch, 2015) learn a shared embedding network along with a distance layer for computing the probability that two examples belong to the same class. Matching Networks (Vinyals et al., 2016) uses a nonparametric classification in the form of attention over nearby examples, which can be interpreted as a form of soft $k$ -nearest neighbors in the embedding space. Prototypical Networks (Snell et al., 2017) make predictions based on distances to nearest class centroids. Relation Networks (Sung et al., 2018) instead learn a more complex neural network distance function on top of the embedding layer. ", + "bbox": [ + 174, + 242, + 825, + 353 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Bayesian Few-shot Classification. More recently, Bayesian FSC approaches that attempt to infer a posterior over task-specific parameters have appeared. Grant et al. (2018) reinterpret MAML as an approximate empirical Bayes algorithm and propose LLAMA, which optimizes the Laplace approximation to the marginal likelihood. Bayesian MAML (Yoon et al., 2018) instead uses Stein Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to approximate the posterior distribution over model parameters. VERSA (Gordon et al., 2019) uses amortized inference networks to obtain an approximate posterior distribution over task-specific parameters. ABML (Ravi & Beatson, 2019) uses a few steps of Bayes by Backprop (Blundell et al., 2015) on the support set to produce an approximate posterior over network parameters. CNAPs (Requeima et al., 2019) modulate taskspecific Feature-wise Linear Modulation (FiLM) (Perez et al., 2018) layer parameters as the output of an adaptation network that takes the support set as input. ", + "bbox": [ + 173, + 368, + 825, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "GPs for Few-shot Learning. There have been relatively few works applying GPs to few-shot learning. Tossou et al. (2020) consider Gaussian processes in the context of few-shot regression with Gaussian likelihoods. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) uses Gaussian processes with least squares classification to perform few-shot classification and learns covariance functions parameterized by deep neural networks. More recently, Titsias et al. (2020) applies GPs to meta-learning by maximizing the mutual information between the query set and a latent representation of the support set. ", + "bbox": [ + 174, + 536, + 825, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 654, + 326, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section we first review Polya-Gamma augmentation for binary classification and the one-vs- ´ each approximation before we introduce our method in Section 4. ", + "bbox": [ + 174, + 685, + 823, + 714 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 POLYA ´ -GAMMA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 729, + 444, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Polya-Gamma augmentation scheme was originally introduced to address Bayesian inference ´ in logistic models (Polson et al., 2013). Suppose we have a vector of logits $\\boldsymbol { \\psi } \\in \\mathbf { \\mathbb { R } } ^ { N }$ with corresponding binary labels $\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }$ . The logistic likelihood is ", + "bbox": [ + 174, + 756, + 825, + 799 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e3e982e6c0f9baa98e741f5cc90a966e02cb28400857304486eb68ec68bcae1b.jpg", + "text": "$$\np ( \\mathbf { y } \\vert \\psi ) = \\prod _ { i = 1 } ^ { N } \\sigma ( \\psi _ { i } ) ^ { y _ { i } } ( 1 - \\sigma ( \\psi _ { i } ) ) ^ { 1 - y _ { i } } = \\prod _ { i = 1 } ^ { N } \\frac { ( e ^ { \\psi _ { i } } ) ^ { y _ { i } } } { 1 + e ^ { \\psi _ { i } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 804, + 679, + 848 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\sigma ( \\cdot )$ is the logistic sigmoid function. Let the prior over $\\psi$ be Gaussian: $p ( \\psi ) = \\mathcal { N } ( \\psi | \\mu , \\Sigma )$ . In Bayesian inference, we are interested in the posterior $p ( \\psi | \\mathbf { y } ) \\propto p ( \\mathbf { y } | \\psi ) p ( \\psi )$ but the form of (1) does not admit analytic computation of the posterior due to non-conjugacy. The main idea of Polya-Gamma augmentation is to introduce auxiliary random variables ´ $\\omega$ to the likelihood such that the original model is recovered when $\\omega$ is marginalized out: $\\begin{array} { r } { p ( \\mathbf { y } \\vert \\psi ) = \\int p ( \\omega ) p ( \\mathbf { y } \\vert \\psi , \\omega ) d \\omega } \\end{array}$ . ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Conditioned on $\\omega \\sim \\mathrm { P G } ( 1 , 0 )$ , the batch likelihood is proportional to a diagonal Gaussian (see Section A for a full derivation): ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3a5894d75b90789adc6d12004abf8179a5863fa7a365b05a01dd891fce90b6bd.jpg", + "text": "$$\np ( \\mathbf { y } \\vert \\psi , \\boldsymbol { \\omega } ) \\propto \\prod _ { i = 1 } ^ { N } e ^ { - \\omega _ { i } \\psi _ { i } ^ { 2 } / 2 } e ^ { \\kappa _ { i } \\psi _ { i } } \\propto \\mathcal { N } ( \\Omega ^ { - 1 } \\kappa \\vert \\psi , \\Omega ^ { - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 137, + 683, + 181 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\kappa _ { i } = y _ { i } - 1 / 2$ and $\\pmb { \\Omega } = \\mathrm { d i a g } ( \\omega )$ . The conditional distribution over $\\psi$ given $\\mathbf { y }$ and $\\omega$ is now tractable: ", + "bbox": [ + 169, + 186, + 825, + 214 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5b78459ed3a3c6430364996574e6c544297f9d4d15e777aec375ee2c0e20e1fb.jpg", + "text": "$$\np ( \\psi | \\mathbf { y } , \\boldsymbol { \\omega } ) \\propto p ( \\mathbf { y } | \\psi , \\boldsymbol { \\omega } ) p ( \\psi ) \\propto \\mathcal { N } ( \\psi | \\tilde { \\boldsymbol { \\Sigma } } ( \\Sigma ^ { - 1 } \\mu + \\kappa ) , \\tilde { \\boldsymbol { \\Sigma } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 212, + 692, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\tilde { \\Sigma } = ( \\Sigma ^ { - 1 } + \\Omega ) ^ { - 1 }$ . The conditional distribution of $\\omega$ given $\\psi$ and $\\mathbf { y }$ can also be easily computed: ", + "bbox": [ + 169, + 234, + 826, + 263 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/59de6b01656fdb2035a5d6345022e91ea9c6ad2a6fb2140f9bdaf065918c0c85.jpg", + "text": "$$\np ( \\omega _ { i } | y _ { i } , \\psi _ { i } ) \\propto \\mathrm { P G } ( \\omega _ { i } | 1 , 0 ) e ^ { - \\omega _ { i } \\psi _ { i } ^ { 2 } / 2 } \\propto \\mathrm { P G } ( \\omega _ { i } | 1 , \\psi _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 323, + 262, + 673, + 282 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the last expression follows from the exponential tilting property of Polya-Gamma random ´ variables. This suggest a Gibbs sampling procedure in which iterates ${ \\bf \\dot { \\boldsymbol { \\omega } } } ^ { ( t ) } \\sim \\dot { p } ( \\boldsymbol { \\omega } | { \\bf y } , \\psi ^ { ( t - 1 ) } )$ and $\\psi ^ { ( t ) } \\sim p ( \\psi | \\mathbf { X } , \\mathbf { y } , \\omega ^ { ( t ) } )$ are drawn sequentially until the Markov chain reaches its stationary distribution, which is the joint posterior $p ( \\boldsymbol { \\psi } , \\boldsymbol { \\omega } | \\mathbf { y } )$ . Fortunately, efficient samplers for the Polya-Gamma ´ distribution have been developed (Windle et al., 2014) to facilitate this. ", + "bbox": [ + 174, + 285, + 825, + 358 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 ONE-VS-EACH APPROXIMATION TO SOFTMAX ", + "text_level": 1, + "bbox": [ + 174, + 375, + 537, + 390 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The one-vs-each (OVE) approximation (Titsias, 2016) was formulated as a lower bound to the softmax likelihood in order to handle classification over a large number of output classes, where computation of the normalizing constant is prohibitive. We employ the OVE approximation not to deal with extreme classification, but rather due to its compatibility with Polya-Gamma augmentation, as ´ we shall soon see. The one-vs-each approximation can be derived by first rewriting the softmax likelihood as follows: ", + "bbox": [ + 173, + 400, + 825, + 484 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e210942b3bd7619017aff1067a426322fe4845d4d68982913634730ab0c4c56f.jpg", + "text": "$$\np ( y = i \\mid { \\bf f } ) \\triangleq { \\frac { e ^ { f _ { i } } } { \\sum _ { j } e ^ { f _ { j } } } } = { \\frac { 1 } { 1 + \\sum _ { j \\neq i } e ^ { - ( f _ { i } - f _ { j } ) } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 341, + 487, + 655, + 526 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbf { f } \\triangleq ( f _ { 1 } , \\ldots , f _ { C } ) ^ { \\intercal }$ are the logits. Since in general $\\begin{array} { r } { \\prod _ { k } ( 1 + \\alpha _ { k } ) \\ge \\left( 1 + \\sum _ { k } \\alpha _ { k } \\right) } \\end{array}$ for $\\alpha _ { k } \\geq 0$ , the softmax likelihood (5) can be bounded as follows: ", + "bbox": [ + 174, + 534, + 823, + 563 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/90861d2f02b324d130c82e1c3f3af6b15ceaf10e2bbe6a6c5d0e64bf0815d56a.jpg", + "text": "$$\np ( y = i \\mid \\mathbf { f } ) \\geq \\prod _ { j \\neq i } { \\frac { 1 } { 1 + e ^ { - ( f _ { i } - f _ { j } ) } } } = \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 568, + 668, + 607 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which is the OVE lower bound. This expression avoids the normalizing constant and factorizes into a product of pairwise sigmoids, which is amenable to Polya-Gamma augmentation for tractable ´ inference. ", + "bbox": [ + 174, + 612, + 825, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 ONE-VS-EACH POLYA ´ -GAMMA GPS ", + "text_level": 1, + "bbox": [ + 174, + 674, + 508, + 691 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we first show how the one-vs-each (OVE) approximation can be interpreted as a pairwise composite likelihood. We then we introduce our method for GP-based Bayesian few-shot classification, which brings together OVE and Polya-Gamma augmentation in a novel combination. ´ ", + "bbox": [ + 174, + 707, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 OVE AS A COMPOSITE LIKELIHOOD ", + "text_level": 1, + "bbox": [ + 174, + 765, + 467, + 780 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Titsias (2016) showed that the OVE approximation shares the same global optimum as the softmax maximum likelihood, suggesting a close relationship between the two. We show here that in fact OVE can be interpreted as a pairwise composite likelihood version of the softmax. Composite likelihoods (Lindsay, 1988; Varin et al., 2011) are a type of approximate likelihood often employed when the exact likelihood is intractable or otherwise difficult to compute. Given a collection of marginal or conditional events $\\{ E _ { 1 } , \\ldots , E _ { K } \\}$ and parameters $\\mathbf { f }$ , a composite likelihood is defined as: ", + "bbox": [ + 173, + 791, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9668b14477129b07e9192a249f8ce4c7428a88c8159f4d41432b0d4f622bf246.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { C L } } ( \\mathbf { f } \\mid y ) \\triangleq \\prod _ { k = 1 } ^ { K } \\mathcal { L } _ { k } ( \\mathbf { f } \\mid y ) ^ { w _ { k } } ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 885, + 598, + 928 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { L } _ { k } ( \\mathbf { f } \\mid y ) \\propto p ( y \\in E _ { k } \\mid \\mathbf { f } )$ and $w _ { k } \\geq 0$ are arbitrary weights. ", + "bbox": [ + 173, + 102, + 611, + 119 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to make the connection to OVE, it will be useful to let the one-hot encoding of the label $y$ be denoted as $\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { C }$ . Define a set of $C ( C - 1 ) / 2$ pairwise conditional events $E _ { i j }$ , one each for all pairs of classes $i \\neq j$ , indicating the event that the model’s output matches the target label for classes $i$ and $j$ conditioned on all the other classes: ", + "bbox": [ + 173, + 125, + 825, + 181 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a25d950c1322db87f6f35072e4e7c35f7b2d81698856e4a5cf68691da1782a29.jpg", + "text": "$$\np ( \\mathbf { y } \\in E _ { i j } \\mid \\mathbf { f } ) \\triangleq p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { \\lnot i j } , \\mathbf { f } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 189, + 614, + 208 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\neg i j$ denotes the set of classes not equal to either $i$ or $j$ . This expression resembles the pseudolikelihood (Besag, 1975), but instead of a single conditional event per output site, the expression in (8) considers all pairs of sites. Stoehr $\\&$ Friel (2015) explored similar composite likelihood generalizations of the pseudolikelihood in the context of random fields. ", + "bbox": [ + 173, + 214, + 825, + 271 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Now suppose that $y _ { c } = 1$ for some class $c \\notin \\{ i , j \\}$ . Then $p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { \\lnot i j } , \\mathbf { f } ) = 1$ due to the one-hot constraint. Otherwise either $y _ { i } = 1$ or $y _ { j } = 1$ . In this case, assume without loss of generality that $y _ { i } = 1$ and $y _ { j } = 0$ and thus ", + "bbox": [ + 174, + 276, + 823, + 320 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f63091135d80d1909645efeac208718d4b4e75984d5fc21fdc6474abaac8ef7f.jpg", + "text": "$$\np ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { - i j } , \\mathbf { f } ) = \\frac { e ^ { f _ { i } } } { e ^ { f _ { i } } + e ^ { f _ { j } } } = \\sigma ( f _ { i } - f _ { j } ) .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 327, + 648, + 362 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The composite likelihood defined in this way with unit component weights is therefore ", + "bbox": [ + 173, + 367, + 743, + 383 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/58dacd071253f6d789146bca499c2eca91a5542e4a8427a5d291e3b6fe64b7f2.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { O V E } } ( \\mathbf { f } \\mid \\mathbf { y } ) = \\prod _ { i } \\prod _ { j \\neq i } p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { - i j } , \\mathbf { f } ) = \\prod _ { i } \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) ^ { y _ { i } } .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 388, + 704, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Alternatively, we may simply write $\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { O V E } } ( \\mathbf { f } \\mid y = i ) = \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) } \\end{array}$ , which is identical to the OVE bound (6). ", + "bbox": [ + 174, + 431, + 823, + 464 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 GP CLASSIFICATION WITH THE OVE LIKELIHOOD ", + "text_level": 1, + "bbox": [ + 173, + 479, + 566, + 496 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now turn our attention to GP classification. Suppose we have access to examples $\\mathbf { X } \\in \\mathbb { R } ^ { N \\times D }$ with corresponding one-hot labels $\\mathbf { Y } \\in \\{ 0 , 1 \\} ^ { N \\times C }$ , where $C$ is the number of classes. We consider the logits jointly as a single vector ", + "bbox": [ + 173, + 506, + 825, + 549 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ff472397405b2fff69ba31b7bc09a13b31fc5be3f0debdd0e8f287fffd0acd39.jpg", + "text": "$$\n\\mathbf { f } \\triangleq ( f _ { 1 } ^ { 1 } , \\hdots , f _ { N } ^ { 1 } , f _ { 1 } ^ { 2 } , \\hdots , f _ { N } ^ { 2 } , \\hdots , f _ { 1 } ^ { C } , \\hdots , f _ { N } ^ { C } ) ^ { \\intercal }\n$$", + "text_format": "latex", + "bbox": [ + 334, + 556, + 661, + 577 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and place an independent GP prior on the logits for each class: $\\mathbf { f } ^ { c } ( \\mathbf { x } ) \\sim \\mathcal { G P } ( m ( \\mathbf { x } ) , k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) )$ . Therefore we have $p ( \\mathbf { f } | \\mathbf { X } ) = \\bar { \\mathcal { N } } ( \\mathbf { f } | \\pmb { \\mu } , \\mathbf { K } )$ , where $\\mu _ { i } ^ { c } = m ( \\mathbf { x } _ { i } )$ and $\\mathbf { K }$ is block diagonal with $K _ { i j } ^ { c } =$ $k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )$ for each block ${ \\bf K } ^ { c }$ . ", + "bbox": [ + 174, + 583, + 825, + 627 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The Polya-Gamma integral identity used to derive (2) does not have a multi-class analogue and thus ´ a direct application of the augmentation scheme to the softmax likelihood is nontrivial. Instead, we propose to directly replace the softmax with the OVE-based composite likelihood function from (10) with unit weights. The posterior over f when using OVE as the likelihood function can be expressed as: ", + "bbox": [ + 173, + 633, + 825, + 703 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3b907a26f1bd9fbcf12fedcbc988dc0b11b59255f1d91a61e8c34221281cebbb.jpg", + "text": "$$\np ( \\mathbf { f } | \\mathbf { X } , \\mathbf { y } ) \\propto p ( \\mathbf { f } | \\mathbf { X } ) \\prod _ { i = 1 } ^ { N } \\prod _ { c ^ { \\prime } \\neq y _ { i } } \\sigma ( f _ { i } ^ { y _ { i } } - f _ { i } ^ { c ^ { \\prime } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 700, + 642, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "to which Polya-Gamma augmentation can be applied as we show in the next section. Our motiva- ´ tion for using a composite likelihood therefore differs from the traditional motivation, which is to avoid the use of a likelihood function which is intractable to evaluate. Instead, we employ a composite likelihood because it makes posterior inference tractable when coupled with Polya-Gamma ´ augmentation. ", + "bbox": [ + 173, + 750, + 825, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Prior work on Bayesian inference with composite likelihoods has shown that the composite posterior is consistent under fairly general conditions for correctly specified models (Miller, 2019) but can produce overly concentrated posteriors (Pauli et al., 2011; Ribatet et al., 2012) since each component likelihood event is treated as independent when in reality there may be significant dependencies. Nevertheless, we show in Section 5 that in practice our method exhibits competitive accuracy and strong calibration relative to baseline few-shot learning algorithms. We leave further theoretical analysis of the OVE composite posterior and its properties for future work. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Compared to choices of likelihoods used by previous approaches, there are several reasons to prefer OVE. Relative to the Gaussian augmentation approach of Girolami & Rogers (2006), Polya-Gamma ´ augmentation has the benefit of fast mixing and the ability of a single value of $\\omega$ to capture much of the marginal distribution over function values1. The stick-breaking construction of Linderman et al. (2015) induces a dependence on the ordering of classes, which leads to undesirable asymmetry. Finally, the logistic-softmax likelihood of Galy-Fajou et al. (2020) requires three augmentations and careful learning of the mean function to avoid a priori underconfidence (see Section F.1 for more details). ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 POSTERIOR INFERENCE VIA GIBBS SAMPLING ", + "text_level": 1, + "bbox": [ + 174, + 231, + 537, + 246 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now describe how we perform tractable posterior inference in our model with Gibbs sampling. Define the matrix $\\mathbf { A } \\ { \\overset { \\Delta } { = } } \\ \\operatorname { O V E - M A T R I X } ( \\mathbf { Y } )$ to be a $C N \\times C N$ sparse block matrix with $C$ row partitions and $C$ column partitions. Each block $\\mathbf { A } _ { c c ^ { \\prime } }$ is a diagonal $N \\times N$ matrix defined as follows: ", + "bbox": [ + 174, + 257, + 823, + 301 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/27d21363187fab33be498e7b781ca214cdc72ceee3720a9e52423a90b00bed25.jpg", + "text": "$$\n\\mathbf { A } _ { c c ^ { \\prime } } \\triangleq \\mathrm { d i a g } ( \\mathbf { Y } _ { \\cdot c ^ { \\prime } } ) - \\mathbb { 1 } [ c = c ^ { \\prime } ] \\mathbf { I } _ { n } ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 306, + 611, + 325 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\mathbf { Y } _ { \\cdot c ^ { \\prime } }$ denotes the $c ^ { \\prime }$ th column of $\\mathbf { Y }$ . Now the binary logit vector $\\pmb { \\psi } \\triangleq \\mathbf { A } \\mathbf { f } \\in \\mathbb { R } ^ { C N }$ will have entries equal to $f _ { i } ^ { y _ { i } } - f _ { i } ^ { c }$ for each unique combination of $c$ and $i$ , of which there are $C N$ in total. The OVE composite likelihood can now be written as $\\begin{array} { r } { \\mathcal { L } ( \\psi | \\mathbf { Y } ) = 2 ^ { N } \\prod _ { j = 1 } ^ { N C } \\sigma ( \\psi _ { j } ) } \\end{array}$ , where the $2 ^ { N }$ term arises from the $N$ cases in which $\\psi _ { j } = 0$ due to comparing the ground truth logit with itself. ", + "bbox": [ + 174, + 330, + 823, + 393 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Analogous to (2), the likelihood of $\\psi$ conditioned on $\\omega$ and $\\mathbf { Y }$ is proportional to a diagonal Gaussian: ", + "bbox": [ + 173, + 398, + 821, + 415 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/63ccdd2f8f594d721e0c141865ba4f53f286772b9ed444697f11fa1db8a57094.jpg", + "text": "$$\n\\mathcal { L } ( \\psi | \\mathbf { Y } , \\omega ) \\propto \\prod _ { j = 1 } ^ { N C } e ^ { - \\omega _ { j } \\psi _ { j } ^ { 2 } / 2 } e ^ { \\kappa _ { j } \\psi _ { j } } \\propto \\mathcal { N } ( \\Omega ^ { - 1 } \\kappa | \\psi , \\Omega ^ { - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 310, + 420, + 686, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\kappa _ { j } ~ = ~ 1 / 2$ and $\\Omega = \\mathrm { d i a g } ( \\omega )$ . By exploiting the fact that $\\psi = \\mathbf { A } \\mathbf { f }$ , we can express the likelihood in terms of $\\mathbf { f }$ and write down the conditional composite posterior as follows: ", + "bbox": [ + 173, + 468, + 823, + 497 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/073269aa65a5eba3370ea1dcaf43de7b569197da28c7c2f7574e73557d6bb305.jpg", + "text": "$$\np ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } , \\omega ) \\propto \\mathcal { N } ( \\Omega ^ { - 1 } \\kappa | \\mathbf { A } \\mathbf { f } , \\Omega ^ { - 1 } ) \\mathcal { N } ( \\mathbf { f } | \\boldsymbol { \\mu } , \\mathbf { K } ) \\propto \\mathcal { N } ( \\mathbf { f } | \\tilde { \\boldsymbol { \\Sigma } } ( \\mathbf { K } ^ { - 1 } \\boldsymbol { \\mu } + \\mathbf { A } ^ { \\top } \\boldsymbol { \\kappa } ) , \\tilde { \\boldsymbol { \\Sigma } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 228, + 501, + 764, + 521 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\tilde { \\boldsymbol { \\Sigma } } = ( \\boldsymbol { \\mathbf { K } } ^ { - 1 } + \\boldsymbol { \\mathbf { A } } ^ { \\top } \\boldsymbol { \\Omega } \\boldsymbol { \\mathbf { A } } ) ^ { - 1 }$ , which is an expression remarkably similar to (3). Analogous to (4), the conditional distribution over $\\omega$ given f and the data becomes $p ( \\omega | \\mathbf { y } , \\mathbf { f } ) = \\mathrm { P G } ( \\omega | \\mathbf { 1 } , \\mathbf { A } \\mathbf { f } )$ . ", + "bbox": [ + 173, + 526, + 823, + 556 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The primary computational bottleneck of posterior inference lies in sampling $\\mathbf { f }$ from (15). Since $\\tilde { \\Sigma }$ is a $C N \\times C N$ matrix, a naive implementation has complexity $\\mathcal { O } ( C ^ { 3 } N ^ { 3 } )$ . By utilizing the matrix inversion lemma and Gaussian sampling techniques summarized in (Doucet, 2010), this can be brought down to $\\mathcal { O } ( C N ^ { 3 } )$ . Details may be found in Section B. ", + "bbox": [ + 174, + 563, + 825, + 621 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 LEARNING COVARIANCE HYPERPARAMETERS FOR FEW-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 171, + 637, + 763, + 651 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now describe how we apply OVE Polya-Gamma augmented GPs to few-shot classification. ´ We assume the standard episodic few-shot setup in which one observes a labeled support set ${ \\boldsymbol { s } } =$ $( \\mathbf { X } , \\mathbf { Y } )$ . Predictions must then be made for a query example $\\left( \\mathbf { x } _ { * } , \\mathbf { y } _ { * } \\right)$ . We consider a zero-mean GP prior over the class logits $\\mathbf { f } ^ { c } ( \\mathbf { x } ) \\sim \\mathcal { G P } ( \\mathbf { 0 } , k _ { \\theta } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) )$ , where $\\pmb \\theta$ are learnable parameters of our covariance function. These could include traditional hyperparameters such as lengthscales or the weights of a deep neural network as in deep kernel learning (Wilson et al., 2016). ", + "bbox": [ + 174, + 662, + 825, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We consider two objectives for learning hyperparameters of the covariance function: the marginal likelihood (ML) and the predictive likelihood (PL). Marginal likelihood measures the likelihood of the hyperparameters given the observed data and is intuitively appealing from a Bayesian perspective. On the other hand, many standard FSC methods optimize for predictive likelihood on the query set (Vinyals et al., 2016; Finn et al., 2017; Snell et al., 2017). Both objectives marginalize over latent functions, thereby making full use of our Bayesian formulation. ", + "bbox": [ + 174, + 753, + 825, + 838 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The details of these objectives and how we compute gradients can be found in Section C. Our learning algorithm for both marginal and predictive likelihood may be found in Section D. Details of computing the posterior predictive distribution $p ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega )$ may be found in Section E. Finally, details of our chosen “cosine” kernel may be found in Section $_ \\mathrm { H }$ . ", + "bbox": [ + 174, + 844, + 823, + 900 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 326, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we present our results on few-shot classification both in terms of accuracy and uncertainty quantification. Additional results comparing the one-vs-each composite likelihood to the softmax, logistic softmax, and Gaussian likelihoods may be found in Section F. ", + "bbox": [ + 174, + 133, + 823, + 175 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "One of our aims is to compare methods based on uncertainty quantification. We therefore developed new benchmark evaluations and tasks: few-shot calibration, robustness, and out-of-episode detection. In order to empirically compare methods, we could not simply borrow the accuracy results from other papers, but instead needed to train each of these baselines ourselves. For all baselines except Bayesian MAML, ABML, and Logistic Softmax GP, we ran the code from (Patacchiola et al., 2020) and verified that the accuracies matched closely to their reported results. We have made PyTorch code for our experiments publicly available2. ", + "bbox": [ + 174, + 183, + 825, + 280 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 FEW-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 297, + 408, + 311 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For our few-shot classification experiments, we follow the training and evaluation protocol of Patacchiola et al. (2020). We train both 1-shot and 5-shot versions of our model in four different settings: Caltech-UCSD Birds (CUB) (Wah et al., 2011), mini-Imagenet with the split proposed by Ravi & Larochelle (2017), as well as two cross-domain transfer tasks. The first transfer task entails training on mini-ImageNet and testing on CUB, and the second measures transfer from Omniglot (Lake et al., 2011) to EMNIST (Cohen et al., 2017). Experimental details and an overview of the baselines we used can be found in Section G. Classification results are shown in Table 1 and 2. We find that our proposed Polya-Gamma OVE GPs yield strong classification results, outperforming the baselines in ´ five of the eight scenarios. ", + "bbox": [ + 173, + 323, + 825, + 449 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/1309722992b5c3a874a367571cd93a505779b4fca1cf275bf846ea6cda75f66d.jpg", + "table_caption": [ + "Table 1: Average accuracy and standard deviation (percentage) on 5-way FSC. Baseline results (through DKT) are from Patacchiola et al. (2020). Evaluation is performed on 3,000 randomly generated test episodes. Standard deviation for the remaining methods are computed by averaging over 5 batches of 600 episodes with different random seeds. The best results are highlighted in bold. " + ], + "table_footnote": [], + "table_body": "
MethodCUBmini-ImageNet
1-shot5-shot1-shot5-shot
FeatureTransfer46.19± 0.6468.40± 0.7939.51 ± 0.2360.51 ± 0.55
Baseline++61.75 ± 0.9578.51 ± 0.5947.15 ± 0.4966.18 ± 0.18
MatchingNet60.19 ± 1.0275.11 ± 0.3548.25 ± 0.6562.71 ± 0.44
ProtoNet52.52 ± 1.9075.93 ± 0.4644.19 ± 1.3064.07 ± 0.65
RelationNet62.52 ± 0.3478.22 ± 0.0748.76 ± 0.1764.20 ± 0.28
MAML56.11 ± 0.6974.84 ± 0.6245.39 ± 0.4961.58 ± 0.53
DKT + Cosine63.37 ± 0.1977.73 ± 0.2648.64 ± 0.4562.85 ± 0.37
BayesianMAML55.93 ± 0.7172.87 ± 0.2644.46 ± 0.3062.60 ± 0.25
Bayesian MAML(Chaser)53.93 ± 0.7271.16 ± 0.3243.74 ± 0.4659.23 ± 0.34
ABML48.80 ± 0.4070.91 ± 0.3240.88 ± 0.2558.19 ± 0.17
Logistic Softmax GP + Cosine (ML)60.23 ± 0.5474.58 ± 0.2546.75 ± 0.2059.93 ± 0.31
Logistic Softmax GP + Cosine (PL)60.07 ±0.2978.14 ± 0.0747.05 ± 0.2066.01 ± 0.25
OVE PG GP + Cosine (ML) [ours]63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
OVE PG GP +Cosine (PL) [ours]60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
", + "bbox": [ + 171, + 530, + 836, + 756 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 UNCERTAINTY QUANTIFICATION THROUGH CALIBRATION ", + "text_level": 1, + "bbox": [ + 173, + 790, + 617, + 804 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We next turn to uncertainty quantification, an important concern for few-shot classifiers. When used in safety-critical applications such as medical diagnosis, it is important for a machine learning system to defer when there is not enough evidence to make a decision. Even in non-critical applications, precise uncertainty quantification helps practitioners in the few-shot setting determine when a class has an adequate amount of labeled data or when more labels are required, and can facilitate active learning. ", + "bbox": [ + 174, + 815, + 825, + 898 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/7d9544c327aa4fe4a030b1dd4d5fdff6563f2b2019326ea07af46af306eef90c.jpg", + "table_caption": [ + "Table 2: Average accuracy and standard deviation (percentage) on 5-way cross-domain FSC, with the same experimental setup as in Table 1. Baseline results (through DKT) are from (Patacchiola et al., 2020). " + ], + "table_footnote": [], + "table_body": "
MethodOmniglot→EMNISTmini-ImageNet→CUB
1-shot5-shot1-shot5-shot
Feature Transfer64.22 ±1.2486.10± 0.8432.77 ± 0.3550.34± 0.27
Baseline++56.84 ± 0.9180.01 ± 0.9239.19 ± 0.1257.31 ± 0.11
MatchingNet75.01 ± 2.0987.41 ± 1.7936.98 ± 0.0650.72 ± 0.36
ProtoNet72.04 ± 0.8287.22 ± 1.0133.27 ± 1.0952.16 ± 0.17
RelationNet75.62 ± 1.0087.84 ± 0.2737.13 ± 0.2051.76 ± 1.48
MAML72.68 ± 1.8583.54 ± 1.7934.01 ±1.2548.83 ± 0.62
DKT + Cosine73.06 ± 2.3688.10 ± 0.7840.22 ± 0.5455.65 ± 0.05
Bayesian MAML63.94 ± 0.4765.26 ± 0.3033.52 ± 0.3651.35 ± 0.16
Bayesian MAML (Chaser)55.04 ± 0.3454.19 ± 0.3236.22 ± 0.5051.53 ± 0.43
ABML73.89 ± 0.2487.28 ± 0.4031.51 ± 0.3247.80 ± 0.51
Logistic Softmax GP + Cosine (ML)62.91 ± 0.4983.80 ± 0.1336.41 ± 0.1850.33 ± 0.13
Logistic Softmax GP + Cosine (PL)70.70 ±0.3686.59 ± 0.1536.73 ± 0.2656.70 ± 0.31
OVE PG GP + Cosine (ML) [ours]68.43± 0.6786.22 ±0.2039.66± 0.1855.71 ± 0.31
OVE PG GP + Cosine (PL) [ours]77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
", + "bbox": [ + 171, + 155, + 841, + 381 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We chose several commonly used metrics for calibration. Expected calibration error (ECE) (Guo et al., 2017) measures the expected binned difference between confidence and accuracy. Maximum calibration error (MCE) is similar to ECE but measures maximum difference instead of expected difference. Brier score (BRI) (Brier, 1950) is a proper scoring rule computed as the squared error between the output probabilities and the one-hot label. For a recent perspective on metrics for uncertainty evaluation, please refer to Ovadia et al. (2019). The results for representative approaches on 5-shot, 5-way CUB can be found in Figure 1. Our OVE PG GPs are the best calibrated overall across the metrics. ", + "bbox": [ + 173, + 407, + 825, + 518 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1830b095a5b4a6ce7ed6b21a00496d1191e5c709cd10ae31e6c6e7a1cada7456.jpg", + "image_caption": [ + "Figure 1: Reliability diagrams, expected calibration error (ECE), maximum calibration error (MCE), and Brier Score (BRI) for 5-shot 5-way tasks on CUB (additional calibration results can be found in Appendix I). Metrics are computed on 3,000 random tasks from the test set. The last two plots are our proposed method. " + ], + "image_footnote": [], + "bbox": [ + 174, + 532, + 823, + 709 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 ROBUSTNESS TO INPUT NOISE ", + "text_level": 1, + "bbox": [ + 176, + 799, + 426, + 814 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Input examples for novel classes in FSC may have been collected under conditions that do not match those observed at training time. For example, labeled support images in a medical diagnosis application may come from a different hospital than the training set. To mimic a simplified version of this scenario, we investigate robustness to input noise. We used the Imagecorruptions package (Michaelis et al., 2019) to apply Gaussian noise, impulse noise, and defocus blur to both the support set and query sets of episodes at test time and evaluated both accuracy and calibration. We used corruption severity of 5 (severe) and evaluated across 1,000 randomly generated tasks on the three datasets involving natural images. The robustness results for Gaussian noise are shown in Figure 2. Full quantitative results tables for each noise type may be found in Section J. We find that in general Bayesian approaches tend to be robust due to their ability to marginalize over hypotheses consistent with the support labels. Our approach is one of the top performing methods across all settings. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/010f14a398ac09e9735aa1817cc4ec54d2d9cb8c053ad8077145d91ee24e15af.jpg", + "image_caption": [ + "Figure 2: Accuracy (↑) and Brier Score (↓) when corrupting both support and query with Gaussian noise on 5-way 5-shot tasks. Quantitative results may be found in Appendix J. " + ], + "image_footnote": [], + "bbox": [ + 174, + 170, + 820, + 297 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.4 OUT-OF-EPISODE DETECTION ", + "text_level": 1, + "bbox": [ + 176, + 381, + 423, + 395 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we measure performance on out-of-episode detection, another application in which uncertainty quantification is important. In this experiment, we used 5-way, 5-shot support sets at test time but incorporated out-of-episode examples into the query set. Each episode had 150 query examples: 15 from each of 5 randomly chosen in-episode classes and 15 from each of 5 randomly chosen outof-episode classes. We then computed the AUROC of binary outlier detection using the negative of the maximum logit as the score. Intuitively, if none of the support classes assign a high logit to the example, it can be classified as an outlier. The results are shown in Figure 3. Our approach generally performs the best across the datasets. ", + "bbox": [ + 173, + 406, + 825, + 518 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/790f796e9c00d9b88a81ad2a638b51abb81455e1bffbd2051471191cde9d6417.jpg", + "image_caption": [ + "Figure 3: Average AUROC (↑) for out-of-episode detection. The AUC is computed separately for each episode and averaged across 1,000 episodes. Bars indicate a $9 5 \\%$ bootstrapped confidence interval. " + ], + "image_footnote": [], + "bbox": [ + 174, + 534, + 818, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 739, + 318, + 756 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we have proposed a Bayesian few-shot classification approach based on Gaussian processes. Our method replaces the ordinary softmax likelihood with a one-vs-each pairwise composite likelihood and applies Polya-Gamma augmentation to perform inference. This allows us to model ´ class logits directly as function values and efficiently marginalize over uncertainty in each few-shot episode. Modeling functions directly enables our approach to avoid the dependence on model size that posterior inference in weight-space based models inherently have. Our approach compares favorably to baseline FSC methods under a variety of dataset and shot configurations, including dataset transfer. We also demonstrate strong uncertainty quantification, robustness to input noise, and out-of-episode detection. We believe that Bayesian modeling is a powerful tool for handling uncertainty and hope that our work will lead to broader adoption of efficient Bayesian inference in the few-shot scenario. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Ryan Adams, Ethan Fetaya, Mike Mozer, Eleni Triantafillou, Kuan-Chieh Wang, and Max Welling for helpful discussions. JS also thanks SK T-Brain for supporting him on an internship that led to precursors of some ideas in this paper. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute (https://www.vectorinstitute. ai/partners). This project is supported by NSERC and the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (DoI/IBC) contract number D16PC00003. 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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", + "bbox": [ + 171, + 814, + 821, + 843 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Jaesik Yoon, Taesup Kim, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian model-agnostic meta-learning. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31, pp. 7332–7342. Curran Associates, Inc., 2018. ", + "bbox": [ + 173, + 858, + 825, + 915 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A DERIVATION OF POLYA ´ -GAMMA AUGMENTED LOGISTIC LIKELIHOOD ", + "text_level": 1, + "bbox": [ + 169, + 101, + 794, + 119 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we show the derivation for the augmented logistic likelihood presented in Section 3.1. First, recall the logistic likelihood: ", + "bbox": [ + 173, + 132, + 820, + 161 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ecc8b03f7b8451ab7e8790552f651198ab886508df5c740244a443338c3d2ae1.jpg", + "text": "$$\np ( \\mathbf { y } \\vert \\psi ) = \\prod _ { i = 1 } ^ { N } \\sigma ( \\psi _ { i } ) ^ { y _ { i } } ( 1 - \\sigma ( \\psi _ { i } ) ) ^ { 1 - y _ { i } } = \\prod _ { i = 1 } ^ { N } \\frac { ( e ^ { \\psi _ { i } } ) ^ { y _ { i } } } { 1 + e ^ { \\psi _ { i } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 167, + 679, + 212 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\sigma ( \\cdot )$ is the logistic sigmoid function. We have a Gaussian prior $p ( \\psi ) = \\mathcal { N } ( \\psi | \\mu , \\Sigma )$ and introduce Polya-Gamma auxiliary random variables ´ $\\omega$ to the likelihood such that the original model is recovered when $\\omega$ is marginalized out: $\\begin{array} { r } { p ( \\mathbf { y } \\vert \\psi ) = \\int p ( \\omega ) p ( \\mathbf { y } \\vert \\psi , \\omega ) d \\omega } \\end{array}$ . ", + "bbox": [ + 173, + 218, + 825, + 262 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The Polya-Gamma distribution ´ $\\omega \\sim \\mathrm { P G } ( b , c )$ can be written as an infinite convolution of Gamma distributions: ", + "bbox": [ + 174, + 266, + 823, + 294 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c25d228f888d309f0dd22bacc4df78dcb07dde7a2763fd22dce476aae8709188.jpg", + "text": "$$\n\\omega \\stackrel { \\cal D } { = } \\frac { 1 } { 2 \\pi ^ { 2 } } \\sum _ { k = 1 } ^ { \\infty } \\frac { \\mathrm { G a } ( b , 1 ) } { ( k - 1 / 2 ) ^ { 2 } + c ^ { 2 } / ( 4 \\pi ^ { 2 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 290, + 627, + 332 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The following integral identity holds for $b > 0$ : ", + "bbox": [ + 173, + 334, + 485, + 349 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/751bb8903fab63363f72cb8f3b88a25b1b8ef90c4780b0e188539197852d8140.jpg", + "text": "$$\n\\frac { ( e ^ { \\psi } ) ^ { a } } { ( 1 + e ^ { \\psi } ) ^ { b } } = 2 ^ { - b } e ^ { \\kappa \\psi } \\int _ { 0 } ^ { \\infty } e ^ { - \\omega \\psi ^ { 2 } / 2 } p ( \\omega ) d \\omega ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 356, + 645, + 391 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\kappa = a - b / 2$ and $\\omega \\sim \\mathrm { P G } ( b , 0 )$ . Specifically, when $a = y$ and $b = 1$ , we recover an individual term of the logistic likelihood (16): ", + "bbox": [ + 174, + 397, + 825, + 426 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7114a3c59361e7cf3b60fb56a5fa3428eba5c0cc21d979747824f0dbcb35bb4e.jpg", + "text": "$$\np ( y | \\psi ) = \\frac { ( e ^ { \\psi } ) ^ { y } } { 1 + e ^ { \\psi } } = \\frac { 1 } { 2 } e ^ { \\kappa \\psi } \\int _ { 0 } ^ { \\infty } e ^ { - \\omega \\psi ^ { 2 } / 2 } p ( \\omega ) d \\omega ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 431, + 663, + 468 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\kappa = y - 1 / 2$ and $\\omega \\sim P G ( 1 , 0 )$ . Conditioned on $\\omega$ , the batch likelihood is proportional to a diagonal Gaussian: ", + "bbox": [ + 174, + 473, + 825, + 502 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/33cfcd5494b2fcc69387f91d3c48e387ccc8a5b1ba064b7ab71707b50ef3384e.jpg", + "text": "$$\np ( \\mathbf { y } \\vert \\psi , \\boldsymbol { \\omega } ) \\propto \\prod _ { i = 1 } ^ { N } e ^ { - \\omega _ { i } \\psi _ { i } ^ { 2 } / 2 } e ^ { \\kappa _ { i } \\psi _ { i } } \\propto \\mathcal { N } ( \\Omega ^ { - 1 } \\kappa \\vert \\psi , \\Omega ^ { - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 508, + 683, + 553 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\kappa _ { i } = y _ { i } - 1 / 2$ and $\\pmb { \\Omega } = \\mathrm { d i a g } ( \\omega )$ . The conditional distribution over $\\psi$ given $\\mathbf { y }$ and $\\omega$ is now tractable: ", + "bbox": [ + 173, + 558, + 821, + 585 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7c2a6f854cc96a993c1633c3b6c4e41f35c8957a6a2705c550ff414ff49146b1.jpg", + "text": "$$\np ( \\psi | \\mathbf { y } , \\boldsymbol { \\omega } ) \\propto p ( \\mathbf { y } | \\psi , \\boldsymbol { \\omega } ) p ( \\psi ) \\propto \\mathcal { N } ( \\psi | \\tilde { \\boldsymbol { \\Sigma } } ( \\Sigma ^ { - 1 } \\mu + \\kappa ) , \\tilde { \\boldsymbol { \\Sigma } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 583, + 691, + 603 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\tilde { \\Sigma } = ( \\Sigma ^ { - 1 } + \\Omega ) ^ { - 1 }$ . ", + "bbox": [ + 174, + 606, + 354, + 622 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B EFFICIENT GIBBS SAMPLING ", + "text_level": 1, + "bbox": [ + 176, + 641, + 450, + 659 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The Gibbs conditional distribution over f is given by: ", + "bbox": [ + 173, + 672, + 526, + 689 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6dcedde296e1a365b163b1b4161c492f2ee582b86b8fe8023022467adb0a2453.jpg", + "text": "$$\np ( \\mathbf { f } | \\mathbf { X } , \\mathbf { y } , \\omega ) = \\mathcal { N } ( \\mathbf { f } | \\tilde { \\boldsymbol { \\Sigma } } ( \\mathbf { K } ^ { - 1 } \\boldsymbol { \\mu } + \\mathbf { A } ^ { \\top } \\boldsymbol { \\kappa } ) , \\tilde { \\boldsymbol { \\Sigma } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 694, + 648, + 714 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\tilde { \\mathbf { \\Sigma } } = ( \\mathbf { K } ^ { - 1 } { + } \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } ) ^ { - 1 }$ . Naively sampling from this distribution requires $\\mathcal { O } ( C ^ { 3 } N ^ { 3 } )$ computation since $\\tilde { \\Sigma }$ is a $C N \\times C N$ matrix. Here we describe a method for sampling from this distribution that requires $\\mathcal { O } ( C N ^ { 3 } )$ computation instead. ", + "bbox": [ + 174, + 722, + 825, + 767 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "First, we note that (22) can be interpreted as the conditional distribution $p ( \\mathbf { f } | \\mathbf { z } = \\Omega ^ { - 1 } \\kappa )$ ) resulting from the following marginal distribution $p ( \\mathbf { f } )$ and conditional $p ( \\mathbf { z } | \\mathbf { f } )$ : ", + "bbox": [ + 173, + 773, + 820, + 803 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/afe9a9c3e49d452e5f0464fb231f21fcf3ec521769d5ca1880d71435aeff9dc2.jpg", + "text": "$$\n\\begin{array} { r } { p ( \\mathbf { f } ) = \\mathcal { N } ( \\mathbf { f } \\vert \\boldsymbol { \\mu } , \\mathbf { K } ) \\qquad } \\\\ { p ( \\mathbf { z } \\vert \\mathbf { f } ) = \\mathcal { N } ( \\mathbf { z } \\vert \\mathbf { A } \\mathbf { f } , \\boldsymbol { \\Omega } ^ { - 1 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 808, + 584, + 848 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where we have made implicit the dependence on $\\mathbf { X } , \\mathbf { Y }$ , and $\\omega$ for brevity of notation. Equivalently, the distribution over f and $\\mathbf { z }$ can be represented by the partitioned Gaussian ", + "bbox": [ + 171, + 852, + 823, + 882 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4b766c8fa0ccafb57890d9231e9f45107568efcf9e61841333ad0f09a041cd17.jpg", + "text": "$$\n\\left[ \\begin{array} { c } { \\mathbf { f } } \\\\ { \\mathbf { z } } \\end{array} \\right] \\sim \\mathcal { N } \\left( \\left[ \\begin{array} { c } { \\ \\mu } \\\\ { \\mathbf { A } \\mu } \\end{array} \\right] , \\left[ \\begin{array} { c c } { \\mathbf { K } } & { \\mathbf { K } \\mathbf { A } ^ { \\top } } \\\\ { \\mathbf { A } \\mathbf { K } } & { \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\boldsymbol { \\Omega } ^ { - 1 } } \\end{array} \\right] \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 886, + 684, + 922 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The conditional distribution $p ( \\mathbf { f } | \\mathbf { z } )$ is given as: ", + "bbox": [ + 174, + 103, + 480, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/027afe56eba0e9f3afd9ca3d4a0c6e6258395359fc410f661b0909b58e9f0bb5.jpg", + "text": "$$\np ( \\mathbf { f } \\vert \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { f } \\vert \\tilde { \\boldsymbol { \\Sigma } } ( \\mathbf { K } ^ { - 1 } \\boldsymbol { \\mu } + \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { z } ) , \\tilde { \\boldsymbol { \\Sigma } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 122, + 632, + 141 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\tilde { \\boldsymbol { \\Sigma } } = ( \\mathbf { K } ^ { - 1 } + \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } ) ^ { - 1 }$ . Note that $p ( \\mathbf { f } | \\mathbf { z } = \\Omega ^ { - 1 } \\boldsymbol { \\kappa } )$ recovers our desired Gibbs conditional distribution from (22). ", + "bbox": [ + 176, + 145, + 823, + 175 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "An efficient approach to conditional Gaussian sampling is due to Hoffman & Ribak (1991) and described in greater clarity by Doucet (2010). The procedure is as follows: ", + "bbox": [ + 173, + 181, + 825, + 210 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. Sample $\\mathbf { f } _ { 0 } \\sim p ( \\mathbf { f } )$ and $\\mathbf { z } _ { 0 } \\sim p ( \\mathbf { z } | \\mathbf { f } )$ . \n2. Return $\\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { A } ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } ( \\Omega ^ { - 1 } \\pmb { \\kappa } - \\mathbf { z } _ { 0 } )$ as the sample from $p ( \\mathbf { f } | \\mathbf { z } )$ ", + "bbox": [ + 205, + 220, + 785, + 256 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\mathbf { K }$ is block diagonal and thus sampling from $p ( \\mathbf { f } )$ requires $\\mathcal { O } ( C N ^ { 3 } )$ time. Af can be computed in $\\mathcal { O } ( C N )$ time, since each entry is the difference between $f _ { i } ^ { y _ { i } }$ and $f _ { i } ^ { c }$ for some $i$ and $c$ . Overall, step 1 requires $\\mathcal { O } ( C N ^ { 3 } )$ time. ", + "bbox": [ + 174, + 265, + 823, + 308 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We now show how to compute ¯f from step 2 in $\\mathcal { O } ( C N ^ { 3 } )$ time. We first expand $( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 }$ ", + "bbox": [ + 173, + 313, + 821, + 329 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1704fba7b1f583265efadabdc3cd9319961691b2bca4d2e712dc3e05867fbb97.jpg", + "text": "$$\n( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } = \\Omega - \\Omega \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ) ^ { - 1 } \\mathbf { A } ^ { \\top } \\Omega\n$$", + "text_format": "latex", + "bbox": [ + 300, + 333, + 696, + 352 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We substitute into the expression for $\\bar { \\bf f }$ : ", + "bbox": [ + 174, + 356, + 429, + 371 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5d6966fd91462eb91a16eeed16f9d8d0e01a2ea8531cb7a51ede650623641d19.jpg", + "text": "$$\n\\begin{array} { r l } & { \\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { A } ^ { \\top } ( \\Omega - \\Omega \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ) ^ { - 1 } \\mathbf { A } ^ { \\top } \\Omega ) ( \\Omega ^ { - 1 } \\kappa - \\mathbf { z } _ { 0 } ) } \\\\ & { \\quad = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { A } ^ { \\top } \\Omega ( \\Omega ^ { - 1 } \\kappa - \\mathbf { z } _ { 0 } ) - \\mathbf { K } \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ) ^ { - 1 } \\mathbf { A } ^ { \\top } \\Omega ( \\Omega ^ { - 1 } \\kappa - \\mathbf { z } _ { 0 } ) } \\\\ & { \\quad = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { v } - \\mathbf { K } \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { A } ^ { \\top } \\Omega \\mathbf { A } ) ^ { - 1 } \\mathbf { v } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 373, + 766, + 435 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where we have defined $\\mathbf { v } \\triangleq \\mathbf { A } ^ { \\top } \\pmb { \\Omega } ( \\Omega ^ { - 1 } \\pmb { \\kappa } - \\mathbf { z } _ { 0 } )$ ", + "bbox": [ + 176, + 439, + 493, + 455 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now let $\\mathbf { d } \\triangleq ( d _ { 1 } ^ { 1 } , \\hdots , d _ { N } ^ { 1 } , d _ { 1 } ^ { 2 } , \\hdots , d _ { N } ^ { 2 } , \\hdots , d _ { 1 } ^ { C } , \\hdots , d _ { N } ^ { C } ) ^ { \\intercal }$ , where $\\begin{array} { r } { d _ { i } ^ { c } = Y _ { i c } \\sum _ { c ^ { \\prime } } \\omega _ { i } ^ { c ^ { \\prime } } } \\end{array}$ . Define $\\mathbf { Y } ^ { \\dagger }$ to be the $C N \\times N$ matrix produced by vertically stacking $\\mathrm { d i a g } ( Y _ { \\cdot c } )$ , and let $\\mathbf { W } ^ { \\dagger }$ be the $C N \\times N$ matrix produced by vertically stacking $\\mathrm { d i a g } ( ( \\omega _ { 1 } ^ { c } , \\dots , \\omega _ { N } ^ { c } ) ^ { \\top } ,$ ). $\\mathbf { A } ^ { \\top } \\mathbf { \\widehat {Omega } } \\mathbf { \\widehat { \\Omega } } \\mathbf { A }$ may then be written as follows: ", + "bbox": [ + 174, + 462, + 825, + 510 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/bc7e2ba7a37cd4c59cb8cf9efa37617deaabd352379951c2c1af81934a5a8f6e.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\boldsymbol { \\Omega } \\mathbf { A } = \\mathbf { D } - \\mathbf { S } \\mathbf { P S } ^ { \\mathrm { ~ I ~ } } , \\mathrm { w i } } \\\\ & { } & { \\mathbf { D } = \\boldsymbol { \\Omega } + \\mathrm { d i a g } ( \\mathbf { d } ) , } \\\\ & { } & { \\mathbf { S } = \\left[ \\begin{array} { l l } { \\mathbf { Y } ^ { \\dagger } } & { \\mathbf { W } ^ { \\dagger } } \\end{array} \\right] , } \\\\ & { } & { \\mathbf { P } = \\left[ \\begin{array} { l l } { \\mathbf { 0 } _ { N } } & { \\mathbf { I } _ { N } } \\\\ { \\mathbf { I } _ { N } } & { \\mathbf { 0 } _ { N } } \\end{array} \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 426, + 516, + 576, + 606 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Substituting (31) into (30): ", + "bbox": [ + 174, + 607, + 351, + 622 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now we expand ", + "bbox": [ + 174, + 648, + 455, + 665 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1e3d88b3a4ff2fa8a08b6a72ac8ee7bf36974f53aa1cbe3a77e76cccf8371a4d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { v } - \\mathbf { K } \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } \\mathbf { v } . } \\\\ & { . . . } \\\\ & { . . . + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } : } \\\\ & { \\mathbf { K } ^ { - 1 } + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } = \\mathbf { E } - \\mathbf { E } \\mathbf { S } ( \\mathbf { S } ^ { \\top } \\mathbf { E } \\mathbf { S } - \\mathbf { P } ^ { - 1 } ) ^ { - 1 } \\mathbf { S } ^ { \\top } \\mathbf { F } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 623, + 700, + 689 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\mathbf { E } = ( \\mathbf { K } ^ { - 1 } + \\mathbf { D } ) ^ { - 1 } = \\mathbf { K } ( \\mathbf { K } + \\mathbf { D } ^ { - 1 } ) ^ { - 1 } \\mathbf { D } ^ { - 1 }$ is a block-diagonal matrix that can be computed in $\\mathcal { O } ( C N ^ { 3 } )$ time, since $\\mathbf { D }$ is diagonal and $\\mathbf { K }$ is block diagonal. Now, substituting (36) back into (35), ", + "bbox": [ + 173, + 690, + 826, + 734 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9f405dc035fe81bf4c83ec960f87831f0014141fcadf81b58975ca79d4362338.jpg", + "text": "$$\n\\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { v } - \\mathbf { K } \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } \\mathbf { E } \\mathbf { v } + \\mathbf { K } \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } \\mathbf { E } \\mathbf { S } ( \\mathbf { S } ^ { \\top } \\mathbf { E } \\mathbf { S } - \\mathbf { P } ^ { - 1 } ) ^ { - 1 } \\mathbf { S } ^ { \\top } \\mathbf { E } \\mathbf { v } .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 734, + 740, + 755 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that $( \\mathbf { S } ^ { \\top } \\mathbf { E } \\mathbf { S } - \\mathbf { P } ^ { - 1 } ) ^ { - 1 }$ is a $2 N \\times 2 N$ matrix and thus can be inverted in $\\mathcal { O } ( N ^ { 3 } )$ time. The overall complexity is therefore $\\mathcal { O } ( C N ^ { 3 } )$ . ", + "bbox": [ + 173, + 758, + 830, + 790 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C MARGINAL LIKELIHOOD AND PREDICTIVE LIKELIHOOD OBJECTIVES ", + "text_level": 1, + "bbox": [ + 171, + 808, + 789, + 825 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Marginal Likelihood (ML). The log marginal likelihood can be written as follows: ", + "bbox": [ + 173, + 838, + 723, + 854 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/26696e6812268754cb66c9543f4312d10ea8983ecc14f7f0a28a7625450f2153.jpg", + "text": "$$\n\\begin{array} { c } { { { \\cal L } _ { \\mathrm { M L } } ( \\theta ; { \\bf X } , { \\bf Y } ) \\triangleq \\log p _ { \\theta } ( { \\bf Y } | { \\bf X } ) = \\log \\displaystyle \\int p ( \\omega ) p _ { \\theta } ( { \\bf Y } | \\omega , { \\bf X } ) d \\omega } } \\\\ { { \\displaystyle = \\log \\displaystyle \\int p ( \\omega ) \\int \\mathcal { L } ( { \\bf f } | { \\bf Y } , \\omega ) p _ { \\theta } ( { \\bf f } | { \\bf X } ) d { \\bf f } d \\omega } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 857, + 745, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The gradient of the log marginal likelihood can be estimated by posterior samples $\\boldsymbol { \\omega } \\sim p _ { \\boldsymbol { \\theta } } ( \\boldsymbol { \\omega } | { \\bf X } , { \\bf Y } )$ . In practice, we use a stochastic training objective based on samples of $\\omega$ from Gibbs chains. We use Fisher’s identity (Douc et al., 2014) to derive the following gradient estimator: ", + "bbox": [ + 173, + 103, + 826, + 146 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c010b0f3619da6af4f4f833fc08d9d1a9be0c044c1a2d97902520caa960c1762.jpg", + "text": "$$\n\\nabla _ { \\theta } L _ { \\mathrm { M L } } = \\int p _ { \\theta } ( \\boldsymbol { \\omega } | \\mathbf { X } , \\mathbf { Y } ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { Y } | \\boldsymbol { \\omega } , \\mathbf { X } ) d \\boldsymbol { \\omega } \\approx \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { Y } | \\mathbf { X } , \\boldsymbol { \\omega } ^ { ( m ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 200, + 148, + 767, + 193 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\omega ^ { ( 1 ) } , \\ldots , \\omega ^ { ( M ) }$ are samples from the posterior Gibbs chain. As suggested by Patacchiola et al. (2020), who applied GPs to FSC via least-squares classification, we merge the support and query sets during learning to take full advantage of the available data within each episode. ", + "bbox": [ + 174, + 195, + 825, + 239 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Predictive Likelihood (PL). The log predictive likelihood for a query example $\\mathbf { x } _ { * }$ is: ", + "bbox": [ + 176, + 244, + 735, + 261 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2ab47f136831924e8442dfd500f339443a9d38d558ded9003cb2ffa5a0c95410.jpg", + "text": "$$\nL _ { \\mathrm { P L } } ( \\theta ; \\mathbf { X } , \\mathbf { Y } , \\mathbf { x } _ { * } , \\mathbf { y } _ { * } ) \\triangleq \\log p \\varrho ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } ) = \\log \\int p ( \\omega ) p \\varrho ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ) d \\omega .\n$$", + "text_format": "latex", + "bbox": [ + 205, + 262, + 764, + 295 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use an approximate gradient estimator again based on posterior samples of $\\omega$ : ", + "bbox": [ + 173, + 296, + 709, + 311 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a8d56b1d20bea67114cc6d9f8b49907b04e92eec68658e33ecbce2725866d9e3.jpg", + "text": "$$\n\\nabla _ { \\theta } L _ { \\mathrm { P L } } \\approx \\int p _ { \\theta } ( \\omega | \\mathbf { X } , \\mathbf { Y } ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } ) d \\omega \\approx \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ^ { ( m ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 314, + 818, + 358 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We note that this is not an unbiased estimator of the gradient, but find it works well in practice. ", + "bbox": [ + 173, + 367, + 794, + 383 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D LEARNING ALGORITHM ", + "text_level": 1, + "bbox": [ + 176, + 402, + 411, + 419 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our learning algorithm for both marginal and predictive likelihood is summarized in Algorithm 1. ", + "bbox": [ + 173, + 433, + 812, + 449 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Algorithm 1 One-vs-Each Polya-Gamma GP Learning ´ ", + "text_level": 1, + "bbox": [ + 174, + 460, + 537, + 477 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/da32a9e5f16d5fbb17e4c47614b931727a8986dfb8d1b2ebc6b16aa87d4d39b1.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input: Objective L ∈ {LML,LPL}, Task distribution T, number of parallel Gibbs chains M, number of steps T,learning rate n. Initialize hyperparameters 0 randomly.
repeat Sample S= (X,Y),Q= (X*,Y*)~ T
if L= LML then X←XUX*,Y←YUY*
end if
A ← OVE-MATRIX(Y) for m = 1 to M do
~ pe(f|X)
fort=1 toTdo
（m）~PG(1b（m))
f（m） ~pe(f|x,Y,ω(m）)
end for end for
if L= LML then
0←0+∑m-1 Velogpe(Y|x,(m)）
else 0←0+∑m=1∑,∀θlogpe(y*j｜x*j,S,ω(m）)
end if until convergence
", + "bbox": [ + 174, + 479, + 831, + 823 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E POSTERIOR PREDICTIVE DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 173, + 847, + 542, + 864 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The posterior predictive distribution for a query example $\\mathbf { x } _ { * }$ conditioned on $\\omega$ is: ", + "bbox": [ + 173, + 877, + 705, + 893 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/ec2400d9384de6297c9651950889dd52777f17ffa860bd7083013a73b47b86e1.jpg", + "text": "$$\np ( { \\bf y } _ { * } | { \\bf x } _ { * } , { \\bf X } , { \\bf Y } , \\omega ) = \\int p ( { \\bf y } _ { * } | { \\bf f } _ { * } ) p ( { \\bf f } _ { * } | { \\bf x } _ { * } , { \\bf X } , { \\bf Y } , \\omega ) d { \\bf f } _ { * } ,\n$$", + "text_format": "latex", + "bbox": [ + 313, + 896, + 683, + 929 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\mathbf { f } _ { * }$ are the query example’s logits. The predictive distribution over $\\mathbf { f } _ { * }$ can be obtained by noting that $\\psi$ and the query logits are jointly Gaussian: ", + "bbox": [ + 173, + 103, + 821, + 132 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3fa752aebaee094bd22e3da4c08d9265a78685acb064479448d14ee69fded031.jpg", + "text": "$$\n\\left[ \\begin{array} { c } { \\psi } \\\\ { \\mathbf { f } _ { * } } \\end{array} \\right] \\sim { \\mathcal { N } } \\left( 0 , \\left[ \\begin{array} { c c } { \\mathbf { A K A ^ { \\top } } + \\boldsymbol { \\Omega } ^ { - 1 } } & { \\mathbf { A K } _ { * } } \\\\ { \\left( \\mathbf { A K } _ { * } \\right) ^ { \\top } } & { \\mathbf { K } _ { * * } } \\end{array} \\right] \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 133, + 663, + 169 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\mathbf { K } _ { * }$ is the $N C \\times C$ block diagonal matrix with blocks $K _ { \\theta } ( \\mathbf { X } , \\mathbf { x } _ { * } )$ and $\\mathbf { K } _ { * * }$ is the $C \\times C$ diagonal matrix with diagonal entries $k _ { \\theta } ( \\mathbf { x } _ { * } , \\mathbf { x } _ { * } )$ . The predictive distribution becomes: ", + "bbox": [ + 174, + 170, + 830, + 199 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$p ( \\mathbf { f } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ) = \\mathcal { N } ( \\mathbf { f } _ { * } | \\mu _ { * } , \\Sigma _ { * } )$ , where ", + "bbox": [ + 333, + 202, + 624, + 218 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/72bda9075e73e9b7e79d735ecd070f9df1c9880468310c7179fb1bd7cf108af3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\pmb { \\jmath } ^ { \\top } ( \\mathbf { A } \\mathbf { x } ) ^ { \\top } \\mathbf { \\Phi } ^ { \\mathbf { a } \\mathbf { x } } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\Omega ^ { - 1 } \\mathbf { \\Lambda } ^ { \\mathrm { a n d } } } \\\\ & { \\pmb { \\mu } _ { * } = ( \\mathbf { A } \\mathbf { K } _ { * } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\Omega ^ { - 1 } \\mathbf { \\Lambda } ^ { \\mathrm { a n d } } } \\\\ & { \\pmb { \\Sigma } _ { * } = \\mathbf { K } _ { * * } - ( \\mathbf { A } \\mathbf { K } _ { * } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\mathbf { A } \\mathbf { K } _ { * } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 212, + 665, + 260 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "With $p ( \\mathbf { f } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega )$ in hand, the integral in (42) can easily be computed numerically for each class $c$ by forming the corresponding OVE linear transformation matrix $\\mathbf { A } ^ { c }$ and then performing 1D Gaussian-Hermite quadrature on each dimension of $\\mathcal { N } ( \\psi _ { * } ^ { c } | \\mathbf { A } ^ { c } \\pmb { \\mu } ^ { * } , \\mathbf { A } ^ { c } \\pmb { \\Sigma } _ { * } \\mathbf { A } ^ { c \\top } )$ . ", + "bbox": [ + 174, + 262, + 820, + 305 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F DETAILED COMPARISON OF LIKELIHOODS ", + "text_level": 1, + "bbox": [ + 173, + 323, + 562, + 340 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section we seek to better understand the behaviors of the softmax, OVE, logistic softmax, and Gaussian likelihoods for classification. For convenience, we summarize the forms of these likelihoods in Table 3. ", + "bbox": [ + 174, + 353, + 823, + 396 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/182a36a68b14416ccd8a15ef39c0fd4a35a36a49885023dde3658ac490d4b56a.jpg", + "table_caption": [ + "Table 3: Likelihoods used in Section F. " + ], + "table_footnote": [], + "table_body": "
LikelihoodC(f|y =c)
Softmaxexp(fe) ∑c exp(fe)
GaussianIIN(2·1[c' =c]-1|μ= fe,σ² =1) c'
Logistic Softmax (LSM)g(fc) ∑eσ(fe)
One-vs-Each (OVE)IIσ(fe- fe) c≠c
", + "bbox": [ + 258, + 430, + 738, + 612 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F.1 HISTOGRAM OF CONFIDENCES ", + "text_level": 1, + "bbox": [ + 174, + 636, + 428, + 650 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We sampled logits from $f _ { c } \\sim \\mathcal { N } ( 0 , 1 )$ and plotted a histogram and kernel density estimate of the maximum output probability m $\\arg ( y = c | \\mathbf { f } )$ for each of the likelihoods shown in Table 3, where $C = 5$ . The results are shown in Figure 4. Logistic softmax is a priori underconfident: it puts little probability mass on confidence above 0.4. This may be due to the use of the sigmoid function which squashes large values of $f$ . Gaussian likelihood and OVE are a priori overconfident in that they put a large amount of probability mass on confident outputs. Note that this is not a complete explanation, because GP hyperparameters such as the prior mean or Gaussian likelihood variance may be able to compensate for these imperfections to some degree. Indeed, we found it helpful to learn a constant mean for the logistic softmax likelihood, as mentioned in Section G.2. ", + "bbox": [ + 173, + 660, + 826, + 787 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F.2 LIKELIHOOD VISUALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 803, + 419, + 818 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In order to visualize the various likelihoods under consideration, we consider a trivial classification task with a single observed example. We assume that there are three classes $C = 3$ ) and the single example belongs to the first class $y = 1 \\mathrm { \\cdot }$ ). We place the following prior on $\\mathbf { f } = \\left( f _ { 1 } , f _ { 2 } , f _ { 3 } \\right) ^ { \\top }$ : ", + "bbox": [ + 173, + 829, + 825, + 875 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8257059dafd89d33d849e0c209b7df0705d90cf922c4a63beb4c1d0767744a8e.jpg", + "text": "$$\n\\begin{array} { r } { p ( \\mathbf { f } ) = \\mathcal { N } \\left( \\mathbf { f } \\bigg | \\mu = \\left[ \\begin{array} { l } { 0 } \\\\ { 0 } \\\\ { 0 } \\end{array} \\right] , \\Sigma = \\left[ \\begin{array} { l l l } { 1 } & { 0 } & { 0 } \\\\ { 0 } & { 1 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right] \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 877, + 668, + 920 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/90f74595999ffa71f93112e5993b98c03df1ad7c6c88dcbaea08535a25ad34fe.jpg", + "image_caption": [ + "Figure 4: Histogram and kernel density estimate of confidence for randomly generated function samples $f _ { c } \\sim \\mathcal { N } ( 0 , 1 )$ . Normalized output probabilities were computed for $C = 5$ and a histogram of $\\operatorname* { m a x } _ { c } p ( y = c | \\mathbf { f } )$ was computed for 50,000 randomly generated simulations. " + ], + "image_footnote": [], + "bbox": [ + 276, + 106, + 718, + 310 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In other words, the prior for $f _ { 1 }$ and $f _ { 2 }$ is a standard normal and $f _ { 3 }$ is clamped at zero (for ease of visualization). The likelihoods are plotted in Figure 5 and the corresponding posteriors are plotted in Figure 6. ", + "bbox": [ + 173, + 421, + 826, + 464 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/c475772f592b02bc778a43d7039536becbe66815b4c507f1cb35ac1a177e623e.jpg", + "image_caption": [ + "Figure 5: Plot of $\\mathcal { L } ( \\mathbf { f } \\mid y = 1 )$ , where $f _ { 3 }$ is clamped to 0. The Gaussian likelihood penalizes configurations far away from $( f _ { 1 } , f _ { 2 } ) = ( 1 , - 1 ) .$ . Logistic softmax is much flatter compared to softmax and has visibly different contours. One-vs-Each is visually similar to the softmax but penalizes $( f _ { 1 } , f _ { 2 } )$ near the origin slightly more. " + ], + "image_footnote": [], + "bbox": [ + 186, + 488, + 813, + 606 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/c80f9c03f1647cfa1a69f4086877bb5121d476ef745a0037b2f946a73d185a49.jpg", + "image_caption": [ + "Figure 6: Plot of posterior $p ( \\mathbf { f } \\mid y = 1 )$ , where $f _ { 3 }$ is clamped to 0. The mode of each posterior distribution is similar, but each differs slightly in shape. Gaussian is more peaked about its mode, while logistic softmax is more spread out. One-vs-Each is similar to softmax, but is slightly more elliptical. " + ], + "image_footnote": [], + "bbox": [ + 187, + 717, + 813, + 834 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We also conducted experiments on a 2D version of the Iris dataset (Fisher, 1936), which contains 150 examples across 3 classes. The first two features of the dataset were retained (sepal length and width). We used a zero-mean GP prior and an RBF kernel $k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) = \\exp \\left( - \\frac { 1 } { 2 } d ( \\mathbf { x } , \\mathbf { \\bar { x } } ^ { \\prime } ) ^ { 2 } \\right)$ , where $d ( \\cdot , \\cdot )$ is Euclidean distance. We considered training set sizes with 1, 2, 3, 4, 5, 10, 15, 20, 25, and 30 examples per class. For each training set size, we performed GP inference on 200 randomly generated train/test splits and compared the predictions across Gaussian, logistic softmax, and onevs-each likelihoods. ", + "bbox": [ + 173, + 131, + 825, + 246 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Predictions at a test point $\\mathbf { x } _ { * }$ were made by applying the (normalized) likelihood to the posterior predictive mean $\\bar { \\mathbf { f } } _ { * }$ . The predictive probabilities for each likelihood is shown in Figure 7 for a randomly generated train/test split with 30 examples per class. Test predictive accuracy, Brier score, expected calibration error, and evidence lower bound (ELBO) results across various training set sizes are shown in Figure 8. ", + "bbox": [ + 173, + 252, + 826, + 323 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The ELBO is computed by treating each likelihood’s posterior $q ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } )$ as an approximation to the softmax posterior $p ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } )$ . ", + "bbox": [ + 173, + 329, + 823, + 358 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/af038d21e2bb247c90c0ff49e91dd4866b064dcc05893eb092d1b9cd39429987.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { E L B O } ( q ) = \\mathbb { E } _ { q } [ \\log p ( \\mathbf { f } | \\mathbf { X } ) ] + \\mathbb { E } _ { q } [ \\log p ( \\mathbf { Y } | \\mathbf { f } ) ] - \\mathbb { E } _ { q } [ \\log q ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } ) ] } \\\\ & { \\quad \\quad \\quad = \\log p ( \\mathbf { x } ) - \\mathrm { K L } ( q ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } ) | | p ( \\mathbf { f } | \\mathbf { X } , \\mathbf { Y } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 367, + 722, + 405 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Even though direct computation of the softmax posterior $p ( \\mathbf { f } | \\mathbf { X } , \\mathbf { y } )$ is intractable, computing the ELBO is tractable. A larger ELBO indicates a lower KL divergence to the softmax posterior. ", + "bbox": [ + 174, + 412, + 823, + 441 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "One-vs-Each performs well for accuracy, Brier score, and ELBO across the training set sizes. Gaussian performs best on expected calibration error through 15 examples per class, beyond which onevs-each is better. ", + "bbox": [ + 174, + 448, + 825, + 491 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/c313125f2f323d8dd5fc6d7b997993186867c8fd3f117136626d8758be027a88.jpg", + "image_caption": [ + "Figure 7: Training points (colored points) and maximum predictive probability for various likelihoods on the Iris dataset. The Gaussian likelihood produces more warped decision boundaries than the others. Logistic softmax tends to produce lower confidence predictions, while one-vs-each produces larger regions of greater confidence than the others. " + ], + "image_footnote": [], + "bbox": [ + 207, + 506, + 790, + 641 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/b1ac3ed75011a27c35369b9894f46e6eb1989c84e9f1d028b3593dfc778ba366.jpg", + "image_caption": [ + "Figure 8: Comparison across likelihoods in terms of test predictive accuracy, Brier score, expected calibration error (computed with 10 bins), and ELBO. Results are averaged over 200 randomly generated splits for each training set size (1, 2, 3, 4, 5, 10, 15, 20, 25, and 30 examples per class). Error bars indicate $9 5 \\%$ confidence intervals. " + ], + "image_footnote": [], + "bbox": [ + 186, + 738, + 812, + 842 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G FEW-SHOT EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 514, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Here we provide more details about our experimental setup for our few-shot classification experiments, which are based on the protocol of (Patacchiola et al., 2020). ", + "bbox": [ + 174, + 133, + 821, + 162 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "G.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 180, + 290, + 195 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We used the four dataset scenarios described below. The first three are the same used by Chen et al. \n(2019) and the final was proposed by Patacchiola et al. (2020). \n• CUB. Caltech-UCSD Birds (CUB) (Wah et al., 2011) consists of 200 classes and 11,788 images. A split of 100 training, 50 validation, and 50 test classes was used (Hilliard et al., 2018; Chen et al., 2019). mini-Imagenet. The mini-Imagenet dataset (Vinyals et al., 2016) consists of 100 classes with 600 images per class. We used the split proposed by Ravi & Larochelle (2017), which has 64 classes for training, 16 for validation, and 20 for test. \nmini-Imagenet $\\scriptstyle \\mathbf { C U B }$ . This cross-domain transfer scenario takes the training split of mini-Imagenet and the validation & test splits of CUB. \nOmniglot EMNIST. We use the same setup as proposed by Patacchiola et al. (2020). Omniglot (Lake et al., 2011) consists of 1,623 classes, each with 20 examples, and is augmented by rotations of 90 degrees to create 6,492 classes, of which 4,114 are used for training. The EMNIST dataset (Cohen et al., 2017), consisting of 62 classes, is split into 31 training and 31 test classes. ", + "bbox": [ + 174, + 207, + 821, + 234 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 215, + 246, + 825, + 445 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "G.2 FEW-SHOT CLASSIFICATION BASELINES ", + "text_level": 1, + "bbox": [ + 174, + 463, + 498, + 477 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Here we explain the few-shot baselines in greater detail. ", + "bbox": [ + 174, + 489, + 540, + 503 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "• Feature Transfer (Chen et al., 2019) involves first training an off-line classifier on the training classes and then training a new classification layer on the episode. Baseline $^ { + + }$ (Chen et al., 2019) is similar to Feature Transfer except it uses a cosine distance module prior to the softmax during fine-tuning. Matching Networks (Vinyals et al., 2016) can be viewed as a soft form of $k$ -nearest neighbors that computes attention and sums over the support examples to form a predictive distribution over classes. \n• Prototypical Networks (Snell et al., 2017) computes class means (prototypes) and forms a predictive distribution based on Euclidean distance to the prototypes. It can be viewed as a Gaussian classifier operating in an embedding space. MAML (Finn et al., 2017) performs one or a few steps of gradient descent on the support set and then makes predictions on the query set, backpropagating through the gradient descent procedure. For this baseline, we simply quote the classification accuracy reported by (Patacchiola et al., 2020). RelationNet (Sung et al., 2018) rather than using a predefined distance metric as in Matching Networks or Prototypical Networks instead learns a deep distance metric as the output of a neural network that accepts as input the latent representation of both examples. It is trained to minimize squared error of output predictions. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) relies on least squares classification (Rifkin & Klautau, 2004) to maintain tractability of Gaussian process posterior inference. In DKT, a separate binary classification task is formed for each class in one-vs-rest fashion by treating labels in $\\{ - \\dot { 1 } , + 1 \\}$ as continuous targets. We include the results of DKT with the cosine kernel as implemented by Patacchiola et al. (2020), which is parameterized slightly differently from the version we used in (47): ", + "bbox": [ + 214, + 518, + 825, + 885 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/849d8a418b08d1777b6164b6ca5fc50c87c4cea0132ac4949d0b4156d2d97e23.jpg", + "text": "$$\nk _ { \\mathrm { d k t } } ^ { \\mathrm { c o s } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha , \\nu ) = \\mathrm { s o f t p l u s } ( \\alpha ) \\cdot \\mathrm { s o f t p l u s } ( \\nu ) \\cdot \\frac { g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\| \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| } .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 893, + 748, + 928 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "• Bayesian MAML (Yoon et al., 2018) relies on Stein Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to get an approximate posterior distribution in weight-space. We compare to both the non-chaser version, which optimizes cross-entropy of query predictions, and the chaser version, which optimizes mean squared error between the approximate posterior on the support set and the approximate posterior on the merged support & query set. The non-chaser version is therefore related to predictive likelihood methods and the chaser version is more analogous to the marginal likelihood methods. For the non-chaser version, we used 20 particles and 1 step of adaptation at both train and test time. For the chaser version, we also used 20 particles. At train time, the chaser took 1 step and the leader 1 additional step. At test time, we used 5 steps of adaptation. Due to the slow performance of this method, we followed the advice of Yoon et al. (2018) and only performed adaptation on the final layer of weights, which may help explain the drop in performance relative to MAML. The authors released Tensorflow code for regression only, so we reimplemented this baseline for classification in PyTorch. ", + "bbox": [ + 217, + 103, + 825, + 297 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Amortized Bayesian Meta-Learning (ABML) (Ravi & Beatson, 2019) performs a few steps of Bayes-by-backprop (Blundell et al., 2015) in order to infer a fully-factorized approximate posterior over the weights. The authors did not release code and so we implemented our own version of ABML in PyTorch. We found the weighting on the inner and outer KL divergences to be important for achieving good performance. We took the negative log likelihood to be mean cross entropy and used an inner KL weight of 0.01 and an outer KL weight of 0.001. These values were arrived upon by doing a small amount of hyperparameter tuning on the Omniglot EMNIST dataset. We used $\\alpha = 1 . 0$ and $\\beta = 0 . 0 1$ for the Gamma prior over the weights. We only applied ABML to the weights of the network; the biases were learned as point estimates. We used 4 steps of adaptation and took 5 samples when computing expectations (using any more than this did not fit into GPU memory). We used the local reparameterization trick (Kingma et al., 2015) and flipout (Wen et al., 2018) when computing expectations in order to reduce variance. In order to match the architecture used by Ravi & Beatson (2019), we trained this baseline with 32 filters throughout the classification network. We trained each 1-shot ABML model for 800 epochs and each 5-shot ABML model for 600 epochs as the learning had not converged within the epoch limits specified in Section G.3. ", + "bbox": [ + 215, + 310, + 825, + 545 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Logistic Softmax GP (Galy-Fajou et al., 2020) is the multi-class Gaussian process classification method that relies on the logistic softmax likelihood. Galy-Fajou et al. (2020) did not consider few-shot, but we use the same objectives described in Section 4.4 to adapt this method to FSC. In addition, we used the cosine kernel (see Section H for a description) that we found to work best with our OVE PG GPs. For this method, we found it important to learn a constant mean function (rather than a zero mean) in order to improve calibration. ", + "bbox": [ + 218, + 558, + 825, + 641 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "G.3 TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 666, + 354, + 681 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "All methods employed the commonly-used Conv4 architecture (Vinyals et al., 2016) (see Table 4 for a detailed specification), except ABML which used 32 filters throughout. All of our experiments used the Adam (Kingma & Ba, 2015) optimizer with learning rate $1 0 ^ { - 3 }$ . During training, all models used epochs consisting of 100 randomly sampled episodes. A single gradient descent step on the encoder network and relevant hyperparameters is made per episode. All 1-shot models are trained for 600 epochs and 5-shot models are trained for 400 epochs, except for ABML which was trained for an extra 200 epochs. Each episode contained 5 classes (5-way) and 16 query examples. At test time, 15 query examples are used for each episode. Early stopping was performed by monitoring accuracy on the validation set. The validation set was not used for retraining. ", + "bbox": [ + 174, + 695, + 825, + 821 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We train both marginal likelihood and predictive likelihood versions of our models. For Polya- ´ Gamma sampling we use the PyPolyaGamma package ´ 3. During training, we use a single step of Gibbs $( T { = } 1 )$ ). For evaluation, we run until $T = 5 0$ . In both training and evaluation, we use $M = 2 0$ parallel Gibbs chains to reduce variance. ", + "bbox": [ + 174, + 829, + 825, + 883 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/fbd5ac073b808a78e40a82dded935d006dbf799ada92509e8f6bfc3729d0b682.jpg", + "table_caption": [ + "Table 4: Specification of Conv4 architecture. Conv2d layers are $3 \\times 3$ with stride 1 and same padding. MaxPool2d layers are $2 \\times 2$ with stride 2 and valid padding. " + ], + "table_footnote": [ + "(a) Omniglot EMNIST dataset. " + ], + "table_body": "
Output SizeLayers
1×28×28Input image
64 × 14 ×14Conv2d BatchNorm2d
ReLU
MaxPool2d
64×7×7Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×3×3Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×1×1Conv2d BatchNorm2d
ReLU
MaxPool2d
64Flatten
", + "bbox": [ + 209, + 140, + 452, + 410 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/f04361cd31f01223776196ac4e7c15fefa2de74a29e05c60c9692754a40516d9.jpg", + "table_caption": [], + "table_footnote": [ + "(b) All other datasets. " + ], + "table_body": "
Output SizeLayers
3× 84×84Input image
64 × 42 × 42Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 × 21 × 21Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 ×10×10Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×5×5Conv2d BatchNorm2d
ReLU
MaxPool2d
1600Flatten
", + "bbox": [ + 522, + 140, + 766, + 410 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "H EFFECT OF KERNEL CHOICE ON CLASSIFICATION ACCURACY ", + "text_level": 1, + "bbox": [ + 171, + 457, + 722, + 474 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In this section, we examine the effect of kernel choice on classification accuracy for our proposed One-vs-Each Polya-Gamma OVE GPs. ´ ", + "bbox": [ + 173, + 489, + 823, + 518 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Cosine Kernel. In the main paper, we showed results for the following kernel, which we refer to as the “cosine” kernel due to its resemblance to cosine similarity: ", + "bbox": [ + 174, + 535, + 821, + 564 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/0597162c7e86cb6ef173bc517e68d0c2850dfd7245ec750e54cfa84627bcd8c6.jpg", + "text": "$$\nk ^ { \\mathrm { { c o s } } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha ) = \\mathrm { { e x p } } ( \\alpha ) \\frac { g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\| \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| } ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 571, + 648, + 608 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $g _ { \\pmb { \\theta } } ( \\cdot )$ is a deep neural network that outputs a fixed-dimensional encoded representation of the input and $\\alpha$ is the scalar log output scale. Both $\\pmb { \\theta }$ and $\\alpha$ are considered hyperparameters and learned simultaneously as shown in Algorithm 1. We found that this kernel works well for a range of datasets and shot settings. We note that the use of cosine similarity is reminiscent of the approach taken by Baseline $^ { + + }$ method of (Chen et al., 2019), which computes the softmax over cosine similarity to class weights. ", + "bbox": [ + 173, + 616, + 825, + 700 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Here we consider three additional kernels: linear, RBF, and normalized RBF. ", + "bbox": [ + 174, + 707, + 678, + 722 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Linear Kernel. The linear kernel is defined as follows: ", + "bbox": [ + 173, + 738, + 549, + 753 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/19061b62e08715e7b747dfd3a68fddf9b3c506155e9bb36dac1bcfdc38ff80ad.jpg", + "text": "$$\nk ^ { \\mathrm { l i n } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha ) = \\frac { 1 } { D } \\exp ( \\alpha ) g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 758, + 643, + 790 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $D$ is the output dimensionality of $g _ { \\pmb { \\theta } } ( \\mathbf { x } )$ . We apply this dimensionality scaling because the dot product between $g _ { \\pmb { \\theta } } ( \\mathbf { x } )$ and $g _ { \\boldsymbol { \\theta } } ( \\mathbf { x } ^ { \\prime } )$ may be large depending on $D$ . ", + "bbox": [ + 171, + 797, + 825, + 827 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "RBF Kernel. The RBF (also known as squared exponential) kernel can be defined as follows: ", + "bbox": [ + 168, + 843, + 799, + 859 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/1b5a4d978ea73c0e84f5c4f2955d0a1ca67a491e20b70fe4df65a26139d321ed.jpg", + "text": "$$\nk ^ { \\mathrm { t b f } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha , \\ell ) = \\mathrm { e x p } ( \\alpha ) \\mathrm { e x p } \\left( - \\frac { 1 } { 2 D \\mathrm { e x p } ( \\ell ) ^ { 2 } } \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) - g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| ^ { 2 } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 263, + 867, + 732, + 901 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\ell$ is the log lengthscale parameter (as with $\\alpha$ , we learn the $\\ell$ alongside $\\pmb { \\theta }$ ). ", + "bbox": [ + 173, + 909, + 696, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Normalized RBF Kernel. Finally, we consider a normalized RBF kernel similar in spirit to the cosine kernel: ", + "bbox": [ + 169, + 102, + 825, + 132 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/7628472cff4207e39dc9752e4806c2ab27fcc703569533fa320cd7f48d7b4b15.jpg", + "text": "$$\nk ^ { \\mathrm { \\scriptsize { t h f - n o r m } } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\theta , \\alpha , \\ell ) = \\exp ( \\alpha ) \\exp \\left( - \\frac { 1 } { 2 \\exp ( \\ell ) ^ { 2 } } \\left\\| \\frac { g _ { \\theta } ( \\mathbf { x } ) } { \\| g _ { \\theta } ( \\mathbf { x } ) \\| } - \\frac { g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) \\| } \\right\\| ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 233, + 136, + 764, + 179 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The results of our Polya-Gamma OVE GPs with different kernels can be found in Tables 5 ´ and 6. In general, we find that the cosine kernel works best overall, with the exception of Omniglot EMNIST, where RBF does best. ", + "bbox": [ + 176, + 190, + 826, + 234 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/8afaa8bc37faf7721b13ba0156273f229e279e8b62a107f3b8c6f8bdb402e506.jpg", + "table_caption": [ + "Table 5: Classification accuracy for Polya-Gamma OVE GPs (our method) using different kernels. ´ Cosine is overall the best, followed closely by linear. RBF-based kernels perform worse, except for the Omniglot EMNIST dataset. Evaluation is performed on 5 randomly generated sets of 600 test episodes. Standard deviation of the mean accuracy is also shown. $\\mathbf { M L } = \\mathbf { I }$ Marginal Likelihood, $\\mathrm { P L } =$ Predictive Likelihood. " + ], + "table_footnote": [], + "table_body": "
CUBmini-ImageNet
KernelObjective1-shot5-shot1-shot5-shot
CosineML63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
LinearML62.48 ± 0.2677.94 ± 0.2150.81 ± 0.3066.66 ± 0.45
RBFML58.49 ± 0.4075.50 ± 0.1850.33 ± 0.2664.62 ± 0.37
RBF (normalized)ML62.75 ± 0.3278.71 ± 0.0850.26 ± 0.3164.84 ± 0.39
CosinePL60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
LinearPL60.44 ± 0.3978.54 ± 0.1947.29 ± 0.3166.66 ± 0.36
RBFPL56.18 ± 0.6977.96 ± 0.1948.06 ± 0.2866.66 ± 0.39
RBF (normalized)PL59.78 ± 0.3478.42 ± 0.1347.51 ± 0.2066.42 ± 0.36
", + "bbox": [ + 176, + 337, + 818, + 481 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/8dee4f3950931273c1819cbccb34a550587b078b33e3abc20c85d59535202354.jpg", + "table_caption": [ + "Table 6: Cross-domain classification accuracy for Polya-Gamma OVE GPs (our method) using dif- ´ ferent kernels. The experimental setup is the same as Table 5. " + ], + "table_footnote": [], + "table_body": "
Omniglot→EMNISTmini-ImageNet-→CUB
KernelObjective1-shot5-shot1-shot5-shot
CosineML68.43 ± 0.6786.22 ± 0.2039.66± 0.1855.71 ± 0.31
LinearML72.42 ± 0.4988.27 ± 0.2039.61 ± 0.1955.07 ± 0.29
RBFML78.05 ± 0.3888.98 ± 0.1636.99 ± 0.0751.75 ± 0.27
RBF (normalized)ML75.51 ± 0.4788.86 ± 0.1638.42 ± 0.1654.20 ± 0.13
CosinePL77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
LinearPL75.87 ± 0.4388.77 ± 0.1036.83 ± 0.2756.46 ± 0.22
RBFPL74.62 ± 0.3589.87 ± 0.1335.06 ± 0.2555.12 ± 0.21
RBF (normalized)PL76.01 ± 0.3189.42 ± 0.1637.50 ± 0.2856.80 ± 0.39
", + "bbox": [ + 176, + 549, + 816, + 694 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "I ADDITIONAL CALIBRATION RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 720, + 516, + 737 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "In Figure 9, we include calibration results for mini-Imagenet and Omniglot EMNIST. They follow similar trends to the results presented in Section 5.2. ", + "bbox": [ + 173, + 751, + 823, + 780 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/3ce955918e7771637ee77a82ff033e887c6e839cbfde137d0515cbb078a00a51.jpg", + "image_caption": [ + "Figure 9: Reliability diagrams, expected calibration error, maximum calibration error, and Brier scores for 5-shot 5-way tasks on mini-Imagenet, Omniglot EMNIST, and mini-Imagenet CUB. Metrics are computed on 3,000 random tasks from the test set. " + ], + "image_footnote": [], + "bbox": [ + 173, + 198, + 825, + 784 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "J QUANTITATIVE ROBUSTNESS TO INPUT NOISE RESULTS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 669, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this section we include quantitative results for the robustness to input noise results presented in Figure 2. Results for Gaussian noise are shown in Table 7, impulse noise in Table 8, and defocus blur in Table 9. ", + "bbox": [ + 173, + 133, + 825, + 175 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/8c580eb14a69fcd72583056ce1402e5a853fa19f6a9506b40015a45142e41464.jpg", + "table_caption": [ + "Table 7: Accuracy $( \\% )$ and Brier Score when applying Gaussian noise corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. " + ], + "table_footnote": [], + "table_body": "
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.450.77522.580.79922.750.799
Baseline++22.600.79823.820.79724.130.797
MatchingNet26.720.80324.800.79723.590.804
ProtoNet32.280.77829.970.78132.300.779
RelationNet25.230.79923.690.80020.000.800
DKT+Cosine29.540.77927.780.79231.940.782
Bayesian MAML22.790.90520.520.96320.460.949
Bayesian MAML (Chaser)20.201.13320.411.11821.391.039
LSM GP + Cosine (ML)27.920.78722.430.79822.360.799
LSM GP + Cosine (PL)31.210.77231.770.76834.740.754
OVE PG GP + Cosine (ML) [ours]32.270.77429.990.77629.970.784
OVE PG GP + Cosine (PL) [ours]33.010.77133.290.76031.410.764
", + "bbox": [ + 171, + 239, + 892, + 439 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/e7ce54d00cf581668309a2e5dafa4af6e030de3745d8ae7245d98b0d6a7f1159.jpg", + "table_caption": [ + "Table 8: Accuracy $( \\% )$ and Brier Score when applying impulse noise corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. " + ], + "table_footnote": [], + "table_body": "
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.200.77623.540.79822.870.799
Baseline++28.050.79023.720.79825.580.795
MatchingNet28.250.79023.800.80323.210.811
ProtoNet32.120.77428.810.78332.700.775
RelationNet25.230.79923.130.80020.000.800
DKT+Cosine29.740.77829.110.78932.260.781
Bayesian MAML22.760.90320.500.97020.560.950
Bayesian MAML (Chaser)20.251.17220.511.11621.451.022
LSM GP + Cosine (ML)28.180.78721.820.79923.640.797
LSM GP + Cosine (PL)32.100.76930.220.77635.090.751
OVE PG GP + Cosine (ML) [ours]31.410.77829.660.77830.280.783
OVE PG GP + Cosine (PL) [ours]33.360.77233.230.76132.060.762
", + "bbox": [ + 171, + 513, + 892, + 712 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/9d3560a9514795efe8a1b393cdc0412ee80df9e42c90518312a8b7f15a30251b.jpg", + "table_caption": [ + "Table 9: Accuracy $( \\% )$ and Brier Score when applying defocus blur corruption of severity 5 to both the support and query set of test-time episodes. Results were evaluated across 1,000 randomly generated 5-shot 5-way tasks. " + ], + "table_footnote": [], + "table_body": "
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer38.030.73433.060.79133.470.792
Baseline++42.550.71035.890.76139.880.740
MatchingNet44.430.68234.430.75435.950.741
ProtoNet46.780.67636.920.73741.450.714
RelationNet40.810.75930.110.79025.690.794
DKT + Cosine45.340.69538.290.73745.170.703
Bayesian MAML42.650.69730.630.80837.320.736
Bayesian MAML (Chaser)40.660.88129.931.12131.331.125
LSM GP + Cosine (ML)45.370.70634.100.76939.660.753
LSM GP + Cosine (PL)48.550.69039.460.73743.150.714
OVE PG GP + Cosine (ML) [ours]46.460.70137.650.77543.480.723
OVE PG GP + Cosine (PL) [ours]49.440.69538.950.78043.660.720
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This elegant treatment of model uncertainty has", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "led to a surge of interest in Bayesian approaches to FSC that infer a posterior distribution over the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 464, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 464, + 622 + ], + "score": 1.0, + "content": "weights of a neural network (Finn et al., 2018; Yoon et al., 2018; Ravi & Beatson, 2019).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 556, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Although conceptually appealing, there are several practical obstacles to applying Bayesian infer-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "ence directly to the weights of a neural network. Bayesian neural networks (BNNs) are expensive", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "from both a computational and memory perspective. 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The prior is directly specified by a mean and covariance function, which may be parameterized", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "by deep neural networks. When used with Gaussian likelihoods, GPs admit closed form expressions", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "for the posterior and predictive distributions. They exchange the computational drawbacks of BNNs", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 81, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 504, + 94 + ], + "score": 1.0, + "content": "for cubic scaling with the number of examples. 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In FSC, where the number of examples is small, this", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 234, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 234, + 105 + ], + "score": 1.0, + "content": "is often an acceptable trade-off.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "When applying GPs to classification with a softmax likelihood, the non-conjugacy of the GP prior", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "renders posterior inference intractable. Many approximate inference methods have been proposed to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "circumvent this, including variational inference and expectation propagation. In this paper we inves-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "tigate a particularly promising class of approaches that augment the GP model with a set of auxiliary", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "random variables, such that when they are marginalized out the original model is recovered (Albert", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "score": 1.0, + "content": "& Chib, 1993; Girolami & Rogers, 2006; Linderman et al., 2015). Such augmentation-based ap-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "proaches typically admit efficient Gibbs sampling procedures for generating posterior samples which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "when combined with Fisher’s identity (Douc et al., 2014) can be used to optimize the parameters of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 249, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 249, + 210 + ], + "score": 1.0, + "content": "the mean and covariance functions.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "In particular, augmentation with Polya-Gamma random variables (Polson et al., 2013) makes in- ´", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "ference tractable in logistic models. Naively, this is useful for handling binary classification, but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "in this paper we show how to extend Polya-Gamma augmentation to multiple classes by using the ´", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "score": 1.0, + "content": "one-vs-each softmax approximation (Titsias, 2016), which can be expressed as a product of logistic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "sigmoids. 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We summarize relevant work here.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 522, + 219, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 221, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 221, + 536 + ], + "score": 1.0, + "content": "2.1 GP CLASSIFICATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Non-augmentation approaches. There are several classes of approaches for applying Gaussian", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "processes to classification. The most straightforward method, known as least squares classifica-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "tion (Rifkin & Klautau, 2004), treats class labels as real-valued observations and performs infer-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "ence with a Gaussian likelihood. The Laplace approximation (Williams & Barber, 1998) constructs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "a Gaussian approximate posterior centered at the posterior mode. Variational approaches (Titsias,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "2009; Matthews et al., 2016) maximize a lower bound on the log marginal likelihood. In expectation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "propagation (Minka, 2001; Kim & Ghahramani, 2006; Hernandez-Lobato & Hernandez-Lobato,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "2016), local Gaussian approximations to the likelihood are fitted iteratively to minimize KL diver-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 227, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 227, + 644 + ], + "score": 1.0, + "content": "gence from the true posterior.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Augmentation approaches. Augmentation-based approaches introduce auxiliary random vari-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "ables such that the original model is recovered when marginalized out. Girolami & Rogers (2006)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "propose a Gaussian augmentation for multinomial probit regression. Linderman et al. (2015) uti-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "lize Polya-Gamma augmentation (Polson et al., 2013) and a stick-breaking construction to decom- ´", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "pose a multinomial distribution into a product of binomials. Galy-Fajou et al. (2020) propose a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "logistic-softmax likelihood for classification and uses Gamma and Poisson augmentation in addition", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 354, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 354, + 734 + ], + "score": 1.0, + "content": "to Polya-Gamma augmentation in order to perform inference. ´", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 504, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "When applying GPs to classification with a softmax likelihood, the non-conjugacy of the GP prior", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "renders posterior inference intractable. 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In this paper we inves-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "tigate a particularly promising class of approaches that augment the GP model with a set of auxiliary", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "random variables, such that when they are marginalized out the original model is recovered (Albert", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "score": 1.0, + "content": "& Chib, 1993; Girolami & Rogers, 2006; Linderman et al., 2015). Such augmentation-based ap-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "proaches typically admit efficient Gibbs sampling procedures for generating posterior samples which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "when combined with Fisher’s identity (Douc et al., 2014) can be used to optimize the parameters of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 249, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 249, + 210 + ], + "score": 1.0, + "content": "the mean and covariance functions.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 111, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "In particular, augmentation with Polya-Gamma random variables (Polson et al., 2013) makes in- ´", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "ference tractable in logistic models. 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We summarize relevant work here.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 486, + 505, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 522, + 219, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 221, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 221, + 536 + ], + "score": 1.0, + "content": "2.1 GP CLASSIFICATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Non-augmentation approaches. There are several classes of approaches for applying Gaussian", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "processes to classification. The most straightforward method, known as least squares classifica-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "tion (Rifkin & Klautau, 2004), treats class labels as real-valued observations and performs infer-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "ence with a Gaussian likelihood. The Laplace approximation (Williams & Barber, 1998) constructs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "a Gaussian approximate posterior centered at the posterior mode. Variational approaches (Titsias,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "2009; Matthews et al., 2016) maximize a lower bound on the log marginal likelihood. In expectation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "propagation (Minka, 2001; Kim & Ghahramani, 2006; Hernandez-Lobato & Hernandez-Lobato,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "2016), local Gaussian approximations to the likelihood are fitted iteratively to minimize KL diver-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 227, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 227, + 644 + ], + "score": 1.0, + "content": "gence from the true posterior.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 543, + 506, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Augmentation approaches. Augmentation-based approaches introduce auxiliary random vari-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "ables such that the original model is recovered when marginalized out. Girolami & Rogers (2006)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "propose a Gaussian augmentation for multinomial probit regression. Linderman et al. (2015) uti-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "lize Polya-Gamma augmentation (Polson et al., 2013) and a stick-breaking construction to decom- ´", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "pose a multinomial distribution into a product of binomials. Galy-Fajou et al. (2020) propose a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "logistic-softmax likelihood for classification and uses Gamma and Poisson augmentation in addition", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 721, + 354, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 354, + 734 + ], + "score": 1.0, + "content": "to Polya-Gamma augmentation in order to perform inference. ´", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 654, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 251, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 252, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 252, + 95 + ], + "score": 1.0, + "content": "2.2 FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 117 + ], + "score": 1.0, + "content": "Meta-learning. 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MAML (Finn et al., 2017) learns initializations of deep neural networks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "score": 1.0, + "content": "that perform well on task-specific losses after one or a few steps of gradient descent by backpropa-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "gating through the gradient descent procedure itself. LEO (Rusu et al., 2019) performs meta-learning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 497, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 497, + 181 + ], + "score": 1.0, + "content": "in a learned low-dimensional latent space from which the parameters of a classifier are generated.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Metric learning. Metric learning approaches learn distances such that input examples can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 201, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 506, + 217 + ], + "score": 1.0, + "content": "meaningfully compared. Siamese Networks (Koch, 2015) learn a shared embedding network along", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "with a distance layer for computing the probability that two examples belong to the same class.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Matching Networks (Vinyals et al., 2016) uses a nonparametric classification in the form of atten-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 396, + 249 + ], + "score": 1.0, + "content": "tion over nearby examples, which can be interpreted as a form of soft", + "type": "text" + }, + { + "bbox": [ + 396, + 237, + 403, + 246 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "-nearest neighbors in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "embedding space. Prototypical Networks (Snell et al., 2017) make predictions based on distances to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "nearest class centroids. Relation Networks (Sung et al., 2018) instead learn a more complex neural", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 337, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 337, + 282 + ], + "score": 1.0, + "content": "network distance function on top of the embedding layer.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 292, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "Bayesian Few-shot Classification. More recently, Bayesian FSC approaches that attempt to infer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "a posterior over task-specific parameters have appeared. Grant et al. (2018) reinterpret MAML", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "as an approximate empirical Bayes algorithm and propose LLAMA, which optimizes the Laplace", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "approximation to the marginal likelihood. Bayesian MAML (Yoon et al., 2018) instead uses Stein", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to approximate the posterior distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "over model parameters. VERSA (Gordon et al., 2019) uses amortized inference networks to obtain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "an approximate posterior distribution over task-specific parameters. ABML (Ravi & Beatson, 2019)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "uses a few steps of Bayes by Backprop (Blundell et al., 2015) on the support set to produce an", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "approximate posterior over network parameters. CNAPs (Requeima et al., 2019) modulate task-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "specific Feature-wise Linear Modulation (FiLM) (Perez et al., 2018) layer parameters as the output", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 345, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 345, + 415 + ], + "score": 1.0, + "content": "of an adaptation network that takes the support set as input.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "GPs for Few-shot Learning. There have been relatively few works applying GPs to few-shot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "learning. Tossou et al. (2020) consider Gaussian processes in the context of few-shot regression", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 448, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "with Gaussian likelihoods. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) uses Gaussian", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "processes with least squares classification to perform few-shot classification and learns covariance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "functions parameterized by deep neural networks. More recently, Titsias et al. (2020) applies GPs to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "meta-learning by maximizing the mutual information between the query set and a latent representa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 199, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 199, + 505 + ], + "score": 1.0, + "content": "tion of the support set.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 200, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "score": 1.0, + "content": "3 BACKGROUND", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 504, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "In this section we first review Polya-Gamma augmentation for binary classification and the one-vs- ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 555, + 371, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 371, + 567 + ], + "score": 1.0, + "content": "each approximation before we introduce our method in Section 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 578, + 272, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 272, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 272, + 591 + ], + "score": 1.0, + "content": "3.1 POLYA ´ -GAMMA AUGMENTATION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "The Polya-Gamma augmentation scheme was originally introduced to address Bayesian inference ´", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 417, + 624 + ], + "score": 1.0, + "content": "in logistic models (Polson et al., 2013). Suppose we have a vector of logits", + "type": "text" + }, + { + "bbox": [ + 418, + 610, + 456, + 623 + ], + "score": 0.93, + "content": "\\boldsymbol { \\psi } \\in \\mathbf { \\mathbb { R } } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "with corre-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 358, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 199, + 635 + ], + "score": 1.0, + "content": "sponding binary labels", + "type": "text" + }, + { + "bbox": [ + 199, + 622, + 250, + 634 + ], + "score": 0.93, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 621, + 358, + 635 + ], + "score": 1.0, + "content": ". 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A common approach to FSC is meta-learning, which seeks to learn a strategy", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "to update neural network parameters when faced with a novel learning task. The Meta-learner", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "LSTM (Ravi & Larochelle, 2017) learns a meta-level LSTM to recurrently output a new set of pa-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "rameters for a base learner. MAML (Finn et al., 2017) learns initializations of deep neural networks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 505, + 160 + ], + "score": 1.0, + "content": "that perform well on task-specific losses after one or a few steps of gradient descent by backpropa-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "gating through the gradient descent procedure itself. LEO (Rusu et al., 2019) performs meta-learning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 497, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 497, + 181 + ], + "score": 1.0, + "content": "in a learned low-dimensional latent space from which the parameters of a classifier are generated.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 102, + 505, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Metric learning. Metric learning approaches learn distances such that input examples can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 201, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 506, + 217 + ], + "score": 1.0, + "content": "meaningfully compared. Siamese Networks (Koch, 2015) learn a shared embedding network along", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "with a distance layer for computing the probability that two examples belong to the same class.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Matching Networks (Vinyals et al., 2016) uses a nonparametric classification in the form of atten-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 396, + 249 + ], + "score": 1.0, + "content": "tion over nearby examples, which can be interpreted as a form of soft", + "type": "text" + }, + { + "bbox": [ + 396, + 237, + 403, + 246 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "-nearest neighbors in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "embedding space. Prototypical Networks (Snell et al., 2017) make predictions based on distances to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "nearest class centroids. Relation Networks (Sung et al., 2018) instead learn a more complex neural", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 337, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 337, + 282 + ], + "score": 1.0, + "content": "network distance function on top of the embedding layer.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 192, + 506, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 292, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "Bayesian Few-shot Classification. More recently, Bayesian FSC approaches that attempt to infer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "a posterior over task-specific parameters have appeared. Grant et al. (2018) reinterpret MAML", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "as an approximate empirical Bayes algorithm and propose LLAMA, which optimizes the Laplace", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "approximation to the marginal likelihood. Bayesian MAML (Yoon et al., 2018) instead uses Stein", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Variational Gradient Descent (SVGD) (Liu & Wang, 2016) to approximate the posterior distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "over model parameters. VERSA (Gordon et al., 2019) uses amortized inference networks to obtain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "an approximate posterior distribution over task-specific parameters. ABML (Ravi & Beatson, 2019)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "uses a few steps of Bayes by Backprop (Blundell et al., 2015) on the support set to produce an", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "approximate posterior over network parameters. CNAPs (Requeima et al., 2019) modulate task-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "specific Feature-wise Linear Modulation (FiLM) (Perez et al., 2018) layer parameters as the output", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 345, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 345, + 415 + ], + "score": 1.0, + "content": "of an adaptation network that takes the support set as input.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 292, + 506, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "GPs for Few-shot Learning. There have been relatively few works applying GPs to few-shot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "learning. Tossou et al. (2020) consider Gaussian processes in the context of few-shot regression", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 448, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "with Gaussian likelihoods. Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) uses Gaussian", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "processes with least squares classification to perform few-shot classification and learns covariance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "functions parameterized by deep neural networks. More recently, Titsias et al. (2020) applies GPs to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "meta-learning by maximizing the mutual information between the query set and a latent representa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 199, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 199, + 505 + ], + "score": 1.0, + "content": "tion of the support set.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 425, + 506, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 200, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 533 + ], + "score": 1.0, + "content": "3 BACKGROUND", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 504, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "In this section we first review Polya-Gamma augmentation for binary classification and the one-vs- ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 555, + 371, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 371, + 567 + ], + "score": 1.0, + "content": "each approximation before we introduce our method in Section 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 542, + 505, + 567 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 578, + 272, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 272, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 272, + 591 + ], + "score": 1.0, + "content": "3.1 POLYA ´ -GAMMA AUGMENTATION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "The Polya-Gamma augmentation scheme was originally introduced to address Bayesian inference ´", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 417, + 624 + ], + "score": 1.0, + "content": "in logistic models (Polson et al., 2013). 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Fortunately, efficient samplers for the Polya-Gamma ´", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 392, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 392, + 285 + ], + "score": 1.0, + "content": "distribution have been developed (Windle et al., 2014) to facilitate this.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 297, + 329, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 330, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 330, + 310 + ], + "score": 1.0, + "content": "3.2 ONE-VS-EACH APPROXIMATION TO SOFTMAX", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "The one-vs-each (OVE) approximation (Titsias, 2016) was formulated as a lower bound to the soft-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "max likelihood in order to handle classification over a large number of output classes, where com-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "putation of the normalizing constant is prohibitive. We employ the OVE approximation not to deal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "with extreme classification, but rather due to its compatibility with Polya-Gamma augmentation, as ´", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "we shall soon see. 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Since in general", + "type": "text" + }, + { + "bbox": [ + 336, + 423, + 455, + 437 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\prod _ { k } ( 1 + \\alpha _ { k } ) \\ge \\left( 1 + \\sum _ { k } \\alpha _ { k } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 422, + 470, + 437 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 470, + 424, + 501, + 435 + ], + "score": 0.9, + "content": "\\alpha _ { k } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 422, + 505, + 437 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 434, + 324, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 324, + 447 + ], + "score": 1.0, + "content": "the softmax likelihood (5) can be bounded as follows:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 450, + 409, + 481 + ], + "lines": [ + { + "bbox": [ + 201, + 450, + 409, + 481 + ], + "spans": [ + { + "bbox": [ + 201, + 450, + 409, + 481 + ], + "score": 0.94, + "content": "p ( y = i \\mid \\mathbf { f } ) \\geq \\prod _ { j \\neq i } { \\frac { 1 } { 1 + e ^ { - ( f _ { i } - f _ { j } ) } } } = \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) ,", + "type": "interline_equation", + "image_path": "90861d2f02b324d130c82e1c3f3af6b15ceaf10e2bbe6a6c5d0e64bf0815d56a.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 450, + 409, + 465.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 201, + 465.5, + 409, + 481.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "which is the OVE lower bound. This expression avoids the normalizing constant and factorizes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "into a product of pairwise sigmoids, which is amenable to Polya-Gamma augmentation for tractable ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 506, + 150, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 150, + 520 + ], + "score": 1.0, + "content": "inference.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 534, + 311, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 312, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 312, + 550 + ], + "score": 1.0, + "content": "4 ONE-VS-EACH POLYA ´ -GAMMA GPS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "In this section, we first show how the one-vs-each (OVE) approximation can be interpreted as a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "pairwise composite likelihood. We then we introduce our method for GP-based Bayesian few-shot", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "classification, which brings together OVE and Polya-Gamma augmentation in a novel combination. ´", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 606, + 286, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 287, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 287, + 619 + ], + "score": 1.0, + "content": "4.1 OVE AS A COMPOSITE LIKELIHOOD", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Titsias (2016) showed that the OVE approximation shares the same global optimum as the softmax", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "maximum likelihood, suggesting a close relationship between the two. 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This expression avoids the normalizing constant and factorizes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "into a product of pairwise sigmoids, which is amenable to Polya-Gamma augmentation for tractable ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 506, + 150, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 150, + 520 + ], + "score": 1.0, + "content": "inference.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 484, + 505, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 534, + 311, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 312, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 312, + 550 + ], + "score": 1.0, + "content": "4 ONE-VS-EACH POLYA ´ -GAMMA GPS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "In this section, we first show how the one-vs-each (OVE) approximation can be interpreted as a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "pairwise composite likelihood. We then we introduce our method for GP-based Bayesian few-shot", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "classification, which brings together OVE and Polya-Gamma augmentation in a novel combination. ´", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 560, + 506, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 606, + 286, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 287, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 287, + 619 + ], + "score": 1.0, + "content": "4.1 OVE AS A COMPOSITE LIKELIHOOD", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Titsias (2016) showed that the OVE approximation shares the same global optimum as the softmax", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "maximum likelihood, suggesting a close relationship between the two. We show here that in fact", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "OVE can be interpreted as a pairwise composite likelihood version of the softmax. Composite", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "likelihoods (Lindsay, 1988; Varin et al., 2011) are a type of approximate likelihood often employed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 672, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 683 + ], + "score": 1.0, + "content": "when the exact likelihood is intractable or otherwise difficult to compute. Given a collection of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 232, + 695 + ], + "score": 1.0, + "content": "marginal or conditional events", + "type": "text" + }, + { + "bbox": [ + 233, + 682, + 292, + 695 + ], + "score": 0.93, + "content": "\\{ E _ { 1 } , \\ldots , E _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 682, + 358, + 695 + ], + "score": 1.0, + "content": "and parameters", + "type": "text" + }, + { + "bbox": [ + 358, + 683, + 364, + 692 + ], + "score": 0.27, + "content": "\\mathbf { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 682, + 506, + 695 + ], + "score": 1.0, + "content": ", a composite likelihood is defined", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 695, + 120, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 120, + 705 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 626, + 506, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 701, + 366, + 735 + ], + "lines": [ + { + "bbox": [ + 245, + 701, + 366, + 735 + ], + "spans": [ + { + "bbox": [ + 245, + 701, + 366, + 735 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { \\mathrm { C L } } ( \\mathbf { f } \\mid y ) \\triangleq \\prod _ { k = 1 } ^ { K } \\mathcal { L } _ { k } ( \\mathbf { f } \\mid y ) ^ { w _ { k } } ,", + "type": "interline_equation", + "image_path": "9668b14477129b07e9192a249f8ce4c7428a88c8159f4d41432b0d4f622bf246.jpg" + } + ] + } + ], + "index": 43.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 701, + 366, + 718.0 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 245, + 718.0, + 366, + 735.0 + ], + "spans": [], + "index": 44 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 374, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 374, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 133, + 97 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 82, + 236, + 95 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { k } ( \\mathbf { f } \\mid y ) \\propto p ( y \\in E _ { k } \\mid \\mathbf { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 80, + 254, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 83, + 286, + 94 + ], + "score": 0.91, + "content": "w _ { k } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 80, + 374, + 97 + ], + "score": 1.0, + "content": "are arbitrary weights.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 99, + 505, + 144 + ], + "lines": [ + { + "bbox": [ + 104, + 98, + 504, + 112 + ], + "spans": [ + { + "bbox": [ + 104, + 98, + 497, + 112 + ], + "score": 1.0, + "content": "In order to make the connection to OVE, it will be useful to let the one-hot encoding of the label", + "type": "text" + }, + { + "bbox": [ + 497, + 101, + 504, + 111 + ], + "score": 0.74, + "content": "y", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 109, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 104, + 109, + 163, + 123 + ], + "score": 1.0, + "content": "be denoted as", + "type": "text" + }, + { + "bbox": [ + 164, + 110, + 215, + 122 + ], + "score": 0.92, + "content": "\\mathbf { y } \\in \\{ 0 , 1 \\} ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 109, + 282, + 123 + ], + "score": 1.0, + "content": ". 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This expression resembles the pseu-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "dolikelihood (Besag, 1975), but instead of a single conditional event per output site, the expression", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 269, + 205 + ], + "score": 1.0, + "content": "in (8) considers all pairs of sites. Stoehr", + "type": "text" + }, + { + "bbox": [ + 270, + 193, + 279, + 202 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Friel (2015) explored similar composite likelihood gen-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 378, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 378, + 215 + ], + "score": 1.0, + "content": "eralizations of the pseudolikelihood in the context of random fields.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 504, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 180, + 234 + ], + "score": 1.0, + "content": "Now suppose that", + "type": "text" + }, + { + "bbox": [ + 181, + 221, + 210, + 232 + ], + "score": 0.91, + "content": "y _ { c } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 219, + 271, + 234 + ], + "score": 1.0, + "content": "for some class", + "type": "text" + }, + { + "bbox": [ + 271, + 220, + 312, + 232 + ], + "score": 0.91, + "content": "c \\notin \\{ i , j \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 219, + 340, + 234 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 340, + 220, + 428, + 233 + ], + "score": 0.92, + "content": "p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { \\lnot i j } , \\mathbf { f } ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "due to the one-hot", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 221, + 244 + ], + "score": 1.0, + "content": "constraint. Otherwise either", + "type": "text" + }, + { + "bbox": [ + 222, + 232, + 251, + 243 + ], + "score": 0.9, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 231, + 263, + 244 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 264, + 233, + 293, + 244 + ], + "score": 0.91, + "content": "y _ { j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 231, + 505, + 244 + ], + "score": 1.0, + "content": ". In this case, assume without loss of generality that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 243, + 218, + 254 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 134, + 254 + ], + "score": 0.91, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 243, + 152, + 254 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 152, + 243, + 181, + 254 + ], + "score": 0.92, + "content": "y _ { j } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 243, + 218, + 254 + ], + "score": 1.0, + "content": "and thus", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 259, + 397, + 287 + ], + "lines": [ + { + "bbox": [ + 213, + 259, + 397, + 287 + ], + "spans": [ + { + "bbox": [ + 213, + 259, + 397, + 287 + ], + "score": 0.95, + "content": "p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { - i j } , \\mathbf { f } ) = \\frac { e ^ { f _ { i } } } { e ^ { f _ { i } } + e ^ { f _ { j } } } = \\sigma ( f _ { i } - f _ { j } ) .", + "type": "interline_equation", + "image_path": "f63091135d80d1909645efeac208718d4b4e75984d5fc21fdc6474abaac8ef7f.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 259, + 397, + 273.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 213, + 273.0, + 397, + 287.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 455, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 455, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 455, + 306 + ], + "score": 1.0, + "content": "The composite likelihood defined in this way with unit component weights is therefore", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 308, + 431, + 338 + ], + "lines": [ + { + "bbox": [ + 180, + 308, + 431, + 338 + ], + "spans": [ + { + "bbox": [ + 180, + 308, + 431, + 338 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { \\mathrm { O V E } } ( \\mathbf { f } \\mid \\mathbf { y } ) = \\prod _ { i } \\prod _ { j \\neq i } p ( y _ { i } , y _ { j } \\mid \\mathbf { y } _ { - i j } , \\mathbf { f } ) = \\prod _ { i } \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) ^ { y _ { i } } .", + "type": "interline_equation", + "image_path": "58dacd071253f6d789146bca499c2eca91a5542e4a8427a5d291e3b6fe64b7f2.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 180, + 308, + 431, + 318.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 180, + 318.0, + 431, + 328.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 180, + 328.0, + 431, + 338.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 504, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 252, + 358 + ], + "score": 1.0, + "content": "Alternatively, we may simply write", + "type": "text" + }, + { + "bbox": [ + 252, + 342, + 401, + 357 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { O V E } } ( \\mathbf { f } \\mid y = i ) = \\prod _ { j \\neq i } \\sigma ( f _ { i } - f _ { j } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 342, + 506, + 358 + ], + "score": 1.0, + "content": ", which is identical to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 174, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 174, + 369 + ], + "score": 1.0, + "content": "OVE bound (6).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 106, + 380, + 347, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 347, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 347, + 394 + ], + "score": 1.0, + "content": "4.2 GP CLASSIFICATION WITH THE OVE LIKELIHOOD", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 503, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 452, + 414 + ], + "score": 1.0, + "content": "We now turn our attention to GP classification. Suppose we have access to examples", + "type": "text" + }, + { + "bbox": [ + 453, + 401, + 503, + 412 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { N \\times D }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 410, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 245, + 426 + ], + "score": 1.0, + "content": "with corresponding one-hot labels", + "type": "text" + }, + { + "bbox": [ + 245, + 412, + 311, + 425 + ], + "score": 0.94, + "content": "\\mathbf { Y } \\in \\{ 0 , 1 \\} ^ { N \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 410, + 342, + 426 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 342, + 414, + 351, + 423 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 410, + 507, + 426 + ], + "score": 1.0, + "content": "is the number of classes. We consider", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 424, + 246, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 246, + 436 + ], + "score": 1.0, + "content": "the logits jointly as a single vector", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 441, + 405, + 457 + ], + "lines": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "spans": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "score": 0.88, + "content": "\\mathbf { f } \\triangleq ( f _ { 1 } ^ { 1 } , \\hdots , f _ { N } ^ { 1 } , f _ { 1 } ^ { 2 } , \\hdots , f _ { N } ^ { 2 } , \\hdots , f _ { 1 } ^ { C } , \\hdots , f _ { N } ^ { C } ) ^ { \\intercal }", + "type": "interline_equation", + "image_path": "ff472397405b2fff69ba31b7bc09a13b31fc5be3f0debdd0e8f287fffd0acd39.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 378, + 476 + ], + "score": 1.0, + "content": "and place an independent GP prior on the logits for each class:", + "type": "text" + }, + { + "bbox": [ + 378, + 462, + 500, + 474 + ], + "score": 0.9, + "content": "\\mathbf { f } ^ { c } ( \\mathbf { x } ) \\sim \\mathcal { G P } ( m ( \\mathbf { x } ) , k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 460, + 505, + 476 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 184, + 488 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + }, + { + "bbox": [ + 184, + 473, + 273, + 486 + ], + "score": 0.93, + "content": "p ( \\mathbf { f } | \\mathbf { X } ) = \\bar { \\mathcal { N } } ( \\mathbf { f } | \\pmb { \\mu } , \\mathbf { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 471, + 304, + 488 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 304, + 474, + 355, + 486 + ], + "score": 0.93, + "content": "\\mu _ { i } ^ { c } = m ( \\mathbf { x } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 471, + 373, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 474, + 384, + 484 + ], + "score": 0.72, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 471, + 476, + 488 + ], + "score": 1.0, + "content": "is block diagonal with", + "type": "text" + }, + { + "bbox": [ + 477, + 475, + 506, + 487 + ], + "score": 0.9, + "content": "K _ { i j } ^ { c } =", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 486, + 223, + 498 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 145, + 498 + ], + "score": 0.92, + "content": "k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 486, + 205, + 497 + ], + "score": 1.0, + "content": "for each block", + "type": "text" + }, + { + "bbox": [ + 206, + 487, + 220, + 496 + ], + "score": 0.86, + "content": "{ \\bf K } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 486, + 223, + 497 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "The Polya-Gamma integral identity used to derive (2) does not have a multi-class analogue and thus ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "a direct application of the augmentation scheme to the softmax likelihood is nontrivial. Instead, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "propose to directly replace the softmax with the OVE-based composite likelihood function from (10)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "with unit weights. 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Our motiva- ´", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "tion for using a composite likelihood therefore differs from the traditional motivation, which is to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "avoid the use of a likelihood function which is intractable to evaluate. 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This expression resembles the pseu-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "dolikelihood (Besag, 1975), but instead of a single conditional event per output site, the expression", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 269, + 205 + ], + "score": 1.0, + "content": "in (8) considers all pairs of sites. Stoehr", + "type": "text" + }, + { + "bbox": [ + 270, + 193, + 279, + 202 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "Friel (2015) explored similar composite likelihood gen-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 378, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 378, + 215 + ], + "score": 1.0, + "content": "eralizations of the pseudolikelihood in the context of random fields.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 170, + 505, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 504, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 180, + 234 + ], + "score": 1.0, + "content": "Now suppose that", + "type": "text" + }, + { + "bbox": [ + 181, + 221, + 210, + 232 + ], + "score": 0.91, + "content": "y _ { c } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 219, + 271, + 234 + ], + "score": 1.0, + "content": "for some class", + "type": "text" + }, + { + "bbox": [ + 271, + 220, + 312, + 232 + ], + "score": 0.91, + "content": "c \\notin \\{ i , j \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 219, + 340, + 234 + ], + "score": 1.0, + "content": ". 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Otherwise either", + "type": "text" + }, + { + "bbox": [ + 222, + 232, + 251, + 243 + ], + "score": 0.9, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 231, + 263, + 244 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 264, + 233, + 293, + 244 + ], + "score": 0.91, + "content": "y _ { j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 231, + 505, + 244 + ], + "score": 1.0, + "content": ". 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Suppose we have access to examples", + "type": "text" + }, + { + "bbox": [ + 453, + 401, + 503, + 412 + ], + "score": 0.91, + "content": "\\mathbf { X } \\in \\mathbb { R } ^ { N \\times D }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 410, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 245, + 426 + ], + "score": 1.0, + "content": "with corresponding one-hot labels", + "type": "text" + }, + { + "bbox": [ + 245, + 412, + 311, + 425 + ], + "score": 0.94, + "content": "\\mathbf { Y } \\in \\{ 0 , 1 \\} ^ { N \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 410, + 342, + 426 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 342, + 414, + 351, + 423 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 410, + 507, + 426 + ], + "score": 1.0, + "content": "is the number of classes. We consider", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 424, + 246, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 246, + 436 + ], + "score": 1.0, + "content": "the logits jointly as a single vector", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 401, + 507, + 436 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 441, + 405, + 457 + ], + "lines": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "spans": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "score": 0.88, + "content": "\\mathbf { f } \\triangleq ( f _ { 1 } ^ { 1 } , \\hdots , f _ { N } ^ { 1 } , f _ { 1 } ^ { 2 } , \\hdots , f _ { N } ^ { 2 } , \\hdots , f _ { 1 } ^ { C } , \\hdots , f _ { N } ^ { C } ) ^ { \\intercal }", + "type": "interline_equation", + "image_path": "ff472397405b2fff69ba31b7bc09a13b31fc5be3f0debdd0e8f287fffd0acd39.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 205, + 441, + 405, + 457 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 378, + 476 + ], + "score": 1.0, + "content": "and place an independent GP prior on the logits for each class:", + "type": "text" + }, + { + "bbox": [ + 378, + 462, + 500, + 474 + ], + "score": 0.9, + "content": "\\mathbf { f } ^ { c } ( \\mathbf { x } ) \\sim \\mathcal { G P } ( m ( \\mathbf { x } ) , k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 460, + 505, + 476 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 184, + 488 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + }, + { + "bbox": [ + 184, + 473, + 273, + 486 + ], + "score": 0.93, + "content": "p ( \\mathbf { f } | \\mathbf { X } ) = \\bar { \\mathcal { N } } ( \\mathbf { f } | \\pmb { \\mu } , \\mathbf { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 471, + 304, + 488 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 304, + 474, + 355, + 486 + ], + "score": 0.93, + "content": "\\mu _ { i } ^ { c } = m ( \\mathbf { x } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 471, + 373, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 374, + 474, + 384, + 484 + ], + "score": 0.72, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 471, + 476, + 488 + ], + "score": 1.0, + "content": "is block diagonal with", + "type": "text" + }, + { + "bbox": [ + 477, + 475, + 506, + 487 + ], + "score": 0.9, + "content": "K _ { i j } ^ { c } =", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 486, + 223, + 498 + ], + "spans": [ + { + "bbox": [ + 107, + 486, + 145, + 498 + ], + "score": 0.92, + "content": "k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 486, + 205, + 497 + ], + "score": 1.0, + "content": "for each block", + "type": "text" + }, + { + "bbox": [ + 206, + 487, + 220, + 496 + ], + "score": 0.86, + "content": "{ \\bf K } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 486, + 223, + 497 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 460, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "The Polya-Gamma integral identity used to derive (2) does not have a multi-class analogue and thus ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "a direct application of the augmentation scheme to the softmax likelihood is nontrivial. Instead, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "propose to directly replace the softmax with the OVE-based composite likelihood function from (10)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "with unit weights. The posterior over f when using OVE as the likelihood function can be expressed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 548, + 120, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 120, + 559 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 502, + 506, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 555, + 393, + 591 + ], + "lines": [ + { + "bbox": [ + 217, + 555, + 393, + 591 + ], + "spans": [ + { + "bbox": [ + 217, + 555, + 393, + 591 + ], + "score": 0.95, + "content": "p ( \\mathbf { f } | \\mathbf { X } , \\mathbf { y } ) \\propto p ( \\mathbf { f } | \\mathbf { X } ) \\prod _ { i = 1 } ^ { N } \\prod _ { c ^ { \\prime } \\neq y _ { i } } \\sigma ( f _ { i } ^ { y _ { i } } - f _ { i } ^ { c ^ { \\prime } } ) ,", + "type": "interline_equation", + "image_path": "3b907a26f1bd9fbcf12fedcbc988dc0b11b59255f1d91a61e8c34221281cebbb.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 555, + 393, + 573.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 217, + 573.0, + 393, + 591.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to which Polya-Gamma augmentation can be applied as we show in the next section. Our motiva- ´", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "tion for using a composite likelihood therefore differs from the traditional motivation, which is to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "avoid the use of a likelihood function which is intractable to evaluate. Instead, we employ a com-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "posite likelihood because it makes posterior inference tractable when coupled with Polya-Gamma ´", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 165, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 165, + 651 + ], + "score": 1.0, + "content": "augmentation.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 594, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Prior work on Bayesian inference with composite likelihoods has shown that the composite posterior", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "is consistent under fairly general conditions for correctly specified models (Miller, 2019) but can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "produce overly concentrated posteriors (Pauli et al., 2011; Ribatet et al., 2012) since each component", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "likelihood event is treated as independent when in reality there may be significant dependencies.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Nevertheless, we show in Section 5 that in practice our method exhibits competitive accuracy and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "strong calibration relative to baseline few-shot learning algorithms. We leave further theoretical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 407, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 407, + 733 + ], + "score": 1.0, + "content": "analysis of the OVE composite posterior and its properties for future work.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 654, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "Compared to choices of likelihoods used by previous approaches, there are several reasons to prefer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "OVE. Relative to the Gaussian augmentation approach of Girolami & Rogers (2006), Polya-Gamma ´", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 418, + 117 + ], + "score": 1.0, + "content": "augmentation has the benefit of fast mixing and the ability of a single value of", + "type": "text" + }, + { + "bbox": [ + 418, + 106, + 427, + 114 + ], + "score": 0.77, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "to capture much of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the marginal distribution over function values1. The stick-breaking construction of Linderman et al.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "(2015) induces a dependence on the ordering of classes, which leads to undesirable asymmetry.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Finally, the logistic-softmax likelihood of Galy-Fajou et al. (2020) requires three augmentations and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "careful learning of the mean function to avoid a priori underconfidence (see Section F.1 for more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 141, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 141, + 172 + ], + "score": 1.0, + "content": "details).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 183, + 329, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 183, + 329, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 329, + 196 + ], + "score": 1.0, + "content": "4.3 POSTERIOR INFERENCE VIA GIBBS SAMPLING", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 504, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "score": 1.0, + "content": "We now describe how we perform tractable posterior inference in our model with Gibbs sampling.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 182, + 230 + ], + "score": 1.0, + "content": "Define the matrix", + "type": "text" + }, + { + "bbox": [ + 182, + 216, + 285, + 228 + ], + "score": 0.54, + "content": "\\mathbf { A } \\ { \\overset { \\Delta } { = } } \\ \\operatorname { O V E - M A T R I X } ( \\mathbf { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 216, + 320, + 230 + ], + "score": 1.0, + "content": "to be a", + "type": "text" + }, + { + "bbox": [ + 320, + 217, + 369, + 227 + ], + "score": 0.9, + "content": "C N \\times C N", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 216, + 476, + 230 + ], + "score": 1.0, + "content": "sparse block matrix with", + "type": "text" + }, + { + "bbox": [ + 476, + 217, + 485, + 227 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "row", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 163, + 241 + ], + "score": 1.0, + "content": "partitions and", + "type": "text" + }, + { + "bbox": [ + 163, + 228, + 172, + 237 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 227, + 295, + 241 + ], + "score": 1.0, + "content": "column partitions. 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Details may be found in Section B.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 105, + 505, + 467, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 469, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 469, + 518 + ], + "score": 1.0, + "content": "4.4 LEARNING COVARIANCE HYPERPARAMETERS FOR FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We now describe how we apply OVE Polya-Gamma augmented GPs to few-shot classification. ´", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 484, + 549 + ], + "score": 1.0, + "content": "We assume the standard episodic few-shot setup in which one observes a labeled support set", + "type": "text" + }, + { + "bbox": [ + 484, + 537, + 505, + 547 + ], + "score": 0.85, + "content": "{ \\boldsymbol { s } } =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 107, + 547, + 137, + 560 + ], + "score": 0.9, + "content": "( \\mathbf { X } , \\mathbf { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 547, + 358, + 560 + ], + "score": 1.0, + "content": ". 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By utilizing the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "matrix inversion lemma and Gaussian sampling techniques summarized in (Doucet, 2010), this can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 371, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 187, + 493 + ], + "score": 1.0, + "content": "be brought down to", + "type": "text" + }, + { + "bbox": [ + 187, + 480, + 225, + 493 + ], + "score": 0.92, + "content": "\\mathcal { O } ( C N ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 481, + 371, + 493 + ], + "score": 1.0, + "content": ". Details may be found in Section B.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 446, + 505, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 505, + 467, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 469, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 469, + 518 + ], + "score": 1.0, + "content": "4.4 LEARNING COVARIANCE HYPERPARAMETERS FOR FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We now describe how we apply OVE Polya-Gamma augmented GPs to few-shot classification. ´", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 484, + 549 + ], + "score": 1.0, + "content": "We assume the standard episodic few-shot setup in which one observes a labeled support set", + "type": "text" + }, + { + "bbox": [ + 484, + 537, + 505, + 547 + ], + "score": 0.85, + "content": "{ \\boldsymbol { s } } =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 107, + 547, + 137, + 560 + ], + "score": 0.9, + "content": "( \\mathbf { X } , \\mathbf { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 547, + 358, + 560 + ], + "score": 1.0, + "content": ". Predictions must then be made for a query example", + "type": "text" + }, + { + "bbox": [ + 359, + 548, + 392, + 559 + ], + "score": 0.92, + "content": "\\left( \\mathbf { x } _ { * } , \\mathbf { y } _ { * } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 547, + 505, + 560 + ], + "score": 1.0, + "content": ". 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These could include traditional hyperparameters such as lengthscales or the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 581, + 433, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 433, + 594 + ], + "score": 1.0, + "content": "weights of a deep neural network as in deep kernel learning (Wilson et al., 2016).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 525, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "We consider two objectives for learning hyperparameters of the covariance function: the marginal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "likelihood (ML) and the predictive likelihood (PL). 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Additional results comparing the one-vs-each composite likelihood to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 426, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 426, + 140 + ], + "score": 1.0, + "content": "softmax, logistic softmax, and Gaussian likelihoods may be found in Section F.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 156 + ], + "score": 1.0, + "content": "One of our aims is to compare methods based on uncertainty quantification. We therefore developed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "new benchmark evaluations and tasks: few-shot calibration, robustness, and out-of-episode detec-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "score": 1.0, + "content": "tion. In order to empirically compare methods, we could not simply borrow the accuracy results", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "from other papers, but instead needed to train each of these baselines ourselves. For all baselines", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "except Bayesian MAML, ABML, and Logistic Softmax GP, we ran the code from (Patacchiola", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "et al., 2020) and verified that the accuracies matched closely to their reported results. We have made", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 324, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 324, + 223 + ], + "score": 1.0, + "content": "PyTorch code for our experiments publicly available2.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 236, + 250, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 252, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 252, + 249 + ], + "score": 1.0, + "content": "5.1 FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "For our few-shot classification experiments, we follow the training and evaluation protocol of Patac-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "score": 1.0, + "content": "chiola et al. (2020). We train both 1-shot and 5-shot versions of our model in four different settings:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "Caltech-UCSD Birds (CUB) (Wah et al., 2011), mini-Imagenet with the split proposed by Ravi &", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "score": 1.0, + "content": "Larochelle (2017), as well as two cross-domain transfer tasks. The first transfer task entails training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "on mini-ImageNet and testing on CUB, and the second measures transfer from Omniglot (Lake et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 325 + ], + "score": 1.0, + "content": "2011) to EMNIST (Cohen et al., 2017). Experimental details and an overview of the baselines we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "used can be found in Section G. Classification results are shown in Table 1 and 2. We find that our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "proposed Polya-Gamma OVE GPs yield strong classification results, outperforming the baselines in ´", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "score": 1.0, + "content": "five of the eight scenarios.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "table", + "bbox": [ + 105, + 420, + 512, + 599 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 366, + 505, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "Table 1: Average accuracy and standard deviation (percentage) on 5-way FSC. Baseline results", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "(through DKT) are from Patacchiola et al. (2020). Evaluation is performed on 3,000 randomly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 388, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 506, + 403 + ], + "score": 1.0, + "content": "generated test episodes. Standard deviation for the remaining methods are computed by averaging", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "over 5 batches of 600 episodes with different random seeds. The best results are highlighted in bold.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 420, + 512, + 599 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 420, + 512, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 512, + 599 + ], + "score": 0.984, + "html": "
MethodCUBmini-ImageNet
1-shot5-shot1-shot5-shot
FeatureTransfer46.19± 0.6468.40± 0.7939.51 ± 0.2360.51 ± 0.55
Baseline++61.75 ± 0.9578.51 ± 0.5947.15 ± 0.4966.18 ± 0.18
MatchingNet60.19 ± 1.0275.11 ± 0.3548.25 ± 0.6562.71 ± 0.44
ProtoNet52.52 ± 1.9075.93 ± 0.4644.19 ± 1.3064.07 ± 0.65
RelationNet62.52 ± 0.3478.22 ± 0.0748.76 ± 0.1764.20 ± 0.28
MAML56.11 ± 0.6974.84 ± 0.6245.39 ± 0.4961.58 ± 0.53
DKT + Cosine63.37 ± 0.1977.73 ± 0.2648.64 ± 0.4562.85 ± 0.37
BayesianMAML55.93 ± 0.7172.87 ± 0.2644.46 ± 0.3062.60 ± 0.25
Bayesian MAML(Chaser)53.93 ± 0.7271.16 ± 0.3243.74 ± 0.4659.23 ± 0.34
ABML48.80 ± 0.4070.91 ± 0.3240.88 ± 0.2558.19 ± 0.17
Logistic Softmax GP + Cosine (ML)60.23 ± 0.5474.58 ± 0.2546.75 ± 0.2059.93 ± 0.31
Logistic Softmax GP + Cosine (PL)60.07 ±0.2978.14 ± 0.0747.05 ± 0.2066.01 ± 0.25
OVE PG GP + Cosine (ML) [ours]63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
OVE PG GP +Cosine (PL) [ours]60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
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When used", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "in safety-critical applications such as medical diagnosis, it is important for a machine learning sys-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "tem to defer when there is not enough evidence to make a decision. Even in non-critical applications,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "precise uncertainty quantification helps practitioners in the few-shot setting determine when a class", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "has an adequate amount of labeled data or when more labels are required, and can facilitate active", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 700, + 145, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 145, + 715 + ], + "score": 1.0, + "content": "learning.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 722, + 379, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 380, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 380, + 734 + ], + "score": 1.0, + "content": "2https://github.com/jakesnell/ove-polya-gamma-gp", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "In this section, we present our results on few-shot classification both in terms of accuracy and un-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 505, + 129 + ], + "score": 1.0, + "content": "certainty quantification. Additional results comparing the one-vs-each composite likelihood to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 426, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 426, + 140 + ], + "score": 1.0, + "content": "softmax, logistic softmax, and Gaussian likelihoods may be found in Section F.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 106, + 505, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 156 + ], + "score": 1.0, + "content": "One of our aims is to compare methods based on uncertainty quantification. We therefore developed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "new benchmark evaluations and tasks: few-shot calibration, robustness, and out-of-episode detec-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "score": 1.0, + "content": "tion. In order to empirically compare methods, we could not simply borrow the accuracy results", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "from other papers, but instead needed to train each of these baselines ourselves. For all baselines", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "except Bayesian MAML, ABML, and Logistic Softmax GP, we ran the code from (Patacchiola", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "et al., 2020) and verified that the accuracies matched closely to their reported results. We have made", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 324, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 324, + 223 + ], + "score": 1.0, + "content": "PyTorch code for our experiments publicly available2.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 146, + 506, + 223 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 236, + 250, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 252, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 252, + 249 + ], + "score": 1.0, + "content": "5.1 FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "For our few-shot classification experiments, we follow the training and evaluation protocol of Patac-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "score": 1.0, + "content": "chiola et al. (2020). We train both 1-shot and 5-shot versions of our model in four different settings:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "Caltech-UCSD Birds (CUB) (Wah et al., 2011), mini-Imagenet with the split proposed by Ravi &", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "score": 1.0, + "content": "Larochelle (2017), as well as two cross-domain transfer tasks. The first transfer task entails training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "on mini-ImageNet and testing on CUB, and the second measures transfer from Omniglot (Lake et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 310, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 325 + ], + "score": 1.0, + "content": "2011) to EMNIST (Cohen et al., 2017). Experimental details and an overview of the baselines we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "used can be found in Section G. Classification results are shown in Table 1 and 2. We find that our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "proposed Polya-Gamma OVE GPs yield strong classification results, outperforming the baselines in ´", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 213, + 357 + ], + "score": 1.0, + "content": "five of the eight scenarios.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 255, + 506, + 357 + ] + }, + { + "type": "table", + "bbox": [ + 105, + 420, + 512, + 599 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 366, + 505, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "Table 1: Average accuracy and standard deviation (percentage) on 5-way FSC. Baseline results", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "(through DKT) are from Patacchiola et al. (2020). Evaluation is performed on 3,000 randomly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 388, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 506, + 403 + ], + "score": 1.0, + "content": "generated test episodes. Standard deviation for the remaining methods are computed by averaging", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "over 5 batches of 600 episodes with different random seeds. The best results are highlighted in bold.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 420, + 512, + 599 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 420, + 512, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 512, + 599 + ], + "score": 0.984, + "html": "
MethodCUBmini-ImageNet
1-shot5-shot1-shot5-shot
FeatureTransfer46.19± 0.6468.40± 0.7939.51 ± 0.2360.51 ± 0.55
Baseline++61.75 ± 0.9578.51 ± 0.5947.15 ± 0.4966.18 ± 0.18
MatchingNet60.19 ± 1.0275.11 ± 0.3548.25 ± 0.6562.71 ± 0.44
ProtoNet52.52 ± 1.9075.93 ± 0.4644.19 ± 1.3064.07 ± 0.65
RelationNet62.52 ± 0.3478.22 ± 0.0748.76 ± 0.1764.20 ± 0.28
MAML56.11 ± 0.6974.84 ± 0.6245.39 ± 0.4961.58 ± 0.53
DKT + Cosine63.37 ± 0.1977.73 ± 0.2648.64 ± 0.4562.85 ± 0.37
BayesianMAML55.93 ± 0.7172.87 ± 0.2644.46 ± 0.3062.60 ± 0.25
Bayesian MAML(Chaser)53.93 ± 0.7271.16 ± 0.3243.74 ± 0.4659.23 ± 0.34
ABML48.80 ± 0.4070.91 ± 0.3240.88 ± 0.2558.19 ± 0.17
Logistic Softmax GP + Cosine (ML)60.23 ± 0.5474.58 ± 0.2546.75 ± 0.2059.93 ± 0.31
Logistic Softmax GP + Cosine (PL)60.07 ±0.2978.14 ± 0.0747.05 ± 0.2066.01 ± 0.25
OVE PG GP + Cosine (ML) [ours]63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
OVE PG GP +Cosine (PL) [ours]60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
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MethodOmniglot→EMNISTmini-ImageNet→CUB
1-shot5-shot1-shot5-shot
Feature Transfer64.22 ±1.2486.10± 0.8432.77 ± 0.3550.34± 0.27
Baseline++56.84 ± 0.9180.01 ± 0.9239.19 ± 0.1257.31 ± 0.11
MatchingNet75.01 ± 2.0987.41 ± 1.7936.98 ± 0.0650.72 ± 0.36
ProtoNet72.04 ± 0.8287.22 ± 1.0133.27 ± 1.0952.16 ± 0.17
RelationNet75.62 ± 1.0087.84 ± 0.2737.13 ± 0.2051.76 ± 1.48
MAML72.68 ± 1.8583.54 ± 1.7934.01 ±1.2548.83 ± 0.62
DKT + Cosine73.06 ± 2.3688.10 ± 0.7840.22 ± 0.5455.65 ± 0.05
Bayesian MAML63.94 ± 0.4765.26 ± 0.3033.52 ± 0.3651.35 ± 0.16
Bayesian MAML (Chaser)55.04 ± 0.3454.19 ± 0.3236.22 ± 0.5051.53 ± 0.43
ABML73.89 ± 0.2487.28 ± 0.4031.51 ± 0.3247.80 ± 0.51
Logistic Softmax GP + Cosine (ML)62.91 ± 0.4983.80 ± 0.1336.41 ± 0.1850.33 ± 0.13
Logistic Softmax GP + Cosine (PL)70.70 ±0.3686.59 ± 0.1536.73 ± 0.2656.70 ± 0.31
OVE PG GP + Cosine (ML) [ours]68.43± 0.6786.22 ±0.2039.66± 0.1855.71 ± 0.31
OVE PG GP + Cosine (PL) [ours]77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
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MethodOmniglot→EMNISTmini-ImageNet→CUB
1-shot5-shot1-shot5-shot
Feature Transfer64.22 ±1.2486.10± 0.8432.77 ± 0.3550.34± 0.27
Baseline++56.84 ± 0.9180.01 ± 0.9239.19 ± 0.1257.31 ± 0.11
MatchingNet75.01 ± 2.0987.41 ± 1.7936.98 ± 0.0650.72 ± 0.36
ProtoNet72.04 ± 0.8287.22 ± 1.0133.27 ± 1.0952.16 ± 0.17
RelationNet75.62 ± 1.0087.84 ± 0.2737.13 ± 0.2051.76 ± 1.48
MAML72.68 ± 1.8583.54 ± 1.7934.01 ±1.2548.83 ± 0.62
DKT + Cosine73.06 ± 2.3688.10 ± 0.7840.22 ± 0.5455.65 ± 0.05
Bayesian MAML63.94 ± 0.4765.26 ± 0.3033.52 ± 0.3651.35 ± 0.16
Bayesian MAML (Chaser)55.04 ± 0.3454.19 ± 0.3236.22 ± 0.5051.53 ± 0.43
ABML73.89 ± 0.2487.28 ± 0.4031.51 ± 0.3247.80 ± 0.51
Logistic Softmax GP + Cosine (ML)62.91 ± 0.4983.80 ± 0.1336.41 ± 0.1850.33 ± 0.13
Logistic Softmax GP + Cosine (PL)70.70 ±0.3686.59 ± 0.1536.73 ± 0.2656.70 ± 0.31
OVE PG GP + Cosine (ML) [ours]68.43± 0.6786.22 ±0.2039.66± 0.1855.71 ± 0.31
OVE PG GP + Cosine (PL) [ours]77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
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\\mathbf { S } \\mathbf { P S } ^ { \\mathrm { ~ I ~ } } , \\mathrm { w i } } \\\\ & { } & { \\mathbf { D } = \\boldsymbol { \\Omega } + \\mathrm { d i a g } ( \\mathbf { d } ) , } \\\\ & { } & { \\mathbf { S } = \\left[ \\begin{array} { l l } { \\mathbf { Y } ^ { \\dagger } } & { \\mathbf { W } ^ { \\dagger } } \\end{array} \\right] , } \\\\ & { } & { \\mathbf { P } = \\left[ \\begin{array} { l l } { \\mathbf { 0 } _ { N } } & { \\mathbf { I } _ { N } } \\\\ { \\mathbf { I } _ { N } } & { \\mathbf { 0 } _ { N } } \\end{array} \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bc7e2ba7a37cd4c59cb8cf9efa37617deaabd352379951c2c1af81934a5a8f6e.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 261, + 409, + 353, + 444.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 261, + 444.5, + 353, + 480.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 215, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 216, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 216, + 495 + ], + "score": 1.0, + "content": "Substituting (31) into (30):", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 279, + 527 + ], + "lines": [ + { + "bbox": [ + 104, + 513, + 173, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 173, + 528 + ], + "score": 1.0, + "content": "Now we expand", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 494, + 429, + 546 + ], + "lines": [ + { + "bbox": [ + 186, + 494, + 422, + 546 + ], + "spans": [ + { + "bbox": [ + 186, + 494, + 422, + 546 + ], + "score": 0.25, + "content": "\\begin{array} { r l } & { \\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { v } - \\mathbf { K } \\mathbf { A } ^ { \\top } \\boldsymbol { \\Omega } \\mathbf { A } ( \\mathbf { K } ^ { - 1 } + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } \\mathbf { v } . } \\\\ & { . . . } \\\\ & { . . . + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } : } \\\\ & { \\mathbf { K } ^ { - 1 } + \\mathbf { D } - \\mathbf { S } \\mathbf { P } \\mathbf { S } ^ { \\top } ) ^ { - 1 } = \\mathbf { E } - \\mathbf { E } \\mathbf { S } ( \\mathbf { S } ^ { \\top } \\mathbf { E } \\mathbf { S } - \\mathbf { P } ^ { - 1 } ) ^ { - 1 } \\mathbf { S } ^ { \\top } \\mathbf { F } } \\end{array}", + "type": "interline_equation", + "image_path": "1e3d88b3a4ff2fa8a08b6a72ac8ee7bf36974f53aa1cbe3a77e76cccf8371a4d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 181, + 494, + 429, + 511.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 181, + 511.3333333333333, + 429, + 528.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 181, + 528.6666666666666, + 429, + 546.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 506, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 133, + 561 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 547, + 311, + 560 + ], + "score": 0.9, + "content": "\\mathbf { E } = ( \\mathbf { K } ^ { - 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\\mathbf { S } \\mathbf { P S } ^ { \\mathrm { ~ I ~ } } , \\mathrm { w i } } \\\\ & { } & { \\mathbf { D } = \\boldsymbol { \\Omega } + \\mathrm { d i a g } ( \\mathbf { d } ) , } \\\\ & { } & { \\mathbf { S } = \\left[ \\begin{array} { l l } { \\mathbf { Y } ^ { \\dagger } } & { \\mathbf { W } ^ { \\dagger } } \\end{array} \\right] , } \\\\ & { } & { \\mathbf { P } = \\left[ \\begin{array} { l l } { \\mathbf { 0 } _ { N } } & { \\mathbf { I } _ { N } } \\\\ { \\mathbf { I } _ { N } } & { \\mathbf { 0 } _ { N } } \\end{array} \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bc7e2ba7a37cd4c59cb8cf9efa37617deaabd352379951c2c1af81934a5a8f6e.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 261, + 409, + 353, + 444.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 261, + 444.5, + 353, + 480.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 215, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 216, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 216, + 495 + ], + "score": 1.0, + "content": "Substituting (31) into (30):", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 480, + 216, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 279, + 527 + ], + "lines": [ + { + "bbox": [ + 104, + 513, + 173, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 173, + 528 + ], + "score": 1.0, + "content": "Now we expand", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 513, + 173, + 528 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 494, + 429, + 546 + ], + "lines": [ + { + "bbox": [ + 186, + 494, + 422, + 546 + ], + "spans": [ + { + "bbox": [ + 186, + 494, + 422, + 546 + ], + "score": 0.25, + "content": "\\begin{array} { r l } & { \\bar { \\mathbf { f } } = \\mathbf { f } _ { 0 } + \\mathbf { K } \\mathbf { v } - 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Input: Objective L ∈ {LML,LPL}, Task distribution T, number of parallel Gibbs chains M, number of steps T,learning rate n. Initialize hyperparameters 0 randomly.
repeat Sample S= (X,Y),Q= (X*,Y*)~ T
if L= LML then X←XUX*,Y←YUY*
end if
A ← OVE-MATRIX(Y) for m = 1 to M do
~ pe(f|X)
fort=1 toTdo
（m）~PG(1b（m))
f（m） ~pe(f|x,Y,ω(m）)
end for end for
if L= LML then
0←0+∑m-1 Velogpe(Y|x,(m)）
else 0←0+∑m=1∑,∀θlogpe(y*j｜x*j,S,ω(m）)
end if until convergence
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We use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 421, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 421, + 117 + ], + "score": 1.0, + "content": "Fisher’s identity (Douc et al., 2014) to derive the following gradient estimator:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 118, + 470, + 153 + ], + "lines": [ + { + "bbox": [ + 123, + 118, + 470, + 153 + ], + "spans": [ + { + "bbox": [ + 123, + 118, + 470, + 153 + ], + "score": 0.94, + "content": "\\nabla _ { \\theta } L _ { \\mathrm { M L } } = \\int p _ { \\theta } ( \\boldsymbol { \\omega } | \\mathbf { X } , \\mathbf { Y } ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { Y } | \\boldsymbol { \\omega } , \\mathbf { X } ) d \\boldsymbol { \\omega } \\approx \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { Y } | \\mathbf { X } , \\boldsymbol { \\omega } ^ { ( m ) } ) ,", + "type": "interline_equation", + "image_path": "c010b0f3619da6af4f4f833fc08d9d1a9be0c044c1a2d97902520caa960c1762.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 123, + 118, + 470, + 129.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 123, + 129.66666666666666, + 470, + 141.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 123, + 141.33333333333331, + 470, + 152.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 134, + 170 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 154, + 198, + 167 + ], + "score": 0.92, + "content": "\\omega ^ { ( 1 ) } , \\ldots , \\omega ^ { ( M ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 153, + 506, + 170 + ], + "score": 1.0, + "content": "are samples from the posterior Gibbs chain. As suggested by Patacchiola", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "et al. (2020), who applied GPs to FSC via least-squares classification, we merge the support and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 468, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 468, + 191 + ], + "score": 1.0, + "content": "query sets during learning to take full advantage of the available data within each episode.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 153, + 506, + 191 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 194, + 450, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 450, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 424, + 208 + ], + "score": 1.0, + "content": "Predictive Likelihood (PL). The log predictive likelihood for a query example", + "type": "text" + }, + { + "bbox": [ + 425, + 197, + 436, + 206 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 193, + 450, + 208 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 193, + 450, + 208 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 208, + 468, + 234 + ], + "lines": [ + { + "bbox": [ + 126, + 208, + 468, + 234 + ], + "spans": [ + { + "bbox": [ + 126, + 208, + 468, + 234 + ], + "score": 0.94, + "content": "L _ { \\mathrm { P L } } ( \\theta ; \\mathbf { X } , \\mathbf { Y } , \\mathbf { x } _ { * } , \\mathbf { y } _ { * } ) \\triangleq \\log p \\varrho ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } ) = \\log \\int p ( \\omega ) p \\varrho ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ) d \\omega .", + "type": "interline_equation", + "image_path": "2ab47f136831924e8442dfd500f339443a9d38d558ded9003cb2ffa5a0c95410.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 126, + 208, + 468, + 234 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 235, + 434, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 434, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 422, + 249 + ], + "score": 1.0, + "content": "We use an approximate gradient estimator again based on posterior samples of", + "type": "text" + }, + { + "bbox": [ + 422, + 237, + 430, + 245 + ], + "score": 0.75, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 234, + 434, + 249 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 234, + 434, + 249 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 249, + 501, + 284 + ], + "lines": [ + { + "bbox": [ + 111, + 249, + 501, + 284 + ], + "spans": [ + { + "bbox": [ + 111, + 249, + 501, + 284 + ], + "score": 0.92, + "content": "\\nabla _ { \\theta } L _ { \\mathrm { P L } } \\approx \\int p _ { \\theta } ( \\omega | \\mathbf { X } , \\mathbf { Y } ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } ) d \\omega \\approx \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\nabla _ { \\theta } \\log p _ { \\theta } ( \\mathbf { y } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ^ { ( m ) } ) .", + "type": "interline_equation", + "image_path": "a8d56b1d20bea67114cc6d9f8b49907b04e92eec68658e33ecbce2725866d9e3.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 111, + 249, + 501, + 260.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 260.6666666666667, + 501, + 272.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 111, + 272.33333333333337, + 501, + 284.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 486, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 487, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 487, + 305 + ], + "score": 1.0, + "content": "We note that this is not an unbiased estimator of the gradient, but find it works well in practice.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 290, + 487, + 305 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 319, + 252, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 253, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 253, + 334 + ], + "score": 1.0, + "content": "D LEARNING ALGORITHM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 343, + 497, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 499, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 499, + 356 + ], + "score": 1.0, + "content": "Our learning algorithm for both marginal and predictive likelihood is summarized in Algorithm 1.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 342, + 499, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 365, + 329, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 330, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 330, + 381 + ], + "score": 1.0, + "content": "Algorithm 1 One-vs-Each Polya-Gamma GP Learning ´", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "table", + "bbox": [ + 107, + 380, + 509, + 652 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 380, + 509, + 652 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 380, + 509, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 380, + 509, + 652 + ], + "score": 0.778, + "html": "
Input: Objective L ∈ {LML,LPL}, Task distribution T, number of parallel Gibbs chains M, number of steps T,learning rate n. Initialize hyperparameters 0 randomly.
repeat Sample S= (X,Y),Q= (X*,Y*)~ T
if L= LML then X←XUX*,Y←YUY*
end if
A ← OVE-MATRIX(Y) for m = 1 to M do
~ pe(f|X)
fort=1 toTdo
（m）~PG(1b（m))
f（m） ~pe(f|x,Y,ω(m）)
end for end for
if L= LML then
0←0+∑m-1 Velogpe(Y|x,(m)）
else 0←0+∑m=1∑,∀θlogpe(y*j｜x*j,S,ω(m）)
end if until convergence
", + "type": "table", + "image_path": "da32a9e5f16d5fbb17e4c47614b931727a8986dfb8d1b2ebc6b16aa87d4d39b1.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 107, + 380, + 509, + 470.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 107, + 470.6666666666667, + 509, + 561.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 107, + 561.3333333333334, + 509, + 652.0 + ], + "spans": [], + "index": 21 + } + ] + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 106, + 671, + 332, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 333, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 333, + 686 + ], + "score": 1.0, + "content": "E POSTERIOR PREDICTIVE DISTRIBUTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 695, + 432, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 695, + 432, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 334, + 710 + ], + "score": 1.0, + "content": "The posterior predictive distribution for a query example", + "type": "text" + }, + { + "bbox": [ + 335, + 698, + 347, + 707 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 695, + 410, + 710 + ], + "score": 1.0, + "content": "conditioned on", + "type": "text" + }, + { + "bbox": [ + 410, + 699, + 419, + 706 + ], + "score": 0.73, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 695, + 432, + 710 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 695, + 432, + 710 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 710, + 418, + 736 + ], + "lines": [ + { + "bbox": [ + 192, + 710, + 418, + 736 + ], + "spans": [ + { + "bbox": [ + 192, + 710, + 418, + 736 + ], + "score": 0.91, + "content": "p ( { \\bf y } _ { * } | { \\bf x } _ { * } , { \\bf X } , { \\bf Y } , \\omega ) = \\int p ( { \\bf y } _ { * } | { \\bf f } _ { * } ) p ( { \\bf f } _ { * } | { \\bf x } _ { * } , { \\bf X } , { \\bf Y } , \\omega ) d { \\bf f } _ { * } ,", + "type": "interline_equation", + "image_path": "ec2400d9384de6297c9651950889dd52777f17ffa860bd7083013a73b47b86e1.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 192, + 710, + 418, + 736 + ], + "spans": [], + "index": 24 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 134, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 144, + 93 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 81, + 412, + 96 + ], + "score": 1.0, + "content": "are the query example’s logits. 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The predictive distribution becomes:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 204, + 160, + 382, + 173 + ], + "lines": [ + { + "bbox": [ + 204, + 159, + 381, + 174 + ], + "spans": [ + { + "bbox": [ + 204, + 160, + 349, + 173 + ], + "score": 0.59, + "content": "p ( \\mathbf { f } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega ) = \\mathcal { N } ( \\mathbf { f } _ { * } | \\mu _ { * } , \\Sigma _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 159, + 381, + 174 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 168, + 407, + 206 + ], + "lines": [ + { + "bbox": [ + 203, + 168, + 407, + 206 + ], + "spans": [ + { + "bbox": [ + 203, + 168, + 407, + 206 + ], + "score": 0.46, + "content": "\\begin{array} { r l } & { \\pmb { \\jmath } ^ { \\top } ( \\mathbf { A } \\mathbf { x } ) ^ { \\top } \\mathbf { \\Phi } ^ { \\mathbf { a } \\mathbf { x } } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\Omega ^ { - 1 } \\mathbf { \\Lambda } ^ { \\mathrm { a n d } } } \\\\ & { \\pmb { \\mu } _ { * } = ( \\mathbf { A } \\mathbf { K } _ { * } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\Omega ^ { - 1 } \\mathbf { \\Lambda } ^ { \\mathrm { a n d } } } \\\\ & { \\pmb { \\Sigma } _ { * } = \\mathbf { K } _ { * * } - ( \\mathbf { A } \\mathbf { K } _ { * } ) ^ { \\top } ( \\mathbf { A } \\mathbf { K } \\mathbf { A } ^ { \\top } + \\Omega ^ { - 1 } ) ^ { - 1 } \\mathbf { A } \\mathbf { K } _ { * } . } \\end{array}", + "type": "interline_equation", + "image_path": "72bda9075e73e9b7e79d735ecd070f9df1c9880468310c7179fb1bd7cf108af3.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 168, + 407, + 187.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 203, + 187.0, + 407, + 206.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 502, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 129, + 222 + ], + "score": 1.0, + "content": "With", + "type": "text" + }, + { + "bbox": [ + 129, + 207, + 203, + 220 + ], + "score": 0.92, + "content": "p ( \\mathbf { f } _ { * } | \\mathbf { x } _ { * } , \\mathbf { X } , \\mathbf { Y } , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 207, + 505, + 222 + ], + "score": 1.0, + "content": "in hand, the integral in (42) can easily be computed numerically for each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 504, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 128, + 231 + ], + "score": 1.0, + "content": "class", + "type": "text" + }, + { + "bbox": [ + 128, + 221, + 134, + 228 + ], + "score": 0.71, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 218, + 392, + 231 + ], + "score": 1.0, + "content": "by forming the corresponding OVE linear transformation matrix", + "type": "text" + }, + { + "bbox": [ + 392, + 219, + 406, + 228 + ], + "score": 0.84, + "content": "\\mathbf { A } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 218, + 504, + 231 + ], + "score": 1.0, + "content": "and then performing 1D", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 425, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 315, + 242 + ], + "score": 1.0, + "content": "Gaussian-Hermite quadrature on each dimension of", + "type": "text" + }, + { + "bbox": [ + 315, + 230, + 421, + 242 + ], + "score": 0.86, + "content": "\\mathcal { N } ( \\psi _ { * } ^ { c } | \\mathbf { A } ^ { c } \\pmb { \\mu } ^ { * } , \\mathbf { A } ^ { c } \\pmb { \\Sigma } _ { * } \\mathbf { A } ^ { c \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 229, + 425, + 242 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 106, + 256, + 344, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 255, + 345, + 272 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 345, + 272 + ], + "score": 1.0, + "content": "F DETAILED COMPARISON OF LIKELIHOODS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 104, + 279, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 506, + 294 + ], + "score": 1.0, + "content": "In this section we seek to better understand the behaviors of the softmax, OVE, logistic softmax,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "and Gaussian likelihoods for classification. For convenience, we summarize the forms of these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 303, + 198, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 198, + 313 + ], + "score": 1.0, + "content": "likelihoods in Table 3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "table", + "bbox": [ + 158, + 341, + 452, + 485 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 226, + 321, + 384, + 333 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 320, + 385, + 333 + ], + "spans": [ + { + "bbox": [ + 225, + 320, + 385, + 333 + ], + "score": 1.0, + "content": "Table 3: Likelihoods used in Section F.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 158, + 341, + 452, + 485 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 158, + 341, + 452, + 485 + ], + "spans": [ + { + "bbox": [ + 158, + 341, + 452, + 485 + ], + "score": 0.984, + "html": "
LikelihoodC(f|y =c)
Softmaxexp(fe) ∑c exp(fe)
GaussianIIN(2·1[c' =c]-1|μ= fe,σ² =1) c'
Logistic Softmax (LSM)g(fc) ∑eσ(fe)
One-vs-Each (OVE)IIσ(fe- fe) c≠c
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The results are shown in Figure 4. Logistic softmax is a priori underconfident: it puts", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "little probability mass on confidence above 0.4. This may be due to the use of the sigmoid function", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 234, + 581 + ], + "score": 1.0, + "content": "which squashes large values of", + "type": "text" + }, + { + "bbox": [ + 234, + 569, + 241, + 580 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 568, + 506, + 581 + ], + "score": 1.0, + "content": ". Gaussian likelihood and OVE are a priori overconfident in that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "they put a large amount of probability mass on confident outputs. Note that this is not a complete", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "explanation, because GP hyperparameters such as the prior mean or Gaussian likelihood variance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "may be able to compensate for these imperfections to some degree. Indeed, we found it helpful to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 612, + 454, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 454, + 624 + ], + "score": 1.0, + "content": "learn a constant mean for the logistic softmax likelihood, as mentioned in Section G.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 257, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 259, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 259, + 649 + ], + "score": 1.0, + "content": "F.2 LIKELIHOOD VISUALIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "In order to visualize the various likelihoods under consideration, we consider a trivial classification", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 414, + 681 + ], + "score": 1.0, + "content": "task with a single observed example. 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We place the following prior on", + "type": "text" + }, + { + "bbox": [ + 406, + 679, + 476, + 693 + ], + "score": 0.92, + "content": "\\mathbf { f } = \\left( f _ { 1 } , f _ { 2 } , f _ { 3 } \\right) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 680, + 481, + 694 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 695, + 409, + 729 + ], + "lines": [ + { + "bbox": [ + 201, + 695, + 409, + 729 + ], + "spans": [ + { + "bbox": [ + 201, + 695, + 409, + 729 + ], + "score": 0.93, + "content": "\\begin{array} { r } { p ( \\mathbf { f } ) = \\mathcal { N } \\left( \\mathbf { f } \\bigg | \\mu = \\left[ \\begin{array} { l } { 0 } \\\\ { 0 } \\\\ { 0 } \\end{array} \\right] , \\Sigma = \\left[ \\begin{array} { l l l } { 1 } & { 0 } & { 0 } \\\\ { 0 } & { 1 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right] \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "8257059dafd89d33d849e0c209b7df0705d90cf922c4a63beb4c1d0767744a8e.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 695, + 409, + 712.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 201, + 712.0, + 409, + 729.0 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 134, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 144, + 93 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 81, + 412, + 96 + ], + "score": 1.0, + "content": "are the query example’s logits. 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For convenience, we summarize the forms of these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 303, + 198, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 198, + 313 + ], + "score": 1.0, + "content": "likelihoods in Table 3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 279, + 506, + 313 + ] + }, + { + "type": "table", + "bbox": [ + 158, + 341, + 452, + 485 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 226, + 321, + 384, + 333 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 320, + 385, + 333 + ], + "spans": [ + { + "bbox": [ + 225, + 320, + 385, + 333 + ], + "score": 1.0, + "content": "Table 3: Likelihoods used in Section F.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 158, + 341, + 452, + 485 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 158, + 341, + 452, + 485 + ], + "spans": [ + { + "bbox": [ + 158, + 341, + 452, + 485 + ], + "score": 0.984, + "html": "
LikelihoodC(f|y =c)
Softmaxexp(fe) ∑c exp(fe)
GaussianIIN(2·1[c' =c]-1|μ= fe,σ² =1) c'
Logistic Softmax (LSM)g(fc) ∑eσ(fe)
One-vs-Each (OVE)IIσ(fe- fe) c≠c
", + "type": "table", + "image_path": "182a36a68b14416ccd8a15ef39c0fd4a35a36a49885023dde3658ac490d4b56a.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 158, + 341, + 452, + 389.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 158, + 389.0, + 452, + 437.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 158, + 437.0, + 452, + 485.0 + ], + "spans": [], + "index": 19 + } + ] + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 262, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 263, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 263, + 516 + ], + "score": 1.0, + "content": "F.1 HISTOGRAM OF CONFIDENCES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 506, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 207, + 538 + ], + "score": 1.0, + "content": "We sampled logits from", + "type": "text" + }, + { + "bbox": [ + 207, + 524, + 264, + 536 + ], + "score": 0.92, + "content": "f _ { c } \\sim \\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "and plotted a histogram and kernel density estimate of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 233, + 550 + ], + "score": 1.0, + "content": "maximum output probability m", + "type": "text" + }, + { + "bbox": [ + 234, + 536, + 295, + 547 + ], + "score": 0.77, + "content": "\\arg ( y = c | \\mathbf { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "for each of the likelihoods shown in Table 3, where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 547, + 137, + 557 + ], + "score": 0.89, + "content": "C = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 546, + 506, + 559 + ], + "score": 1.0, + "content": ". The results are shown in Figure 4. Logistic softmax is a priori underconfident: it puts", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "little probability mass on confidence above 0.4. This may be due to the use of the sigmoid function", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 234, + 581 + ], + "score": 1.0, + "content": "which squashes large values of", + "type": "text" + }, + { + "bbox": [ + 234, + 569, + 241, + 580 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 568, + 506, + 581 + ], + "score": 1.0, + "content": ". Gaussian likelihood and OVE are a priori overconfident in that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "they put a large amount of probability mass on confident outputs. Note that this is not a complete", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "explanation, because GP hyperparameters such as the prior mean or Gaussian likelihood variance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "may be able to compensate for these imperfections to some degree. Indeed, we found it helpful to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 612, + 454, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 454, + 624 + ], + "score": 1.0, + "content": "learn a constant mean for the logistic softmax likelihood, as mentioned in Section G.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 524, + 506, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 257, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 259, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 259, + 649 + ], + "score": 1.0, + "content": "F.2 LIKELIHOOD VISUALIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "In order to visualize the various likelihoods under consideration, we consider a trivial classification", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 414, + 681 + ], + "score": 1.0, + "content": "task with a single observed example. We assume that there are three classes", + "type": "text" + }, + { + "bbox": [ + 415, + 668, + 443, + 679 + ], + "score": 0.84, + "content": "C = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 668, + 505, + 681 + ], + "score": 1.0, + "content": ") and the single", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 679, + 481, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 680, + 243, + 694 + ], + "score": 1.0, + "content": "example belongs to the first class", + "type": "text" + }, + { + "bbox": [ + 244, + 681, + 270, + 693 + ], + "score": 0.87, + "content": "y = 1 \\mathrm { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 680, + 406, + 694 + ], + "score": 1.0, + "content": "). We place the following prior on", + "type": "text" + }, + { + "bbox": [ + 406, + 679, + 476, + 693 + ], + "score": 0.92, + "content": "\\mathbf { f } = \\left( f _ { 1 } , f _ { 2 } , f _ { 3 } \\right) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 680, + 481, + 694 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 657, + 505, + 694 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 695, + 409, + 729 + ], + "lines": [ + { + "bbox": [ + 201, + 695, + 409, + 729 + ], + "spans": [ + { + "bbox": [ + 201, + 695, + 409, + 729 + ], + "score": 0.93, + "content": "\\begin{array} { r } { p ( \\mathbf { f } ) = \\mathcal { N } \\left( \\mathbf { f } \\bigg | \\mu = \\left[ \\begin{array} { l } { 0 } \\\\ { 0 } \\\\ { 0 } \\end{array} \\right] , \\Sigma = \\left[ \\begin{array} { l l l } { 1 } & { 0 } & { 0 } \\\\ { 0 } & { 1 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right] \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "8257059dafd89d33d849e0c209b7df0705d90cf922c4a63beb4c1d0767744a8e.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 201, + 695, + 409, + 712.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 201, + 712.0, + 409, + 729.0 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 169, + 84, + 440, + 246 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 169, + 84, + 440, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 169, + 84, + 440, + 246 + ], + "spans": [ + { + "bbox": [ + 169, + 84, + 440, + 246 + ], + "score": 0.963, + "type": "image", + "image_path": "90f74595999ffa71f93112e5993b98c03df1ad7c6c88dcbaea08535a25ad34fe.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 169, + 84, + 440, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 169, + 138.0, + 440, + 192.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 169, + 192.0, + 440, + 246.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 257, + 505, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "Figure 4: Histogram and kernel density estimate of confidence for randomly generated function", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 141, + 282 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 269, + 196, + 281 + ], + "score": 0.91, + "content": "f _ { c } \\sim \\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 268, + 409, + 282 + ], + "score": 1.0, + "content": ". 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The likelihoods are plotted in Figure 5 and the corresponding posteriors are plotted", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 356, + 156, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 156, + 370 + ], + "score": 1.0, + "content": "in Figure 6.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 114, + 387, + 498, + 480 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 387, + 498, + 480 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 114, + 387, + 498, + 480 + ], + "spans": [ + { + "bbox": [ + 114, + 387, + 498, + 480 + ], + "score": 0.966, + "type": "image", + "image_path": "c475772f592b02bc778a43d7039536becbe66815b4c507f1cb35ac1a177e623e.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 114, + 387, + 498, + 418.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 114, + 418.0, + 498, + 449.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 114, + 449.0, + 498, + 480.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 488, + 505, + 533 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 174, + 501 + ], + "score": 1.0, + "content": "Figure 5: Plot of", + "type": "text" + }, + { + "bbox": [ + 174, + 488, + 223, + 500 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\mathbf { f } \\mid y = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 488, + 254, + 501 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 254, + 489, + 265, + 500 + ], + "score": 0.86, + "content": "f _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "is clamped to 0. The Gaussian likelihood penalizes configu-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 499, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 194, + 511 + ], + "score": 1.0, + "content": "rations far away from", + "type": "text" + }, + { + "bbox": [ + 194, + 500, + 270, + 511 + ], + "score": 0.92, + "content": "( f _ { 1 } , f _ { 2 } ) = ( 1 , - 1 ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 499, + 506, + 511 + ], + "score": 1.0, + "content": ". Logistic softmax is much flatter compared to softmax and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 510, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 472, + 523 + ], + "score": 1.0, + "content": "has visibly different contours. 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Logistic softmax tends to produce lower confidence predictions, while one-vs-each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 352, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 352, + 562 + ], + "score": 1.0, + "content": "produces larger regions of greater confidence than the others.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + } + ], + "index": 24.75 + }, + { + "type": "image", + "bbox": [ + 114, + 585, + 497, + 667 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 585, + 497, + 667 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 114, + 585, + 497, + 667 + ], + "spans": [ + { + "bbox": [ + 114, + 585, + 497, + 667 + ], + "score": 0.943, + "type": "image", + "image_path": "b1ac3ed75011a27c35369b9894f46e6eb1989c84e9f1d028b3593dfc778ba366.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 114, + 585, + 497, + 612.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 114, + 612.3333333333334, + 497, + 639.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 114, + 639.6666666666667, + 497, + 667.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 675, + 505, + 720 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "Figure 8: Comparison across likelihoods in terms of test predictive accuracy, Brier score, expected", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 687, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 698 + ], + "score": 1.0, + "content": "calibration error (computed with 10 bins), and ELBO. Results are averaged over 200 randomly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 710 + ], + "score": 1.0, + "content": "generated splits for each training set size (1, 2, 3, 4, 5, 10, 15, 20, 25, and 30 examples per class).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 288, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 182, + 720 + ], + "score": 1.0, + "content": "Error bars indicate", + "type": "text" + }, + { + "bbox": [ + 183, + 709, + 203, + 719 + ], + "score": 0.85, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 708, + 288, + 720 + ], + "score": 1.0, + "content": "confidence intervals.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + } + ], + "index": 31.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 315, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 317, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 317, + 96 + ], + "score": 1.0, + "content": "G FEW-SHOT EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "score": 1.0, + "content": "Here we provide more details about our experimental setup for our few-shot classification experi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 380, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 380, + 129 + ], + "score": 1.0, + "content": "ments, which are based on the protocol of (Patacchiola et al., 2020).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 178, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 179, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 179, + 155 + ], + "score": 1.0, + "content": "G.1 DATASETS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 503, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 504, + 177 + ], + "score": 1.0, + "content": "We used the four dataset scenarios described below. The first three are the same used by Chen et al.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 174, + 358, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 358, + 187 + ], + "score": 1.0, + "content": "(2019) and the final was proposed by Patacchiola et al. (2020).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 132, + 195, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 133, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 133, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "• CUB. Caltech-UCSD Birds (CUB) (Wah et al., 2011) consists of 200 classes and 11,788", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "images. A split of 100 training, 50 validation, and 50 test classes was used (Hilliard et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 218, + 244, + 230 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 244, + 230 + ], + "score": 1.0, + "content": "2018; Chen et al., 2019).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 140, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 140, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "mini-Imagenet. The mini-Imagenet dataset (Vinyals et al., 2016) consists of 100 classes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "with 600 images per class. We used the split proposed by Ravi & Larochelle (2017), which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 255, + 384, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 384, + 268 + ], + "score": 1.0, + "content": "has 64 classes for training, 16 for validation, and 20 for test.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 137, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 137, + 271, + 205, + 284 + ], + "score": 1.0, + "content": "mini-Imagenet", + "type": "text" + }, + { + "bbox": [ + 206, + 271, + 237, + 282 + ], + "score": 0.28, + "content": "\\scriptstyle \\mathbf { C U B }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 271, + 505, + 284 + ], + "score": 1.0, + "content": ". This cross-domain transfer scenario takes the training split of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 282, + 364, + 294 + ], + "spans": [ + { + "bbox": [ + 141, + 282, + 364, + 294 + ], + "score": 1.0, + "content": "mini-Imagenet and the validation & test splits of CUB.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 138, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 138, + 297, + 185, + 311 + ], + "score": 1.0, + "content": "Omniglot", + "type": "text" + }, + { + "bbox": [ + 185, + 299, + 198, + 308 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "EMNIST. We use the same setup as proposed by Patacchiola et al. (2020).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "Omniglot (Lake et al., 2011) consists of 1,623 classes, each with 20 examples, and is aug-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "mented by rotations of 90 degrees to create 6,492 classes, of which 4,114 are used for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "training. The EMNIST dataset (Cohen et al., 2017), consisting of 62 classes, is split into", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 340, + 267, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 267, + 355 + ], + "score": 1.0, + "content": "31 training and 31 test classes.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 305, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 305, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 305, + 380 + ], + "score": 1.0, + "content": "G.2 FEW-SHOT CLASSIFICATION BASELINES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 331, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 333, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 333, + 401 + ], + "score": 1.0, + "content": "Here we explain the few-shot baselines in greater detail.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 131, + 411, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 133, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 133, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "• Feature Transfer (Chen et al., 2019) involves first training an off-line classifier on the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 422, + 441, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 441, + 434 + ], + "score": 1.0, + "content": "training classes and then training a new classification layer on the episode.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 140, + 437, + 178, + 449 + ], + "score": 1.0, + "content": "Baseline", + "type": "text" + }, + { + "bbox": [ + 178, + 438, + 195, + 448 + ], + "score": 0.79, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "(Chen et al., 2019) is similar to Feature Transfer except it uses a cosine dis-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 448, + 357, + 461 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 357, + 461 + ], + "score": 1.0, + "content": "tance module prior to the softmax during fine-tuning.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 140, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 140, + 463, + 438, + 476 + ], + "score": 1.0, + "content": "Matching Networks (Vinyals et al., 2016) can be viewed as a soft form of", + "type": "text" + }, + { + "bbox": [ + 438, + 464, + 444, + 474 + ], + "score": 0.84, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "-nearest neigh-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "bors that computes attention and sums over the support examples to form a predictive dis-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 486, + 231, + 497 + ], + "spans": [ + { + "bbox": [ + 142, + 486, + 231, + 497 + ], + "score": 1.0, + "content": "tribution over classes.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 132, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 132, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "• Prototypical Networks (Snell et al., 2017) computes class means (prototypes) and forms", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "a predictive distribution based on Euclidean distance to the prototypes. It can be viewed as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 523, + 362, + 536 + ], + "spans": [ + { + "bbox": [ + 141, + 523, + 362, + 536 + ], + "score": 1.0, + "content": "a Gaussian classifier operating in an embedding space.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 139, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 139, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "MAML (Finn et al., 2017) performs one or a few steps of gradient descent on the support", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 141, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "set and then makes predictions on the query set, backpropagating through the gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "descent procedure. For this baseline, we simply quote the classification accuracy reported", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 572, + 259, + 583 + ], + "spans": [ + { + "bbox": [ + 141, + 572, + 259, + 583 + ], + "score": 1.0, + "content": "by (Patacchiola et al., 2020).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 139, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 139, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "RelationNet (Sung et al., 2018) rather than using a predefined distance metric as in Match-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "ing Networks or Prototypical Networks instead learns a deep distance metric as the output", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 141, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "of a neural network that accepts as input the latent representation of both examples. It is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 620, + 366, + 632 + ], + "spans": [ + { + "bbox": [ + 141, + 620, + 366, + 632 + ], + "score": 1.0, + "content": "trained to minimize squared error of output predictions.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 138, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 138, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) relies on least squares classifica-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 141, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "tion (Rifkin & Klautau, 2004) to maintain tractability of Gaussian process posterior infer-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 141, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "ence. In DKT, a separate binary classification task is formed for each class in one-vs-rest", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 141, + 668, + 253, + 680 + ], + "score": 1.0, + "content": "fashion by treating labels in", + "type": "text" + }, + { + "bbox": [ + 254, + 668, + 294, + 680 + ], + "score": 0.93, + "content": "\\{ - \\dot { 1 } , + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "as continuous targets. We include the results of DKT", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 142, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 142, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "with the cosine kernel as implemented by Patacchiola et al. (2020), which is parameterized", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 143, + 691, + 353, + 702 + ], + "spans": [ + { + "bbox": [ + 143, + 691, + 353, + 702 + ], + "score": 1.0, + "content": "slightly differently from the version we used in (47):", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 708, + 458, + 735 + ], + "lines": [ + { + "bbox": [ + 188, + 708, + 458, + 735 + ], + "spans": [ + { + "bbox": [ + 188, + 708, + 458, + 735 + ], + "score": 0.91, + "content": "k _ { \\mathrm { d k t } } ^ { \\mathrm { c o s } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha , \\nu ) = \\mathrm { s o f t p l u s } ( \\alpha ) \\cdot \\mathrm { s o f t p l u s } ( \\nu ) \\cdot \\frac { g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\| \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| } .", + "type": "interline_equation", + "image_path": "849d8a418b08d1777b6164b6ca5fc50c87c4cea0132ac4949d0b4156d2d97e23.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 188, + 708, + 458, + 717.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 188, + 717.0, + 458, + 726.0 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 188, + 726.0, + 458, + 735.0 + ], + "spans": [], + "index": 47 + } + ] + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 315, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 317, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 317, + 96 + ], + "score": 1.0, + "content": "G FEW-SHOT EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "score": 1.0, + "content": "Here we provide more details about our experimental setup for our few-shot classification experi-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 380, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 380, + 129 + ], + "score": 1.0, + "content": "ments, which are based on the protocol of (Patacchiola et al., 2020).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 106, + 504, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 143, + 178, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 179, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 179, + 155 + ], + "score": 1.0, + "content": "G.1 DATASETS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "list", + "bbox": [ + 107, + 164, + 503, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 504, + 177 + ], + "score": 1.0, + "content": "We used the four dataset scenarios described below. The first three are the same used by Chen et al.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 174, + 358, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 358, + 187 + ], + "score": 1.0, + "content": "(2019) and the final was proposed by Patacchiola et al. (2020).", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 133, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "• CUB. Caltech-UCSD Birds (CUB) (Wah et al., 2011) consists of 200 classes and 11,788", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "images. A split of 100 training, 50 validation, and 50 test classes was used (Hilliard et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 218, + 244, + 230 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 244, + 230 + ], + "score": 1.0, + "content": "2018; Chen et al., 2019).", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 140, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "mini-Imagenet. The mini-Imagenet dataset (Vinyals et al., 2016) consists of 100 classes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "with 600 images per class. We used the split proposed by Ravi & Larochelle (2017), which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 255, + 384, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 384, + 268 + ], + "score": 1.0, + "content": "has 64 classes for training, 16 for validation, and 20 for test.", + "type": "text" + } + ], + "index": 11, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 137, + 271, + 205, + 284 + ], + "score": 1.0, + "content": "mini-Imagenet", + "type": "text" + }, + { + "bbox": [ + 206, + 271, + 237, + 282 + ], + "score": 0.28, + "content": "\\scriptstyle \\mathbf { C U B }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 271, + 505, + 284 + ], + "score": 1.0, + "content": ". This cross-domain transfer scenario takes the training split of", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 282, + 364, + 294 + ], + "spans": [ + { + "bbox": [ + 141, + 282, + 364, + 294 + ], + "score": 1.0, + "content": "mini-Imagenet and the validation & test splits of CUB.", + "type": "text" + } + ], + "index": 13, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 138, + 297, + 185, + 311 + ], + "score": 1.0, + "content": "Omniglot", + "type": "text" + }, + { + "bbox": [ + 185, + 299, + 198, + 308 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "EMNIST. We use the same setup as proposed by Patacchiola et al. (2020).", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "Omniglot (Lake et al., 2011) consists of 1,623 classes, each with 20 examples, and is aug-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "mented by rotations of 90 degrees to create 6,492 classes, of which 4,114 are used for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "training. The EMNIST dataset (Cohen et al., 2017), consisting of 62 classes, is split into", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 340, + 267, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 267, + 355 + ], + "score": 1.0, + "content": "31 training and 31 test classes.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + } + ], + "index": 4.5, + "bbox_fs": [ + 106, + 163, + 504, + 187 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 195, + 505, + 353 + ], + "lines": [], + "index": 12, + "bbox_fs": [ + 133, + 195, + 505, + 355 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 305, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 305, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 305, + 380 + ], + "score": 1.0, + "content": "G.2 FEW-SHOT CLASSIFICATION BASELINES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 331, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 333, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 333, + 401 + ], + "score": 1.0, + "content": "Here we explain the few-shot baselines in greater detail.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 387, + 333, + 401 + ] + }, + { + "type": "list", + "bbox": [ + 131, + 411, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 133, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 133, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "• Feature Transfer (Chen et al., 2019) involves first training an off-line classifier on the", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 422, + 441, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 441, + 434 + ], + "score": 1.0, + "content": "training classes and then training a new classification layer on the episode.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 140, + 437, + 178, + 449 + ], + "score": 1.0, + "content": "Baseline", + "type": "text" + }, + { + "bbox": [ + 178, + 438, + 195, + 448 + ], + "score": 0.79, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "(Chen et al., 2019) is similar to Feature Transfer except it uses a cosine dis-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 448, + 357, + 461 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 357, + 461 + ], + "score": 1.0, + "content": "tance module prior to the softmax during fine-tuning.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 140, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 140, + 463, + 438, + 476 + ], + "score": 1.0, + "content": "Matching Networks (Vinyals et al., 2016) can be viewed as a soft form of", + "type": "text" + }, + { + "bbox": [ + 438, + 464, + 444, + 474 + ], + "score": 0.84, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "-nearest neigh-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "bors that computes attention and sums over the support examples to form a predictive dis-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 486, + 231, + 497 + ], + "spans": [ + { + "bbox": [ + 142, + 486, + 231, + 497 + ], + "score": 1.0, + "content": "tribution over classes.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 132, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "• Prototypical Networks (Snell et al., 2017) computes class means (prototypes) and forms", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "a predictive distribution based on Euclidean distance to the prototypes. It can be viewed as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 523, + 362, + 536 + ], + "spans": [ + { + "bbox": [ + 141, + 523, + 362, + 536 + ], + "score": 1.0, + "content": "a Gaussian classifier operating in an embedding space.", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 139, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "MAML (Finn et al., 2017) performs one or a few steps of gradient descent on the support", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 141, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "set and then makes predictions on the query set, backpropagating through the gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "descent procedure. For this baseline, we simply quote the classification accuracy reported", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 572, + 259, + 583 + ], + "spans": [ + { + "bbox": [ + 141, + 572, + 259, + 583 + ], + "score": 1.0, + "content": "by (Patacchiola et al., 2020).", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 139, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "RelationNet (Sung et al., 2018) rather than using a predefined distance metric as in Match-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 141, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "ing Networks or Prototypical Networks instead learns a deep distance metric as the output", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 141, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "of a neural network that accepts as input the latent representation of both examples. It is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 620, + 366, + 632 + ], + "spans": [ + { + "bbox": [ + 141, + 620, + 366, + 632 + ], + "score": 1.0, + "content": "trained to minimize squared error of output predictions.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + }, + { + "bbox": [ + 138, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 138, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "Deep Kernel Transfer (DKT) (Patacchiola et al., 2020) relies on least squares classifica-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 141, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "tion (Rifkin & Klautau, 2004) to maintain tractability of Gaussian process posterior infer-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 141, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "ence. 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We com-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "pare to both the non-chaser version, which optimizes cross-entropy of query predictions,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and the chaser version, which optimizes mean squared error between the approximate pos-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "terior on the support set and the approximate posterior on the merged support & query set.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "The non-chaser version is therefore related to predictive likelihood methods and the chaser", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 142, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "version is more analogous to the marginal likelihood methods. For the non-chaser version,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 142, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "we used 20 particles and 1 step of adaptation at both train and test time. For the chaser", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 171, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 141, + 171, + 506, + 182 + ], + "score": 1.0, + "content": "version, we also used 20 particles. At train time, the chaser took 1 step and the leader 1", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 141, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "additional step. At test time, we used 5 steps of adaptation. Due to the slow performance of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 141, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "this method, we followed the advice of Yoon et al. (2018) and only performed adaptation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 141, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "on the final layer of weights, which may help explain the drop in performance relative to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "MAML. The authors released Tensorflow code for regression only, so we reimplemented", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 225, + 311, + 237 + ], + "spans": [ + { + "bbox": [ + 141, + 225, + 311, + 237 + ], + "score": 1.0, + "content": "this baseline for classification in PyTorch.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 132, + 246, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 133, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 133, + 245, + 506, + 259 + ], + "score": 1.0, + "content": "• Amortized Bayesian Meta-Learning (ABML) (Ravi & Beatson, 2019) performs a few", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "steps of Bayes-by-backprop (Blundell et al., 2015) in order to infer a fully-factorized ap-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "proximate posterior over the weights. The authors did not release code and so we imple-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "mented our own version of ABML in PyTorch. We found the weighting on the inner and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 140, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "outer KL divergences to be important for achieving good performance. We took the neg-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "ative log likelihood to be mean cross entropy and used an inner KL weight of 0.01 and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 140, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 140, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "an outer KL weight of 0.001. These values were arrived upon by doing a small amount", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 318, + 335 + ], + "score": 1.0, + "content": "of hyperparameter tuning on the Omniglot", + "type": "text" + }, + { + "bbox": [ + 318, + 324, + 330, + 333 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 322, + 447, + 335 + ], + "score": 1.0, + "content": "EMNIST dataset. We used", + "type": "text" + }, + { + "bbox": [ + 447, + 323, + 486, + 333 + ], + "score": 0.89, + "content": "\\alpha = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 183, + 345 + ], + "score": 0.92, + "content": "\\beta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "for the Gamma prior over the weights. We only applied ABML to the weights", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "of the network; the biases were learned as point estimates. We used 4 steps of adaptation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 140, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 140, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "and took 5 samples when computing expectations (using any more than this did not fit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "into GPU memory). We used the local reparameterization trick (Kingma et al., 2015) and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "flipout (Wen et al., 2018) when computing expectations in order to reduce variance. In or-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "der to match the architecture used by Ravi & Beatson (2019), we trained this baseline with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "32 filters throughout the classification network. We trained each 1-shot ABML model for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 142, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "800 epochs and each 5-shot ABML model for 600 epochs as the learning had not converged", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 421, + 336, + 434 + ], + "spans": [ + { + "bbox": [ + 142, + 421, + 336, + 434 + ], + "score": 1.0, + "content": "within the epoch limits specified in Section G.3.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 134, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 132, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 132, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "• Logistic Softmax GP (Galy-Fajou et al., 2020) is the multi-class Gaussian process classi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "fication method that relies on the logistic softmax likelihood. Galy-Fajou et al. (2020) did", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 465, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 142, + 465, + 505, + 476 + ], + "score": 1.0, + "content": "not consider few-shot, but we use the same objectives described in Section 4.4 to adapt this", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 476, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 476, + 505, + 487 + ], + "score": 1.0, + "content": "method to FSC. In addition, we used the cosine kernel (see Section H for a description) that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "we found to work best with our OVE PG GPs. For this method, we found it important to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 497, + 497, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 497, + 510 + ], + "score": 1.0, + "content": "learn a constant mean function (rather than a zero mean) in order to improve calibration.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 217, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 219, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 219, + 542 + ], + "score": 1.0, + "content": "G.3 TRAINING DETAILS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "All methods employed the commonly-used Conv4 architecture (Vinyals et al., 2016) (see Table 4", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "for a detailed specification), except ABML which used 32 filters throughout. All of our experiments", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 369, + 587 + ], + "score": 1.0, + "content": "used the Adam (Kingma & Ba, 2015) optimizer with learning rate", + "type": "text" + }, + { + "bbox": [ + 369, + 573, + 390, + 585 + ], + "score": 0.9, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 572, + 505, + 587 + ], + "score": 1.0, + "content": ". During training, all models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "used epochs consisting of 100 randomly sampled episodes. A single gradient descent step on the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "encoder network and relevant hyperparameters is made per episode. All 1-shot models are trained", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "for 600 epochs and 5-shot models are trained for 400 epochs, except for ABML which was trained", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "for an extra 200 epochs. Each episode contained 5 classes (5-way) and 16 query examples. At test", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 643 + ], + "score": 1.0, + "content": "time, 15 query examples are used for each episode. Early stopping was performed by monitoring", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 415, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 415, + 653 + ], + "score": 1.0, + "content": "accuracy on the validation set. The validation set was not used for retraining.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "We train both marginal likelihood and predictive likelihood versions of our models. For Polya- ´", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Gamma sampling we use the PyPolyaGamma package ´ 3. During training, we use a single step of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 679, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 133, + 691 + ], + "score": 1.0, + "content": "Gibbs", + "type": "text" + }, + { + "bbox": [ + 134, + 679, + 157, + 690 + ], + "score": 0.76, + "content": "( T { = } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 679, + 275, + 691 + ], + "score": 1.0, + "content": "). For evaluation, we run until", + "type": "text" + }, + { + "bbox": [ + 276, + 679, + 307, + 689 + ], + "score": 0.89, + "content": "T = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 679, + 468, + 691 + ], + "score": 1.0, + "content": ". 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We com-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "pare to both the non-chaser version, which optimizes cross-entropy of query predictions,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and the chaser version, which optimizes mean squared error between the approximate pos-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "terior on the support set and the approximate posterior on the merged support & query set.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "The non-chaser version is therefore related to predictive likelihood methods and the chaser", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 142, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "version is more analogous to the marginal likelihood methods. For the non-chaser version,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 142, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "we used 20 particles and 1 step of adaptation at both train and test time. For the chaser", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 171, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 141, + 171, + 506, + 182 + ], + "score": 1.0, + "content": "version, we also used 20 particles. At train time, the chaser took 1 step and the leader 1", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 141, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "additional step. At test time, we used 5 steps of adaptation. Due to the slow performance of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 141, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "this method, we followed the advice of Yoon et al. (2018) and only performed adaptation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 141, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "on the final layer of weights, which may help explain the drop in performance relative to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "MAML. The authors released Tensorflow code for regression only, so we reimplemented", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 225, + 311, + 237 + ], + "spans": [ + { + "bbox": [ + 141, + 225, + 311, + 237 + ], + "score": 1.0, + "content": "this baseline for classification in PyTorch.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 131, + 82, + 506, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 246, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 133, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 133, + 245, + 506, + 259 + ], + "score": 1.0, + "content": "• Amortized Bayesian Meta-Learning (ABML) (Ravi & Beatson, 2019) performs a few", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "steps of Bayes-by-backprop (Blundell et al., 2015) in order to infer a fully-factorized ap-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "proximate posterior over the weights. The authors did not release code and so we imple-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "mented our own version of ABML in PyTorch. We found the weighting on the inner and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 140, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "outer KL divergences to be important for achieving good performance. We took the neg-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "ative log likelihood to be mean cross entropy and used an inner KL weight of 0.01 and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 140, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 140, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "an outer KL weight of 0.001. These values were arrived upon by doing a small amount", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 318, + 335 + ], + "score": 1.0, + "content": "of hyperparameter tuning on the Omniglot", + "type": "text" + }, + { + "bbox": [ + 318, + 324, + 330, + 333 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 322, + 447, + 335 + ], + "score": 1.0, + "content": "EMNIST dataset. We used", + "type": "text" + }, + { + "bbox": [ + 447, + 323, + 486, + 333 + ], + "score": 0.89, + "content": "\\alpha = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 183, + 345 + ], + "score": 0.92, + "content": "\\beta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "for the Gamma prior over the weights. We only applied ABML to the weights", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "of the network; the biases were learned as point estimates. We used 4 steps of adaptation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 140, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 140, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "and took 5 samples when computing expectations (using any more than this did not fit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "into GPU memory). We used the local reparameterization trick (Kingma et al., 2015) and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "flipout (Wen et al., 2018) when computing expectations in order to reduce variance. In or-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "der to match the architecture used by Ravi & Beatson (2019), we trained this baseline with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "32 filters throughout the classification network. We trained each 1-shot ABML model for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 142, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "800 epochs and each 5-shot ABML model for 600 epochs as the learning had not converged", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 421, + 336, + 434 + ], + "spans": [ + { + "bbox": [ + 142, + 421, + 336, + 434 + ], + "score": 1.0, + "content": "within the epoch limits specified in Section G.3.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 22, + "bbox_fs": [ + 133, + 245, + 506, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 442, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 132, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 132, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "• Logistic Softmax GP (Galy-Fajou et al., 2020) is the multi-class Gaussian process classi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "fication method that relies on the logistic softmax likelihood. Galy-Fajou et al. (2020) did", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 465, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 142, + 465, + 505, + 476 + ], + "score": 1.0, + "content": "not consider few-shot, but we use the same objectives described in Section 4.4 to adapt this", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 476, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 476, + 505, + 487 + ], + "score": 1.0, + "content": "method to FSC. In addition, we used the cosine kernel (see Section H for a description) that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "we found to work best with our OVE PG GPs. For this method, we found it important to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 497, + 497, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 497, + 510 + ], + "score": 1.0, + "content": "learn a constant mean function (rather than a zero mean) in order to improve calibration.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 132, + 442, + 506, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 528, + 217, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 219, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 219, + 542 + ], + "score": 1.0, + "content": "G.3 TRAINING DETAILS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "All methods employed the commonly-used Conv4 architecture (Vinyals et al., 2016) (see Table 4", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "for a detailed specification), except ABML which used 32 filters throughout. All of our experiments", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 369, + 587 + ], + "score": 1.0, + "content": "used the Adam (Kingma & Ba, 2015) optimizer with learning rate", + "type": "text" + }, + { + "bbox": [ + 369, + 573, + 390, + 585 + ], + "score": 0.9, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 572, + 505, + 587 + ], + "score": 1.0, + "content": ". During training, all models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "used epochs consisting of 100 randomly sampled episodes. A single gradient descent step on the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "encoder network and relevant hyperparameters is made per episode. All 1-shot models are trained", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "for 600 epochs and 5-shot models are trained for 400 epochs, except for ABML which was trained", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "for an extra 200 epochs. Each episode contained 5 classes (5-way) and 16 query examples. At test", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 643 + ], + "score": 1.0, + "content": "time, 15 query examples are used for each episode. Early stopping was performed by monitoring", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 639, + 415, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 415, + 653 + ], + "score": 1.0, + "content": "accuracy on the validation set. The validation set was not used for retraining.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 552, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "We train both marginal likelihood and predictive likelihood versions of our models. For Polya- ´", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "Gamma sampling we use the PyPolyaGamma package ´ 3. During training, we use a single step of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 679, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 133, + 691 + ], + "score": 1.0, + "content": "Gibbs", + "type": "text" + }, + { + "bbox": [ + 134, + 679, + 157, + 690 + ], + "score": 0.76, + "content": "( T { = } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 679, + 275, + 691 + ], + "score": 1.0, + "content": "). For evaluation, we run until", + "type": "text" + }, + { + "bbox": [ + 276, + 679, + 307, + 689 + ], + "score": 0.89, + "content": "T = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 679, + 468, + 691 + ], + "score": 1.0, + "content": ". In both training and evaluation, we use", + "type": "text" + }, + { + "bbox": [ + 469, + 679, + 504, + 689 + ], + "score": 0.88, + "content": "M = 2 0", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 690, + 272, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 272, + 702 + ], + "score": 1.0, + "content": "parallel Gibbs chains to reduce variance.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 657, + 506, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 111, + 277, + 325 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 378, + 93 + ], + "score": 1.0, + "content": "Table 4: Specification of Conv4 architecture. Conv2d layers are", + "type": "text" + }, + { + "bbox": [ + 379, + 81, + 404, + 91 + ], + "score": 0.88, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "with stride 1 and same", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 89, + 405, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 89, + 241, + 105 + ], + "score": 1.0, + "content": "padding. MaxPool2d layers are", + "type": "text" + }, + { + "bbox": [ + 241, + 92, + 265, + 102 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 89, + 405, + 105 + ], + "score": 1.0, + "content": "with stride 2 and valid padding.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 111, + 277, + 325 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 111, + 277, + 325 + ], + "spans": [ + { + "bbox": [ + 128, + 111, + 277, + 325 + ], + "score": 0.974, + "html": "
Output SizeLayers
1×28×28Input image
64 × 14 ×14Conv2d BatchNorm2d
ReLU
MaxPool2d
64×7×7Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×3×3Conv2d
BatchNorm2d
ReLU
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64×1×1Conv2d BatchNorm2d
ReLU
MaxPool2d
64Flatten
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Output SizeLayers
3× 84×84Input image
64 × 42 × 42Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 × 21 × 21Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 ×10×10Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×5×5Conv2d BatchNorm2d
ReLU
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Output SizeLayers
1×28×28Input image
64 × 14 ×14Conv2d BatchNorm2d
ReLU
MaxPool2d
64×7×7Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×3×3Conv2d
BatchNorm2d
ReLU
MaxPool2d
64×1×1Conv2d BatchNorm2d
ReLU
MaxPool2d
64Flatten
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Output SizeLayers
3× 84×84Input image
64 × 42 × 42Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 × 21 × 21Conv2d
BatchNorm2d
ReLU
MaxPool2d
64 ×10×10Conv2d
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64×5×5Conv2d BatchNorm2d
ReLU
MaxPool2d
1600Flatten
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In the main paper, we showed results for the following kernel, which we refer to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 435, + 369, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 369, + 449 + ], + "score": 1.0, + "content": "as the “cosine” kernel due to its resemblance to cosine similarity:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 424, + 505, + 449 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 453, + 397, + 482 + ], + "lines": [ + { + "bbox": [ + 213, + 453, + 397, + 482 + ], + "spans": [ + { + "bbox": [ + 213, + 453, + 397, + 482 + ], + "score": 0.94, + "content": "k ^ { \\mathrm { { c o s } } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha ) = \\mathrm { { e x p } } ( \\alpha ) \\frac { g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) \\| \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| } ,", + "type": "interline_equation", + "image_path": "0597162c7e86cb6ef173bc517e68d0c2850dfd7245ec750e54cfa84627bcd8c6.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 453, + 397, + 467.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 213, + 467.5, + 397, + 482.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 133, + 501 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 489, + 155, + 501 + ], + "score": 0.92, + "content": "g _ { \\pmb { \\theta } } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "is a deep neural network that outputs a fixed-dimensional encoded representation of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 146, + 512 + ], + "score": 1.0, + "content": "input and", + "type": "text" + }, + { + "bbox": [ + 146, + 502, + 154, + 510 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 501, + 294, + 512 + ], + "score": 1.0, + "content": "is the scalar log output scale. Both", + "type": "text" + }, + { + "bbox": [ + 294, + 501, + 301, + 510 + ], + "score": 0.79, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 501, + 318, + 512 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 502, + 327, + 510 + ], + "score": 0.82, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "are considered hyperparameters and learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "simultaneously as shown in Algorithm 1. We found that this kernel works well for a range of datasets", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "and shot settings. We note that the use of cosine similarity is reminiscent of the approach taken by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 141, + 546 + ], + "score": 1.0, + "content": "Baseline", + "type": "text" + }, + { + "bbox": [ + 142, + 534, + 158, + 543 + ], + "score": 0.85, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "method of (Chen et al., 2019), which computes the softmax over cosine similarity to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 543, + 164, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 164, + 557 + ], + "score": 1.0, + "content": "class weights.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 488, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 415, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 415, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 415, + 573 + ], + "score": 1.0, + "content": "Here we consider three additional kernels: linear, RBF, and normalized RBF.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 560, + 415, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 336, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 336, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 336, + 599 + ], + "score": 1.0, + "content": "Linear Kernel. The linear kernel is defined as follows:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 585, + 336, + 599 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 601, + 394, + 626 + ], + "lines": [ + { + "bbox": [ + 216, + 601, + 394, + 626 + ], + "spans": [ + { + "bbox": [ + 216, + 601, + 394, + 626 + ], + "score": 0.94, + "content": "k ^ { \\mathrm { l i n } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha ) = \\frac { 1 } { D } \\exp ( \\alpha ) g _ { \\pmb { \\theta } } ( \\mathbf { x } ) ^ { \\top } g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) ,", + "type": "interline_equation", + "image_path": "19061b62e08715e7b747dfd3a68fddf9b3c506155e9bb36dac1bcfdc38ff80ad.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 216, + 601, + 394, + 626 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 632, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 133, + 646 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 633, + 144, + 642 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 631, + 272, + 646 + ], + "score": 1.0, + "content": "is the output dimensionality of", + "type": "text" + }, + { + "bbox": [ + 272, + 632, + 297, + 645 + ], + "score": 0.92, + "content": "g _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 631, + 505, + 646 + ], + "score": 1.0, + "content": ". We apply this dimensionality scaling because the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 642, + 387, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 190, + 656 + ], + "score": 1.0, + "content": "dot product between", + "type": "text" + }, + { + "bbox": [ + 191, + 644, + 215, + 655 + ], + "score": 0.92, + "content": "g _ { \\pmb { \\theta } } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 642, + 233, + 656 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 644, + 261, + 655 + ], + "score": 0.92, + "content": "g _ { \\boldsymbol { \\theta } } ( \\mathbf { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 642, + 372, + 656 + ], + "score": 1.0, + "content": "may be large depending on", + "type": "text" + }, + { + "bbox": [ + 373, + 644, + 382, + 653 + ], + "score": 0.84, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 642, + 387, + 656 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 631, + 505, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 668, + 489, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 490, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 490, + 682 + ], + "score": 1.0, + "content": "RBF Kernel. The RBF (also known as squared exponential) kernel can be defined as follows:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 667, + 490, + 682 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 687, + 448, + 714 + ], + "lines": [ + { + "bbox": [ + 161, + 687, + 448, + 714 + ], + "spans": [ + { + "bbox": [ + 161, + 687, + 448, + 714 + ], + "score": 0.94, + "content": "k ^ { \\mathrm { t b f } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\pmb { \\theta } , \\alpha , \\ell ) = \\mathrm { e x p } ( \\alpha ) \\mathrm { e x p } \\left( - \\frac { 1 } { 2 D \\mathrm { e x p } ( \\ell ) ^ { 2 } } \\| g _ { \\pmb { \\theta } } ( \\mathbf { x } ) - g _ { \\pmb { \\theta } } ( \\mathbf { x } ^ { \\prime } ) \\| ^ { 2 } \\right) ,", + "type": "interline_equation", + "image_path": "1b5a4d978ea73c0e84f5c4f2955d0a1ca67a491e20b70fe4df65a26139d321ed.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 161, + 687, + 448, + 714 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 426, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 426, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 133, + 734 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 721, + 139, + 730 + ], + "score": 0.8, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 719, + 303, + 734 + ], + "score": 1.0, + "content": "is the log lengthscale parameter (as with", + "type": "text" + }, + { + "bbox": [ + 303, + 725, + 310, + 730 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 719, + 365, + 734 + ], + "score": 1.0, + "content": ", we learn the", + "type": "text" + }, + { + "bbox": [ + 366, + 721, + 371, + 730 + ], + "score": 0.81, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 719, + 413, + 734 + ], + "score": 1.0, + "content": "alongside", + "type": "text" + }, + { + "bbox": [ + 413, + 721, + 420, + 730 + ], + "score": 0.73, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 719, + 426, + 734 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 719, + 426, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Normalized RBF Kernel. Finally, we consider a normalized RBF kernel similar in spirit to the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 164, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 164, + 105 + ], + "score": 1.0, + "content": "cosine kernel:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 108, + 468, + 142 + ], + "lines": [ + { + "bbox": [ + 143, + 108, + 468, + 142 + ], + "spans": [ + { + "bbox": [ + 143, + 108, + 468, + 142 + ], + "score": 0.94, + "content": "k ^ { \\mathrm { \\scriptsize { t h f - n o r m } } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\theta , \\alpha , \\ell ) = \\exp ( \\alpha ) \\exp \\left( - \\frac { 1 } { 2 \\exp ( \\ell ) ^ { 2 } } \\left\\| \\frac { g _ { \\theta } ( \\mathbf { x } ) } { \\| g _ { \\theta } ( \\mathbf { x } ) \\| } - \\frac { g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) \\| } \\right\\| ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "7628472cff4207e39dc9752e4806c2ab27fcc703569533fa320cd7f48d7b4b15.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 143, + 108, + 468, + 119.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 143, + 119.33333333333333, + 468, + 130.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 143, + 130.66666666666666, + 468, + 142.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "The results of our Polya-Gamma OVE GPs with different kernels can be found in Tables 5 ´", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "and 6. In general, we find that the cosine kernel works best overall, with the exception of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 174, + 286, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 145, + 186 + ], + "score": 1.0, + "content": "Omniglot", + "type": "text" + }, + { + "bbox": [ + 146, + 175, + 156, + 184 + ], + "score": 0.78, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 174, + 286, + 186 + ], + "score": 1.0, + "content": "EMNIST, where RBF does best.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "table", + "bbox": [ + 108, + 267, + 501, + 381 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 195, + 505, + 251 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "Table 5: Classification accuracy for Polya-Gamma OVE GPs (our method) using different kernels. ´", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Cosine is overall the best, followed closely by linear. RBF-based kernels perform worse, except for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 160, + 231 + ], + "score": 1.0, + "content": "the Omniglot", + "type": "text" + }, + { + "bbox": [ + 160, + 219, + 170, + 228 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "EMNIST dataset. Evaluation is performed on 5 randomly generated sets of 600 test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 369, + 243 + ], + "score": 1.0, + "content": "episodes. Standard deviation of the mean accuracy is also shown.", + "type": "text" + }, + { + "bbox": [ + 370, + 229, + 398, + 239 + ], + "score": 0.32, + "content": "\\mathbf { M L } = \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 228, + 482, + 243 + ], + "score": 1.0, + "content": "Marginal Likelihood,", + "type": "text" + }, + { + "bbox": [ + 483, + 229, + 505, + 240 + ], + "score": 0.68, + "content": "\\mathrm { P L } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 198, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 198, + 252 + ], + "score": 1.0, + "content": "Predictive Likelihood.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 108, + 267, + 501, + 381 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 267, + 501, + 381 + ], + "spans": [ + { + "bbox": [ + 108, + 267, + 501, + 381 + ], + "score": 0.983, + "html": "
CUBmini-ImageNet
KernelObjective1-shot5-shot1-shot5-shot
CosineML63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
LinearML62.48 ± 0.2677.94 ± 0.2150.81 ± 0.3066.66 ± 0.45
RBFML58.49 ± 0.4075.50 ± 0.1850.33 ± 0.2664.62 ± 0.37
RBF (normalized)ML62.75 ± 0.3278.71 ± 0.0850.26 ± 0.3164.84 ± 0.39
CosinePL60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
LinearPL60.44 ± 0.3978.54 ± 0.1947.29 ± 0.3166.66 ± 0.36
RBFPL56.18 ± 0.6977.96 ± 0.1948.06 ± 0.2866.66 ± 0.39
RBF (normalized)PL59.78 ± 0.3478.42 ± 0.1347.51 ± 0.2066.42 ± 0.36
", + "type": "table", + "image_path": "8afaa8bc37faf7721b13ba0156273f229e279e8b62a107f3b8c6f8bdb402e506.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 108, + 267, + 501, + 305.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 108, + 305.0, + 501, + 343.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 108, + 343.0, + 501, + 381.0 + ], + "spans": [], + "index": 15 + } + ] + } + ], + "index": 12.0 + }, + { + "type": "table", + "bbox": [ + 108, + 435, + 500, + 550 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 397, + 503, + 420 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Table 6: Cross-domain classification accuracy for Polya-Gamma OVE GPs (our method) using dif- ´", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 408, + 354, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 354, + 420 + ], + "score": 1.0, + "content": "ferent kernels. The experimental setup is the same as Table 5.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 435, + 500, + 550 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 435, + 500, + 550 + ], + "spans": [ + { + "bbox": [ + 108, + 435, + 500, + 550 + ], + "score": 0.983, + "html": "
Omniglot→EMNISTmini-ImageNet-→CUB
KernelObjective1-shot5-shot1-shot5-shot
CosineML68.43 ± 0.6786.22 ± 0.2039.66± 0.1855.71 ± 0.31
LinearML72.42 ± 0.4988.27 ± 0.2039.61 ± 0.1955.07 ± 0.29
RBFML78.05 ± 0.3888.98 ± 0.1636.99 ± 0.0751.75 ± 0.27
RBF (normalized)ML75.51 ± 0.4788.86 ± 0.1638.42 ± 0.1654.20 ± 0.13
CosinePL77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
LinearPL75.87 ± 0.4388.77 ± 0.1036.83 ± 0.2756.46 ± 0.22
RBFPL74.62 ± 0.3589.87 ± 0.1335.06 ± 0.2555.12 ± 0.21
RBF (normalized)PL76.01 ± 0.3189.42 ± 0.1637.50 ± 0.2856.80 ± 0.39
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Finally, we consider a normalized RBF kernel similar in spirit to the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 164, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 164, + 105 + ], + "score": 1.0, + "content": "cosine kernel:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 108, + 468, + 142 + ], + "lines": [ + { + "bbox": [ + 143, + 108, + 468, + 142 + ], + "spans": [ + { + "bbox": [ + 143, + 108, + 468, + 142 + ], + "score": 0.94, + "content": "k ^ { \\mathrm { \\scriptsize { t h f - n o r m } } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ; \\theta , \\alpha , \\ell ) = \\exp ( \\alpha ) \\exp \\left( - \\frac { 1 } { 2 \\exp ( \\ell ) ^ { 2 } } \\left\\| \\frac { g _ { \\theta } ( \\mathbf { x } ) } { \\| g _ { \\theta } ( \\mathbf { x } ) \\| } - \\frac { g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) } { \\| g _ { \\theta } ( \\mathbf { x } ^ { \\prime } ) \\| } \\right\\| ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "7628472cff4207e39dc9752e4806c2ab27fcc703569533fa320cd7f48d7b4b15.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 143, + 108, + 468, + 119.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 143, + 119.33333333333333, + 468, + 130.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 143, + 130.66666666666666, + 468, + 142.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "The results of our Polya-Gamma OVE GPs with different kernels can be found in Tables 5 ´", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 505, + 176 + ], + "score": 1.0, + "content": "and 6. 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RBF-based kernels perform worse, except for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 160, + 231 + ], + "score": 1.0, + "content": "the Omniglot", + "type": "text" + }, + { + "bbox": [ + 160, + 219, + 170, + 228 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "EMNIST dataset. Evaluation is performed on 5 randomly generated sets of 600 test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 369, + 243 + ], + "score": 1.0, + "content": "episodes. Standard deviation of the mean accuracy is also shown.", + "type": "text" + }, + { + "bbox": [ + 370, + 229, + 398, + 239 + ], + "score": 0.32, + "content": "\\mathbf { M L } = \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 228, + 482, + 243 + ], + "score": 1.0, + "content": "Marginal Likelihood,", + "type": "text" + }, + { + "bbox": [ + 483, + 229, + 505, + 240 + ], + "score": 0.68, + "content": "\\mathrm { P L } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 198, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 198, + 252 + ], + "score": 1.0, + "content": "Predictive Likelihood.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 108, + 267, + 501, + 381 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 267, + 501, + 381 + ], + "spans": [ + { + "bbox": [ + 108, + 267, + 501, + 381 + ], + "score": 0.983, + "html": "
CUBmini-ImageNet
KernelObjective1-shot5-shot1-shot5-shot
CosineML63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
LinearML62.48 ± 0.2677.94 ± 0.2150.81 ± 0.3066.66 ± 0.45
RBFML58.49 ± 0.4075.50 ± 0.1850.33 ± 0.2664.62 ± 0.37
RBF (normalized)ML62.75 ± 0.3278.71 ± 0.0850.26 ± 0.3164.84 ± 0.39
CosinePL60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
LinearPL60.44 ± 0.3978.54 ± 0.1947.29 ± 0.3166.66 ± 0.36
RBFPL56.18 ± 0.6977.96 ± 0.1948.06 ± 0.2866.66 ± 0.39
RBF (normalized)PL59.78 ± 0.3478.42 ± 0.1347.51 ± 0.2066.42 ± 0.36
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Omniglot→EMNISTmini-ImageNet-→CUB
KernelObjective1-shot5-shot1-shot5-shot
CosineML68.43 ± 0.6786.22 ± 0.2039.66± 0.1855.71 ± 0.31
LinearML72.42 ± 0.4988.27 ± 0.2039.61 ± 0.1955.07 ± 0.29
RBFML78.05 ± 0.3888.98 ± 0.1636.99 ± 0.0751.75 ± 0.27
RBF (normalized)ML75.51 ± 0.4788.86 ± 0.1638.42 ± 0.1654.20 ± 0.13
CosinePL77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
LinearPL75.87 ± 0.4388.77 ± 0.1036.83 ± 0.2756.46 ± 0.22
RBFPL74.62 ± 0.3589.87 ± 0.1335.06 ± 0.2555.12 ± 0.21
RBF (normalized)PL76.01 ± 0.3189.42 ± 0.1637.50 ± 0.2856.80 ± 0.39
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Results were evaluated across 1,000 randomly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 228, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 228, + 182 + ], + "score": 1.0, + "content": "generated 5-shot 5-way tasks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 190, + 546, + 348 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 190, + 546, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 546, + 348 + ], + "score": 0.985, + "html": "
MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.450.77522.580.79922.750.799
Baseline++22.600.79823.820.79724.130.797
MatchingNet26.720.80324.800.79723.590.804
ProtoNet32.280.77829.970.78132.300.779
RelationNet25.230.79923.690.80020.000.800
DKT+Cosine29.540.77927.780.79231.940.782
Bayesian MAML22.790.90520.520.96320.460.949
Bayesian MAML (Chaser)20.201.13320.411.11821.391.039
LSM GP + Cosine (ML)27.920.78722.430.79822.360.799
LSM GP + Cosine (PL)31.210.77231.770.76834.740.754
OVE PG GP + Cosine (ML) [ours]32.270.77429.990.77629.970.784
OVE PG GP + Cosine (PL) [ours]33.010.77133.290.76031.410.764
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.200.77623.540.79822.870.799
Baseline++28.050.79023.720.79825.580.795
MatchingNet28.250.79023.800.80323.210.811
ProtoNet32.120.77428.810.78332.700.775
RelationNet25.230.79923.130.80020.000.800
DKT+Cosine29.740.77829.110.78932.260.781
Bayesian MAML22.760.90320.500.97020.560.950
Bayesian MAML (Chaser)20.251.17220.511.11621.451.022
LSM GP + Cosine (ML)28.180.78721.820.79923.640.797
LSM GP + Cosine (PL)32.100.76930.220.77635.090.751
OVE PG GP + Cosine (ML) [ours]31.410.77829.660.77830.280.783
OVE PG GP + Cosine (PL) [ours]33.360.77233.230.76132.060.762
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.450.77522.580.79922.750.799
Baseline++22.600.79823.820.79724.130.797
MatchingNet26.720.80324.800.79723.590.804
ProtoNet32.280.77829.970.78132.300.779
RelationNet25.230.79923.690.80020.000.800
DKT+Cosine29.540.77927.780.79231.940.782
Bayesian MAML22.790.90520.520.96320.460.949
Bayesian MAML (Chaser)20.201.13320.411.11821.391.039
LSM GP + Cosine (ML)27.920.78722.430.79822.360.799
LSM GP + Cosine (PL)31.210.77231.770.76834.740.754
OVE PG GP + Cosine (ML) [ours]32.270.77429.990.77629.970.784
OVE PG GP + Cosine (PL) [ours]33.010.77133.290.76031.410.764
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.200.77623.540.79822.870.799
Baseline++28.050.79023.720.79825.580.795
MatchingNet28.250.79023.800.80323.210.811
ProtoNet32.120.77428.810.78332.700.775
RelationNet25.230.79923.130.80020.000.800
DKT+Cosine29.740.77829.110.78932.260.781
Bayesian MAML22.760.90320.500.97020.560.950
Bayesian MAML (Chaser)20.251.17220.511.11621.451.022
LSM GP + Cosine (ML)28.180.78721.820.79923.640.797
LSM GP + Cosine (PL)32.100.76930.220.77635.090.751
OVE PG GP + Cosine (ML) [ours]31.410.77829.660.77830.280.783
OVE PG GP + Cosine (PL) [ours]33.360.77233.230.76132.060.762
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer38.030.73433.060.79133.470.792
Baseline++42.550.71035.890.76139.880.740
MatchingNet44.430.68234.430.75435.950.741
ProtoNet46.780.67636.920.73741.450.714
RelationNet40.810.75930.110.79025.690.794
DKT + Cosine45.340.69538.290.73745.170.703
Bayesian MAML42.650.69730.630.80837.320.736
Bayesian MAML (Chaser)40.660.88129.931.12131.331.125
LSM GP + Cosine (ML)45.370.70634.100.76939.660.753
LSM GP + Cosine (PL)48.550.69039.460.73743.150.714
OVE PG GP + Cosine (ML) [ours]46.460.70137.650.77543.480.723
OVE PG GP + Cosine (PL) [ours]49.440.69538.950.78043.660.720
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer38.030.73433.060.79133.470.792
Baseline++42.550.71035.890.76139.880.740
MatchingNet44.430.68234.430.75435.950.741
ProtoNet46.780.67636.920.73741.450.714
RelationNet40.810.75930.110.79025.690.794
DKT + Cosine45.340.69538.290.73745.170.703
Bayesian MAML42.650.69730.630.80837.320.736
Bayesian MAML (Chaser)40.660.88129.931.12131.331.125
LSM GP + Cosine (ML)45.370.70634.100.76939.660.753
LSM GP + Cosine (PL)48.550.69039.460.73743.150.714
OVE PG GP + Cosine (ML) [ours]46.460.70137.650.77543.480.723
OVE PG GP + Cosine (PL) [ours]49.440.69538.950.78043.660.720
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MethodCUBmini-ImageNet
1-shot5-shot1-shot5-shot
FeatureTransfer46.19± 0.6468.40± 0.7939.51 ± 0.2360.51 ± 0.55
Baseline++61.75 ± 0.9578.51 ± 0.5947.15 ± 0.4966.18 ± 0.18
MatchingNet60.19 ± 1.0275.11 ± 0.3548.25 ± 0.6562.71 ± 0.44
ProtoNet52.52 ± 1.9075.93 ± 0.4644.19 ± 1.3064.07 ± 0.65
RelationNet62.52 ± 0.3478.22 ± 0.0748.76 ± 0.1764.20 ± 0.28
MAML56.11 ± 0.6974.84 ± 0.6245.39 ± 0.4961.58 ± 0.53
DKT + Cosine63.37 ± 0.1977.73 ± 0.2648.64 ± 0.4562.85 ± 0.37
BayesianMAML55.93 ± 0.7172.87 ± 0.2644.46 ± 0.3062.60 ± 0.25
Bayesian MAML(Chaser)53.93 ± 0.7271.16 ± 0.3243.74 ± 0.4659.23 ± 0.34
ABML48.80 ± 0.4070.91 ± 0.3240.88 ± 0.2558.19 ± 0.17
Logistic Softmax GP + Cosine (ML)60.23 ± 0.5474.58 ± 0.2546.75 ± 0.2059.93 ± 0.31
Logistic Softmax GP + Cosine (PL)60.07 ±0.2978.14 ± 0.0747.05 ± 0.2066.01 ± 0.25
OVE PG GP + Cosine (ML) [ours]63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
OVE PG GP +Cosine (PL) [ours]60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
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MethodOmniglot→EMNISTmini-ImageNet→CUB
1-shot5-shot1-shot5-shot
Feature Transfer64.22 ±1.2486.10± 0.8432.77 ± 0.3550.34± 0.27
Baseline++56.84 ± 0.9180.01 ± 0.9239.19 ± 0.1257.31 ± 0.11
MatchingNet75.01 ± 2.0987.41 ± 1.7936.98 ± 0.0650.72 ± 0.36
ProtoNet72.04 ± 0.8287.22 ± 1.0133.27 ± 1.0952.16 ± 0.17
RelationNet75.62 ± 1.0087.84 ± 0.2737.13 ± 0.2051.76 ± 1.48
MAML72.68 ± 1.8583.54 ± 1.7934.01 ±1.2548.83 ± 0.62
DKT + Cosine73.06 ± 2.3688.10 ± 0.7840.22 ± 0.5455.65 ± 0.05
Bayesian MAML63.94 ± 0.4765.26 ± 0.3033.52 ± 0.3651.35 ± 0.16
Bayesian MAML (Chaser)55.04 ± 0.3454.19 ± 0.3236.22 ± 0.5051.53 ± 0.43
ABML73.89 ± 0.2487.28 ± 0.4031.51 ± 0.3247.80 ± 0.51
Logistic Softmax GP + Cosine (ML)62.91 ± 0.4983.80 ± 0.1336.41 ± 0.1850.33 ± 0.13
Logistic Softmax GP + Cosine (PL)70.70 ±0.3686.59 ± 0.1536.73 ± 0.2656.70 ± 0.31
OVE PG GP + Cosine (ML) [ours]68.43± 0.6786.22 ±0.2039.66± 0.1855.71 ± 0.31
OVE PG GP + Cosine (PL) [ours]77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
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Input: Objective L ∈ {LML,LPL}, Task distribution T, number of parallel Gibbs chains M, number of steps T,learning rate n. Initialize hyperparameters 0 randomly.
repeat Sample S= (X,Y),Q= (X*,Y*)~ T
if L= LML then X←XUX*,Y←YUY*
end if
A ← OVE-MATRIX(Y) for m = 1 to M do
~ pe(f|X)
fort=1 toTdo
（m）~PG(1b（m))
f（m） ~pe(f|x,Y,ω(m）)
end for end for
if L= LML then
0←0+∑m-1 Velogpe(Y|x,(m)）
else 0←0+∑m=1∑,∀θlogpe(y*j｜x*j,S,ω(m）)
end if until convergence
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LikelihoodC(f|y =c)
Softmaxexp(fe) ∑c exp(fe)
GaussianIIN(2·1[c' =c]-1|μ= fe,σ² =1) c'
Logistic Softmax (LSM)g(fc) ∑eσ(fe)
One-vs-Each (OVE)IIσ(fe- fe) c≠c
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Output SizeLayers
1×28×28Input image
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ReLU
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ReLU
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Omniglot→EMNISTmini-ImageNet-→CUB
KernelObjective1-shot5-shot1-shot5-shot
CosineML68.43 ± 0.6786.22 ± 0.2039.66± 0.1855.71 ± 0.31
LinearML72.42 ± 0.4988.27 ± 0.2039.61 ± 0.1955.07 ± 0.29
RBFML78.05 ± 0.3888.98 ± 0.1636.99 ± 0.0751.75 ± 0.27
RBF (normalized)ML75.51 ± 0.4788.86 ± 0.1638.42 ± 0.1654.20 ± 0.13
CosinePL77.00 ± 0.5087.52 ± 0.1937.49 ± 0.1157.23 ± 0.31
LinearPL75.87 ± 0.4388.77 ± 0.1036.83 ± 0.2756.46 ± 0.22
RBFPL74.62 ± 0.3589.87 ± 0.1335.06 ± 0.2555.12 ± 0.21
RBF (normalized)PL76.01 ± 0.3189.42 ± 0.1637.50 ± 0.2856.80 ± 0.39
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CUBmini-ImageNet
KernelObjective1-shot5-shot1-shot5-shot
CosineML63.98 ± 0.4377.44 ± 0.1850.02 ± 0.3564.58 ± 0.31
LinearML62.48 ± 0.2677.94 ± 0.2150.81 ± 0.3066.66 ± 0.45
RBFML58.49 ± 0.4075.50 ± 0.1850.33 ± 0.2664.62 ± 0.37
RBF (normalized)ML62.75 ± 0.3278.71 ± 0.0850.26 ± 0.3164.84 ± 0.39
CosinePL60.11 ± 0.2679.07 ± 0.0548.00 ± 0.2467.14 ± 0.23
LinearPL60.44 ± 0.3978.54 ± 0.1947.29 ± 0.3166.66 ± 0.36
RBFPL56.18 ± 0.6977.96 ± 0.1948.06 ± 0.2866.66 ± 0.39
RBF (normalized)PL59.78 ± 0.3478.42 ± 0.1347.51 ± 0.2066.42 ± 0.36
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.200.77623.540.79822.870.799
Baseline++28.050.79023.720.79825.580.795
MatchingNet28.250.79023.800.80323.210.811
ProtoNet32.120.77428.810.78332.700.775
RelationNet25.230.79923.130.80020.000.800
DKT+Cosine29.740.77829.110.78932.260.781
Bayesian MAML22.760.90320.500.97020.560.950
Bayesian MAML (Chaser)20.251.17220.511.11621.451.022
LSM GP + Cosine (ML)28.180.78721.820.79923.640.797
LSM GP + Cosine (PL)32.100.76930.220.77635.090.751
OVE PG GP + Cosine (ML) [ours]31.410.77829.660.77830.280.783
OVE PG GP + Cosine (PL) [ours]33.360.77233.230.76132.060.762
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer30.450.77522.580.79922.750.799
Baseline++22.600.79823.820.79724.130.797
MatchingNet26.720.80324.800.79723.590.804
ProtoNet32.280.77829.970.78132.300.779
RelationNet25.230.79923.690.80020.000.800
DKT+Cosine29.540.77927.780.79231.940.782
Bayesian MAML22.790.90520.520.96320.460.949
Bayesian MAML (Chaser)20.201.13320.411.11821.391.039
LSM GP + Cosine (ML)27.920.78722.430.79822.360.799
LSM GP + Cosine (PL)31.210.77231.770.76834.740.754
OVE PG GP + Cosine (ML) [ours]32.270.77429.990.77629.970.784
OVE PG GP + Cosine (PL) [ours]33.010.77133.290.76031.410.764
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MethodCUBmini-ImageNetmini-ImageNet-→CUB
Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)Acc. (↑)Brier (↓)
Feature Transfer38.030.73433.060.79133.470.792
Baseline++42.550.71035.890.76139.880.740
MatchingNet44.430.68234.430.75435.950.741
ProtoNet46.780.67636.920.73741.450.714
RelationNet40.810.75930.110.79025.690.794
DKT + Cosine45.340.69538.290.73745.170.703
Bayesian MAML42.650.69730.630.80837.320.736
Bayesian MAML (Chaser)40.660.88129.931.12131.331.125
LSM GP + Cosine (ML)45.370.70634.100.76939.660.753
LSM GP + Cosine (PL)48.550.69039.460.73743.150.714
OVE PG GP + Cosine (ML) [ours]46.460.70137.650.77543.480.723
OVE PG GP + Cosine (PL) [ours]49.440.69538.950.78043.660.720
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Our main insight is that the input reactant and reagent molecules can be jointly represented as a graph, and the process of generating product molecules from reactant molecules can be formulated as a sequence of graph transformations. To this end, we propose Graph Transformation Policy Network (GTPN) − a novel generic method that combines the strengths of graph neural networks and reinforcement learning to learn the reactions directly from data with minimal chemical knowledge. Compared to previous methods, GTPN has some appealing properties such as: end-to-end learning, and making no assumption about the length or the order of graph transformations. In order to guide model search through the complex discrete space of sets of bond changes effectively, we extend the standard policy gradient loss by adding useful constraints. Evaluation results show that GTPN improves the top-1 accuracy over the current state-of-the-art method by about $3 \%$ on the large USPTO dataset. Our model’s performances and prediction errors are also analyzed carefully in the paper. + +# 1 INTRODUCTION + +Chemical reaction product prediction is a fundamental problem in organic chemistry. It paves the way for planning syntheses of new substances (Chen & Baldi, 2009). For decades, huge effort has been spent to solve this problem. However, most methods still depend on the handcrafted reaction rules (Chen & Baldi, 2009; Kayala & Baldi, 2011; Wei et al., 2016) or heuristically extracted reaction templates (Segler & Waller, 2017; Coley et al., 2017), thus are not well generalizable to unseen reactions. + +A reaction can be regarded as a set (or unordered sequence) of graph transformations in which reactants represented as molecular graphs are transformed into products by modifying the bonds between some atom pairs (Jochum et al., 1980; Ugi et al., 1979). See Fig. 1 for an illustration. We call an atom pair $( u , v )$ that changes its connectivity during reaction and its new bond $b$ a reaction triple $( u , v , b )$ . The reaction product prediction problem now becomes predicting a set of reaction triples given the input reactants and reagents. We argue that in order to solve this problem well, an intelligent system should have two key capabilities: (a) Understanding the molecular graph structure of the input reactants and reagents so that it can identify possible reactivity patterns (i.e., atom pairs with changing connectivity). (b) Knowing how to choose from these reactivity patterns a correct set of reaction triples to generate the desired products. + +Recent state-of-the-art methods (Jin et al., 2017; Bradshaw et al., 2018) have built the first capability by leveraging graph neural networks (Duvenaud et al., 2015; Hamilton et al., 2017; Pham et al., 2017; Gilmer et al., 2017). However, these methods are either unaware of the valid sets of reaction triples (Jin et al., 2017) or limited to sequences of reaction triples with a predefined orders (Bradshaw et al., 2018). The main challenge is that the space of all possible configurations of reaction triples is extremely large and non-differentiable. Moreover, a small change in the predicted set of reaction triples can lead to very different reaction products and a little mistake can produce invalid prediction. + +In this paper, we propose a novel method called Graph Transformation Policy Network (GTPN) that addresses the aforementioned challenges. Our model consists of three main components: a graph neural network (GNN), a node pair prediction network (NPPN) and a policy network (PN). Starting from the initial graph of reactant and reagent molecules, our model iteratively alternates between modeling an input graph using GNN and predicting a reaction triple using NPPN and PN to generate a new intermediate graph as input for the next step until it decides to stop. The final generated graph is considered as the predicted products of the reaction. Importantly, GTPN does not assume any fixed number or any order of bond changes but learn these properties itself. One can view GTPN as a reinforcement learning (RL) agent that operates on a complex and non-differentiable space of sets of reaction triples. To guide our model towards learning a diverse yet robust-to-small-changes policy, we customize our loss function by adding some useful constraints to the standard policy gradient loss (Mnih et al., 2016). + +![](images/a12f14d83ce2155ad6d7941ebcf1a5e44c5921b556fe473e90e0c85202a223ba.jpg) +Figure 1: A sample reaction represented as a set of graph transformations from reactants (leftmost) to products (rightmost). Atoms are labeled with their type (Carbon, Oxygen,...) and their index (1, 2,...) in the molecular graph. The atom pairs that change connectivity and their new bonds (if existed) are highlighted in green. There are two bond changes in this case: 1) The double bond between O:1 and C:2 becomes single. 2) A new single bond between C:2 and C:10 is added. + +To the best of our knowledge, GTPN is the most generic approach for the reaction product prediction problem so far in the sense that: i) It combines graph neural networks and reinforcement learning into a unified framework and trains everything end-to-end; ii) It does not use any handcrafted or heuristically extracted reaction rules/templates to predict the products. Instead, it automatically learns various types of reactions from the training data and can generalize to unseen reactions; iii) It can interpret how the products are formed via the sequence of reaction triples it generates. + +We evaluate GTPN on two large public datasets named $U S P T O – I 5 k$ and USPTO. Our method significantly outperforms all baselines in the top-1 accuracy, achieving new state-of-the-art results of $8 2 . 3 9 \%$ and $8 3 . 2 0 \%$ on $U S P T O – I 5 k$ and USPTO, respectively. In addition, we also provide comprehensive analyses about the performance of GTPN and about different types of errors our model could make. + +# 2 METHOD + +2.1 CHEMICAL REACTION AS MARKOV DECISION PROCESS OF GRAPH TRANSFORMATIONS + +A reaction occurs when reactant molecules interact with each other in the presence (or absence) of reagent molecules to form new product molecules by breaking or adding some of their bonds. Our main insight is that reaction product prediction can be formulated as predicting a sequence of such bond changes given the reactant and reagent molecules as input. A bond change is characterized by the atom pair (where the change happens) and the new bond type (what is the change). We call this atom pair a reaction atom pair and call this atom pair with the new bond type a reaction triple. + +More formally, we represent the entire system of input reactant and reagent molecules as a labeled graph $\mathcal { G } = ( \dot { \mathcal { V } } , \mathcal { E } )$ with multiple connected components, each of which corresponds to a molecule. Nodes in $\nu$ are atoms labeled with their atomic numbers and edges in $\mathcal { E }$ are bonds labeled with their bond types. Given $\mathcal { G }$ as input, we predict a sequence of reaction triples that transforms $\mathcal { G }$ into a graph of product molecules $\mathcal { G } ^ { \prime }$ . + +As reactions vary in number of transformation steps, we represent the sequence of reaction triples as $( \xi , u , v , b ) ^ { 0 } , ( \dot { \xi } , u , v , b ) ^ { 1 } , . . . , ( \xi , u , v , b ) ^ { T - 1 }$ or $( \dot { \xi } , \dot { u } , v , b ) ^ { \dot { 0 } : T }$ for short. Here $T$ is the maximum number of steps, $( u , v )$ is a pair of nodes, $b$ is the new edge type of $( u , v )$ , and $\xi$ is a binary signal that indicates the end of the sequence. If the sequence ends at $\dot { T } _ { \mathrm { e n d } } < \dot { T } , \dot { \xi } ^ { 0 } , . . . \xi ^ { \dot { T } _ { \mathrm { c n d } } - 1 }$ will be 1 and $\xi ^ { T _ { \mathrm { e n d } } } , . . . , \xi ^ { T - 1 }$ will be 0. At every step $\tau$ , if $\xi ^ { \tau } = 1$ end , we apply the predicted edge change $( u , v , b ) ^ { \tau }$ on the current graph $\mathcal { G } ^ { \tau }$ to create a new intermediate graph $\mathcal { G } ^ { \tau + 1 }$ as input for the next step $\tau + 1$ . This iterative process of graph transformation can be formulated as a Markov Decision Process (MDP) characterized by a tuple $( S , { \mathcal { A } } , P , R , \gamma )$ , in which $s$ is a set of states, $\mathcal { A }$ is a set of actions, $P$ is a state transition function, $R$ is a reward function, and $\gamma$ is a discount factor. Since the process is finite and contains no loop, we set the discount factor $\gamma$ to be 1. The rest of the MDP tuple are defined as follows: + +![](images/1228c94e7af5a5048db2eb5fe16475e451da8a13e5b75b55f27945f853ca3bb0.jpg) +Figure 2: Workflow of a Graph Transformation Policy Network (GTPN). At every step of the forward pass, our model performs 7 major functions: 1) Computing the atom representation vectors, 2) Computing the most possible $K$ reaction atom pairs, 3) Predicting the continuation signal $\xi$ , 4) Predicting the reaction atom pair $( u , v )$ , 5) Predicting a new bond $b$ of this atom pair, 6) Updating the atom representation vectors, and 7) Updating the recurrent state. + +• State: A state $s ^ { \tau } \in \boldsymbol { S }$ is an intermediate graph $\mathcal { G } ^ { \tau }$ generated at step $\tau$ $( 0 \leq \tau < T )$ ). When $\tau = 0$ , we denote $s ^ { 0 } = \mathcal { G } ^ { 0 } = \mathcal { G }$ . +Action: An action $a ^ { \tau } \in { \mathcal { A } }$ performed at step $\tau$ is the tuple $( \xi , u , v , b ) ^ { \tau }$ . The action is composed of three consecutive sub-actions: $\xi ^ { \tau }$ , $( u , v ) ^ { \tau }$ , and $b ^ { \tau }$ . If $\xi ^ { \tau } = 0$ , our model will ignore the next sub-actions $( u , v ) ^ { \tau }$ and $b ^ { \tau }$ , and all the future actions $( \xi , u , v , b ) ^ { \tau + 1 : T }$ . Note that setting $\xi ^ { \tau }$ to be the first sub-action is useful in case a reaction does not happen, i.e., $\xi ^ { 0 } = 0$ +• State Transition: If $\xi ^ { \tau } = 1$ , the current graph $\mathcal { G } ^ { \tau }$ is modified based on the reaction triple $( u , v , b ) ^ { \tau }$ to generate a new intermediate graph $\mathcal { G } ^ { \tau + 1 }$ . We do not incorporate chemical rules such as valency check during state transition because the current bond change may result in invalid intermediate molecules $\mathcal { G } ^ { \tau }$ , but later, other bond changes may compensate it to create the valid final products $\mathcal { G } ^ { T _ { \mathrm { e n d } } }$ . Reward: We use both immediate rewards and delayed rewards to encourage our model to learn the optimal policy faster. At every step $\tau$ , if the model predicts $\xi ^ { \tau }$ , $( u , v ) ^ { \tau }$ or $b ^ { \tau }$ correctly, it will receive a positive reward for each correct sub-action. Otherwise, a negative reward is given. After the prediction process has terminated, if the generated products are exactly the same as the groundtruth products, we give the model a positive reward, otherwise a negative reward. The concrete reward values are provided in Appendix A.3. + +# 2.2 GRAPH TRANSFORMATION POLICY NETWORK + +In this section, we describe the architecture of our model − a Graph Transformation Policy Network (GTPN). GTPN has three main components namely a Graph Neural Network (GNN), a Node Pair Prediciton Network (NPPN), and a Policy Network (PN). Each component is responsible for one or several key functions shown in Fig. 2: GNN performs functions 1 and 6; NPPN performs function 2; and PN performs functions 3, 4 and 5. Apart from these components, GTPN also has a Recurrent Neural Network (RNN) to keep track of the past transformations. The hidden state $^ { h }$ of this RNN is used by NPPN and PN to make accurate prediction. + +# 2.2.1 GRAPH NEURAL NETWORK + +To model the intermediate graph $\mathcal { G } ^ { \tau }$ at step $\tau$ , we compute the node state vector $\mathbf { \boldsymbol { x } } _ { i } ^ { \tau }$ of every node $i$ in $\mathcal { G } ^ { \tau }$ by using a variant of the Message Passing Neural Networks (Gilmer et al., 2017): + +$$ +\begin{array} { r l r } { \pmb { x } _ { i } ^ { \tau } } & { = } & { \mathrm { M e s s a g e P a s s i n g } ^ { m } \left( \pmb { x } _ { i } ^ { \tau - 1 } , \pmb { v } _ { i } , \sqrt { \tau } \left( i \right) \right) } \end{array} +$$ + +where $m$ is the number of message passing steps; ${ \mathbf { } } v _ { i }$ is the feature vector of node $i$ ; $\mathcal { N } ^ { \tau } ( i )$ is the set of all neighbor nodes of node $i$ ; and $\pmb { x } _ { i } ^ { \tau - 1 }$ is the state vector of node $i$ at the previous step. When $\tau = 0$ $\pmb { x } _ { i } ^ { \tau - 1 }$ is initialized from ${ \mathbf { } } v _ { i }$ using a neural network. Details about the MessagePassing $( . )$ function are provided in Appendix A.1. + +# 2.2.2 NODE PAIR PREDICTION NETWORK + +In order to predict how likely an atom pair $( i , j )$ of the intermediate graph $\mathcal { G } ^ { \tau }$ will change its bond, we assign $( i , j )$ with a score $s _ { i j } ^ { \tau } \in \mathbb { R }$ . If $s _ { i j } ^ { \tau }$ is high, $( i , j )$ is more probably a reaction atom pair, otherwise, less probably. Similar to (Jin et al., 2017), we use two different networks called “local” network and “global” network for this task. In case of the “local” network, $s _ { i j } ^ { \tau }$ is computed as: + +$$ +\begin{array} { r c l } { { z _ { i j } ^ { \tau } } } & { { = } } & { { \sigma \left( W _ { 1 } \left[ { { h ^ { \tau } } ^ { - 1 } } , ( { x _ { i } ^ { \tau } } + { { \bf { x } } _ { j } ^ { \tau } } ) , { { e _ { i j } } } \right] + b _ { 1 } \right) } } \\ { { s _ { i j } ^ { \tau } } } & { { = } } & { { f ^ { \mathrm { a t o m p a i r } } \left( { z _ { i j } ^ { \tau } } \right) } } \end{array} +$$ + +where $f ^ { \mathrm { a t o m } } \operatorname { p a i r }$ is a neural network; $\sigma$ is a nonlinear activation function (e.g., ReLU); $[ . ]$ denotes vector concatenation; $W _ { 1 }$ and $b _ { 1 }$ are parameters; $\pmb { h } ^ { \tau - 1 }$ is the hidden state of the RNN at the previous step; and $e _ { i j }$ is the representation vector of the bond between $( i , j )$ . If there is no bond between $( i , j )$ we assume that its bond type is “NULL”. We consider $z _ { i j }$ as the representation vector for the atom pair $( i , j )$ . + +The “global” network leverages self-attention (Vaswani et al., 2017; Wang et al., 2018) to detect compatibility between atom $i$ and all other atoms before computing the scores: + +$$ +\begin{array} { r c l } { { r _ { i j } ^ { \tau } } } & { { = } } & { { \sigma \left( V _ { 1 } \left[ ( { \bf { x } } _ { i } ^ { \tau } + { \bf { x } } _ { j } ^ { \tau } ) , e _ { i j } \right] + c _ { 1 } \right) } } \\ { { a _ { i j } ^ { \tau } } } & { { = } } & { { \mathrm { s o f t m a x } \left( V _ { 2 } r _ { i j } ^ { \tau } + c _ { 2 } \right) } } \\ { { c _ { i } ^ { \tau } } } & { { = } } & { { \displaystyle \sum _ { j \in \mathcal { V } } a _ { i j } { \bf { x } } _ { j } ^ { \tau } } } \\ { { z _ { i j } ^ { \tau } } } & { { = } } & { { \sigma \left( W _ { 1 } \left[ { \bf { h } } ^ { \tau - 1 } , ( { \bf { x } } _ { i } ^ { \tau } + { \bf { x } } _ { j } ^ { \tau } ) , ( c _ { i } ^ { \tau } + c _ { j } ^ { \tau } ) , e _ { i j } \right] + b _ { 1 } \right) } } \\ { { s _ { i j } ^ { \tau } } } & { { = } } & { { f _ { \mathrm { c o m p a i r } } ^ { \mathrm { a t m } } \left( z _ { i j } ^ { \tau } \right) } } \\ { { . } } & { { } } & { { . } } \end{array} +$$ + +where $a _ { i j }$ is the attention score from node $\imath$ to every other node $j$ ; $c _ { i }$ is the context vector of atom $i$ that summarizes the information from all other atoms. + +During experiments, we tried both options mentioned above and saw that the “global” network clearly outperforms the “local” network so we set the “global” network as a default module in our model. In addition, since reagents never change their form during a reaction, we explicitly exclude all atom pairs that have either atoms belong to the reagents. This leads to better results than not using reagent information. Detailed analyses are provided in Appendix A.5. + +Top- $K$ atom pairs Because the number of atom pairs that actually participate in a reaction is very small (usually smaller than 10) compared to the total number of atom pairs of the input molecules (usually hundreds or thousands), it is much more efficient to identify reaction triples from a small subset of highly probable reaction atom pairs. For that reason, we extract $K$ $K \ll | \mathcal { V } | ^ { 2 } )$ atom pairs with the highest scores. Later, we will predict reaction triples taken from these $K$ atom pairs only. We denote the set of top- $K$ atom pairs, their corresponding scores, and representation vectors as $\left\{ ( u _ { k } , v _ { k } ) | k = \overline { { 1 , K } } \right\}$ , $\left\{ \dot { s } _ { u _ { k } v _ { k } } | k = \overline { { 1 , K } } \right\}$ and $Z _ { K } = \mathsf { \bar { \{ } } z _ { u _ { k } v _ { k } } \mathsf { \bar { | } } k = \overline { { 1 , K } } \mathsf \}$ , respectively. + +# 2.2.3 POLICY NETWORK + +Predicting continuation signal To account for varying number of transformation steps, PN generates a continuation signal $\xi ^ { \tau } \in \{ 0 , 1 \}$ to indicate whether prediction should continue or terminate. + +$\xi ^ { \tau }$ is drawn from a Bernoulli distribution: + +$$ +\begin{array} { l l l } { p \left( \xi ^ { \tau } = 1 \right) } & { = } & { \mathrm { s i g m o i d } \left( f ^ { \mathrm { s i g n a l } } \left( \left[ { h ^ { \tau - 1 } } , g \left( Z _ { K } ^ { \tau } \right) \right] \right) \right) } \end{array} +$$ + +where $\pmb { h } ^ { \tau - 1 }$ is the previous RNN state; $Z _ { K } ^ { \tau }$ is the set of representation vectors of the top $K$ atom pairs at the current step; $f ^ { \mathrm { s i g n a l } }$ is a neural network; $g$ is a function that maps an unordered set of inputs to an output vector. For simplicity, we use a mean function: + +$$ +z _ { K } ^ { \tau - 1 } = g \left( Z _ { K } ^ { \tau } \right) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } W z _ { u _ { k } v _ { k } } ^ { \tau - 1 } +$$ + +Predicting atom pair At the next sub-step, PN predicts which atom pair changes its bond during the reaction by sampling from the top- $K$ atom pairs with probability: + +$$ +p \left( ( u _ { k } , v _ { k } ) ^ { \tau } \right) = \operatorname { s o f t m a x } _ { K } \left( s _ { u _ { k } v _ { k } } ^ { \tau } \right) +$$ + +where $s _ { u _ { k } v _ { k } } ^ { \tau }$ is the score of the atom pair $( u _ { k } , v _ { k } ) ^ { \tau }$ computed in Eq. (5). After predicting the atom pair $( u , v ) ^ { \tau }$ , we will mask it to ensure that it could not be in the top $K$ again at future steps. + +Predicting bond type Given an atom pair $( u , v ) ^ { \tau }$ sampled from the previous sub-step, we predict a new bond type $b ^ { \tau }$ between $u$ and $v$ to get a complete reaction triple $( u , v , b ) ^ { \tau }$ using the probability: + +$$ +\begin{array} { r } { p ( b ^ { \tau } \vert ( u , v ) ^ { \tau } ) = \mathrm { s o f t m a x } _ { B } ( f ^ { \mathrm { b o n d } } ( [ h ^ { \tau - 1 } , z _ { u v } ^ { \tau } , ( e _ { b } - e _ { b ^ { \mathrm { o l d } } } ) ] ) ) } \end{array} +$$ + +where $B$ is the total number of bond types; $z _ { u v } ^ { \tau }$ is the representation vector of $( u , v ) ^ { \tau }$ computed in Eq. (4); $b ^ { \mathrm { o l d } }$ is the old bond of $( u , v )$ ; $e _ { b ^ { \mathrm { o l d } } }$ and $e _ { b }$ are the embedding vectors corresponding to the bond type $b ^ { \mathrm { o l d } }$ and $b$ , respectively; and $f ^ { \mathrm { b o n d } }$ is a neural network. + +# 2.3 UPDATING STATES + +After predicting a complete reaction triple $( u , v , b ) ^ { \tau }$ , our model updates: i) the new recurrent hidden state $\pmb { h } ^ { \tau }$ , and ii) the new node representation vectors $\pmb { x } _ { i } ^ { \tau + 1 }$ of the new intermediate graph $\mathcal { G } ^ { \tau + 1 }$ for $i \in \nu$ . These updates are presented in Appendix A.2. + +# 2.4 TRAINING + +Loss function plays a central role in achieving fast training and high performance. We design the following loss: + +$$ +{ \mathcal { L } } = \lambda _ { 1 } { \mathcal { L } } ^ { \mathrm { { A 2 C } } } + \lambda _ { 2 } { \mathcal { L } } ^ { \mathrm { { v a l u e } } } + \lambda _ { 3 } { \mathcal { L } } ^ { \mathrm { { a t o m p a i r } } } + \lambda _ { 4 } { \mathcal { L } } ^ { \mathrm { { o v e r l e n g t h } } } + \lambda _ { 5 } { \mathcal { L } } ^ { \mathrm { { i n } } \mathrm { { t o p } } K } +$$ + +where $\mathcal { L } ^ { \mathrm { A 2 C } }$ is the Advantage Actor-Critic (A2C) loss (Mnih et al., 2016) to account for the correct sequence of reaction triples; $\mathcal { L } ^ { \mathrm { v a l u e } }$ is the loss for estimating the value function used in A2C; $\mathcal { L } ^ { \mathrm { a t o m } }$ pair accounts for binary change in the bond of an atom pair; Lover length penalizes long predicted sequences; and $\mathcal { L } ^ { \mathrm { i n t o p } ~ K }$ is the rank loss to force a ground-truth reaction atom pair to appear in the top- $K$ ; and $\lambda _ { 1 } , . . . , \lambda _ { 5 } > 0$ are tunable coefficients. The component losses are explained in the following. + +# 2.4.1 REACTION TRIPLE LOSS + +The loss follows a policy gradient method known as Advantage Actor-Critic (A2C): + +$$ +\begin{array} { r c l } { { \mathcal { L } ^ { \mathrm { A 2 C } } } } & { { = } } & { { \displaystyle - \sum _ { \tau = 0 } ^ { T _ { \mathrm { e n d } } - 1 } \left( A _ { \mathrm { s i g n a l } } ^ { \tau } \log p \left( \xi ^ { \tau } \right) + A _ { \mathrm { a t o m p a i r } } ^ { \tau } \log p \left( \left( u , v \right) ^ { \tau } \right) + A _ { \mathrm { b o n d } } ^ { \tau } \log p \left( b ^ { \tau } \right) \right) } } \\ { { } } & { { } } & { { \displaystyle - A _ { \mathrm { s i g n a l } } ^ { T _ { \mathrm { e n d } } } \log \pi \left( \xi ^ { T _ { \mathrm { e n d } } } \right) } } \end{array} +$$ + +where $T _ { \mathrm { e n d } }$ is the first step that $\xi = 0$ ; $A _ { \mathrm { s i g n a l } }$ , $A _ { \mathrm { a t o m } \operatorname { p a i r } }$ and $A _ { \mathrm { b o n d } }$ are called advantages. To compute these advantages, we use the unbiased estimations called Temporal Different errors, defined as: + +$$ +\begin{array} { r l r } { A _ { \mathrm { s i g n a l } } ^ { \tau } } & { = } & { r _ { \mathrm { s i g n a l } } ^ { \tau } + \gamma V _ { \phi } \left( Z _ { K } ^ { \tau + 1 } \right) - V _ { \phi } \left( Z _ { K } ^ { \tau } \right) \quad } \\ { A _ { \mathrm { a t o m p a i r } } ^ { \tau } } & { = } & { r _ { \mathrm { a t o m p a i r } } ^ { \tau } + \gamma V _ { \phi } \left( Z _ { K } ^ { \tau + 1 } \right) - V _ { \phi } \left( Z _ { K } ^ { \tau } \right) \quad } \\ { A _ { \mathrm { b o n d } } ^ { \tau } } & { = } & { r _ { \mathrm { b o n d } } ^ { \tau } + \gamma V _ { \phi } \left( Z _ { K } ^ { \tau + 1 } \right) - V _ { \phi } \left( Z _ { K } ^ { \tau } \right) \quad \quad } \end{array} +$$ + +where $r _ { \mathrm { s i g n a l } } ^ { \tau }$ , $r _ { \mathrm { a t o m } \ p \mathrm { a i r } } ^ { \tau }$ , $r _ { \mathrm { b o n d } } ^ { \tau }$ are immediate rewards at step $\tau$ ; at the final step $\tau = T _ { \mathrm { e n d } }$ , the model receives additional delayed rewards; $\gamma$ is the discount factor; and $V _ { \phi }$ is the parametric value function. We train $V _ { \phi }$ using the following mean square error loss: + +$$ +\mathcal { L } ^ { \mathrm { v a l u e } } \ = \ \sum _ { \tau = 0 } ^ { T _ { \mathrm { e n d } } } \left\| V _ { \phi } \left( Z _ { K } ^ { \tau } \right) - R ^ { \tau } \right\| ^ { 2 } +$$ + +where $R ^ { \tau }$ is the return at step $\tau$ . + +Episode termination during training Although the loss defined in Eq. (9) is correct, it is not good to use in practice because: i) If our model selects a wrong sub-action at any sub-step of the step Twrong $\mathit { T _ { \mathrm { w r o n g } } } < \mathit { T _ { \mathrm { e n d } } } )$ , the whole predicted sequence will be incorrect regardless of what will be predicted from $T _ { \mathrm { w r o n g } } + 1$ to $T _ { \mathrm { e n d } }$ . Therefore, computing the loss for actions from $T _ { \mathrm { w r o n g } } + 1$ to $T _ { \mathrm { e n d } }$ is redundant. ii) More importantly, the incorrect updates of the graph structure at subsequent steps from $T _ { \mathrm { w r o n g } } + 1$ to $T _ { \mathrm { e n d } }$ will lead to cumulative prediction errors which make the training of our model much more difficult. + +To resolve this issue, during training, we use a binary vector $\zeta \in \{ 0 , 1 \} ^ { 3 T }$ to keep track of the first wrong sub-action: $\zeta ^ { t } = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } t \leq t _ { \mathrm { f i r s t ~ w r o n g } } } \\ { 0 } & { { \mathrm { i f ~ } } t > t _ { \mathrm { f i r s t ~ w r o n g } } } \end{array} \right. }$ if t ≤ tfirst wrong where tfirst wrong denotes the sub-step at which our model chooses a wrong sub-action the first time. The actor-critic loss in Eq. (9) now becomes: + +$$ +\mathcal { L } ^ { \mathrm { A 2 C } } = - \sum _ { \tau = 0 } ^ { T } \left( \zeta ^ { \tau } A _ { \mathrm { s i g n a l } } ^ { \tau } \log p \left( \xi ^ { \tau } \right) + \zeta ^ { ( \tau + 1 ) } A _ { \mathrm { a t o m p a i r } } ^ { \tau } \log p \left( ( u , v ) ^ { \tau } \right) + \zeta ^ { ( \tau + 2 ) } A _ { \mathrm { b o n d } } ^ { \tau } \log p \left( b ^ { \tau } \right) \right) . +$$ + +where $T$ is the maximum number of steps. Similarly, we change the value loss into: + +$$ +\mathcal { L } ^ { \mathrm { v a l u e } } = \sum _ { \tau = 0 } ^ { T } \zeta ^ { \tau } \left\| V _ { \phi } \left( Z _ { K } ^ { \tau } \right) - R ^ { \tau } \right\| ^ { 2 } +$$ + +# 2.4.2 REACTION ATOM PAIR LOSS + +To train our model to assign higher scores to reaction atom pairs and lower to non-reaction atom pairs, we use the following cross-entropy loss function: + +$$ +\mathcal { L } ^ { \mathrm { a t o m \ p a i r } } = - \sum _ { \tau = 0 } ^ { T _ { \mathrm { \bar { m } s t w o u p } } } \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathcal { V } , j \neq i } \eta _ { i j \tau } \left( y _ { i j } \log p _ { i j } + ( 1 - y _ { i j } ) \log ( 1 - p _ { i j } ) \right) +$$ + +where $\begin{array} { r } { T _ { \mathrm { f i r s t w r o n g } } = \biggl \lfloor \frac { t _ { \mathrm { f i r s t w r o n g } } } { 3 } \biggr \rfloor ; \eta _ { i j t } \in \{ 0 , 1 \} } \end{array}$ is a mask of the atom pair $( i , j )$ at step $\tau$ ; $y _ { i j } \in \{ 0 , 1 \}$ is the label indicating whether the atom pair $( i , j )$ is a reaction atom pair or not; $p _ { i j } = \mathrm { s i g m o i d } ( s _ { i j } )$ (see Eq. (5)). + +# 2.4.3 CONSTRAINT ON THE SEQUENCE LENGTH + +One major difficulty of the chemical reaction prediction problem is to know exactly when to stop prediction so we can make accurate inference. By forcing the model to stop immediately when making wrong prediction, we can prevent cumulative error and significantly reduce variance during training. But it also comes with a cost: The model cannot learn (because it does not have to learn) when to stop. This phenomenon can be visualized easily as the model predicts 1 for the signal at every step $\tau$ during inference. In order to make the model aware of the correct sequence length during training, we define a loss that punishes the model if it produces a longer sequence than the ground truth sequence: + +
Dataset#reactions#changes#molecules#atoms#bonds
USPTO-15ktrain10,500111112.3112013.641100134.931110134.7
valid1,500111112.3112013.67194134.55199134.2
test3,000111112.3111613.67198134.951102134.7
USPTOtrain409,03511612.2212914.891150139.761165138.6
valid30,00011612.2212514.891150139.671158138.5
test40.00011612.2212214.891150139.871162138.7
+ +Table 1: Statistics of $U S P T O – I 5 k$ and USPTO datasets. “changes” means bond changes, “molecules” means reactants and reagents in a reaction; “atoms” and “bonds” are defined for a molecule. Apart from “#reactions”, other columns are presented in the format “min | max | mean”. + +$$ +\mathcal { L } ^ { \mathrm { o v e r } \mathrm { l e n g t h } } = - \sum _ { T _ { \mathrm { e n d } } ^ { \mathrm { g t } } \leq \tau < T _ { \mathrm { e n d } } } \log p \left( \xi ^ { \tau } = 0 \right) +$$ + +wherewhen $T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ end step of the ground-trut. The reason is that forcing nce. with s in Eq. (16) is not applied is not theoretically correct $T _ { \mathrm { e n d } } \leq T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ $\xi ^ { \tau } = 1$ $T _ { \mathrm { e n d } } \leq \tau < T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ because all the signals after $T _ { \mathrm { e n d } }$ are assumed to be 0. The incentive to force $T _ { \mathrm { e n d } }$ close to $T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ when it is smaller than $T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ has already been included in the advantages in Eq. (14). + +# 2.4.4 CONSTRAINT ON THE TOP- $K$ ATOM PAIRS + +Ideally, the loss from Eq. (15) pushes a reaction atom pair $( \tilde { u } , \tilde { v } ) ^ { \tau }$ into the top- $K$ atom pairs at each step $\tau < T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ . However, this is not guaranteed, especially when τ $\tau$ comes close to $T _ { \mathrm { e n d } } ^ { \mathrm { g t } }$ . To encourage the ground-truth reaction atom pair with the highest score to appear in the top $K$ , we introduce an additional rank-based loss: + +$$ +{ \mathcal { L } } ^ { \mathrm { i n } \mathrm { t o p } K } = - \sum _ { \tau = 0 } ^ { T _ { \mathrm { f i r s t w r o n g } } } \log p \left( ( { \tilde { u } } , { \tilde { v } } ) ^ { \tau } \operatorname { i n } \mathrm { t o p } K \right) +$$ + +where $p \left( ( \tilde { u } , \tilde { v } ) ^ { \tau } \right.$ in top $K$ ) is computed as: + +$$ +\begin{array} { r l r } { p \left( ( \tilde { u } , \tilde { v } ) ^ { \tau } \mathrm { i n } \mathrm { t o p } K \right) } & { = } & { \frac { \exp \left( s _ { \tilde { u } \tilde { v } } ^ { \tau } \right) } { \exp \left( s _ { \tilde { u } \tilde { v } } ^ { \tau } \right) + \sum _ { k = 1 } ^ { K } \exp \left( s _ { u _ { k } v _ { k } } ^ { \tau } \right) } } \end{array} +$$ + +# 3 EXPERIMENTS + +# 3.1 DATASET + +We evaluate our model on two standard datasets USPTO-15k (15K reactions) and USPTO (480K reactions) which have been used in previous works (Jin et al., 2017; Schwaller et al., 2018; Bradshaw et al., 2018). Details about these datasets are given in Table 1. The USPTO dataset contains reactant, reagent and product molecules represented as SMILES strings. Using RDKit1, we convert the SMILES strings into molecule objects and store them as graphs. For each reaction, every atom in the reactant and reagent molecules is identified with a unique “atom map number”. This identity is the same in the products. Using this knowledge, we compare every atom pair in the input molecules with the correspondent in the product molecules to obtain a ground-truth set of reaction triples for training. In USPTO-15k, the ground-truth sets of reaction triples was precomputed by (Jin et al., 2017). + +
ModelUSPTO-15kUSPTO
C@6C@8C@10C@6C@8C@10
WLN* (Jin et al., 2017)81.686.189.189.892.093.3
WLN (Jin et al., 2017)88.4591.6593.3490.9793.9895.26
CLN (Pham et al., 2017)88.6891.6393.0790.7293.5794.80
Our GNN88.9292.0093.5791.2494.1795.33
+ +Table 2: Results for reaction atom pair prediction. $C @ k$ is coverage at $k$ . Best results are highlighted in bold. $\mathrm { W L N ^ { \star } }$ is the original model from (Jin et al., 2017) while WLN is our re-implemented version. Except for $\mathrm { W L N ^ { \star } }$ , other models explicitly use reagent information. + +![](images/14668a9481c915b9fba00c8d77e2952ac553712c8edb6065fb0fd1071dd67232.jpg) +Figure 3: Coverage $@ k$ and Recall $@ k$ with respect to $k$ for the USPTO dataset. + +# 3.2 REACTION ATOM PAIR PREDICTION + +In this section, we test our model’s ability to identify reaction atom pairs by formulating it as a ranking problem with the scores computed in Eq. (5). Similar to (Jin et al., 2017), we use Coverage $@ k$ as the evaluation metric, which is the proportion of reactions that have all groundtruth reaction atom pairs appear in the top $k$ predicted atom pairs. + +We compare our proposed graph neural network (GNN) with Weisfeiler-Lehman Network (WLN) (Jin et al., 2017) and Column Network (CLN) (Pham et al., 2017). Since our GNN explicitly uses reagent information to compute the scores of atom pairs, we modify the implementation of WLN and CLN accordingly for fair comparison. From Table 2, we observe that our GNN clearly outperforms WLN and CLN in all cases. We attribute this improvement to the use of a separate node state vector $\boldsymbol { x } _ { i } ^ { t }$ (different from the node feature vector ${ \mathbf { } } v _ { i }$ ) for updating the structural information of a node (see Eq. (21)). The other two models, on the other hand, only use a single vector to store both the node features and structure, hence, some information may be lost. In addition, using explicit reagent information boosts the prediction accuracy, which improves the WLN by $1 - 7 \%$ depending on the metrics. The presence of reagent information reduces the number of atom pairs to be searched on and contributes to the likelihood of reaction atom pairs. Further results are presented in Appendix A.5. + +# 3.3 TOP- $K$ ATOM PAIR EXTRACTION + +The performance of our model depends on the number of selected top atom pairs $K$ . The value of $K$ presents a trade-off between coverage and efficiency. In addition to the metric Coverage $@ k$ in Sec. 3.2, we use Recall $@ k$ which is the proportion of correct atom pairs that appear in top $k$ to find the good $K$ . Fig. 3 shows Coverage $@ k$ and Recall $@ k$ for the USPTO dataset with respect to $k$ . We see that both curves increase rapidly when $k < 1 0$ and stablize when $k > 1 0$ . We also ran experiments with $k = 1 0$ , 15, 20 and observed that their prediction results are quite similar. Hence, in what follows we select $K = 1 0$ for efficiency. + +Table 3: Results for reaction prediction. $P \ @ k$ is precision at $k$ . State-of-the-art results from (Jin et al., 2017) are written in italic. Results from (Schwaller et al., 2018) are marked with ? and they are computed on a slightly different version of USPTO that contains only single-product reactions. Best results are highlighted in bold. $\diamondsuit$ : With beam search (beam width $= 2 0$ ), $\lessgtr$ : Invalid product removal, ♣: Duplicated product removal. + +
ModelUSPTO-15kUSPTO
P@1P@3P@5P@1P@3P@5
WLDN (Jin et al., 2017)76.785.686.879.687.789.2
Seq2Seq (Schwaller et al., 2018)---80.3*86.2*87.5*
GTPN72.311171.26--
GTPN74.5682.6284.2373.2580.5683.53
GTPN474.5683.1984.9773.2584.3185.76
GTPN82.3985.6086.6883.2084.9785.90
GTPN**82.3985.7386.7883.2086.0386.48
+ +# 3.4 REACTION PRODUCT PREDICTION + +This experiment validates GTPN on full reaction product prediction against the recent state-of-the-art methods (Jin et al., 2017; Schwaller et al., 2018) using the accuracy metric. The recent method ELECTRO (Bradshaw et al., 2018) is not compatible here because it was only evaluated on a subset of USPTO limited to linear chain topology. Comparison against ELECTRO is reported separately in Appendix A.6. Table 3 shows the prediction results. We produce multiple reaction product candidates by using beam search decoding with beam width $N = 2 0$ . Details about beam search and its behaviors are presented in Appendix A.4. + +In brief, we compute the normalized-over-length log probabilities of $N$ predicted sequences of reaction triples and sort these values in descending order to get a rank list of $N$ possible reaction outcomes. Given a predicted sequence of reaction triples $( \overline { { u } } , v , b ) ^ { 0 : T }$ , we can generate reaction products from input reactants simply by replacing the old bond of $( u , v ) ^ { \tau }$ with $b ^ { \tau }$ . However, these products are not guaranteed to be valid (e.g., maximum valence constraint violation or aromatic molecules cannot be kekulized) so we post-process the outputs by removing all invalid products. The removal increases the top-1 accuracy by about $8 \%$ and $10 \%$ on $U S P T O – I 5 k$ and USPTO, respectively. Due to the permutation invariance of the predicted sequence of reaction triples, some product candidates are duplicate and will also be removed. This does not lead to any change in $P \ @ { \cal I }$ but slightly improves $P \ @ 3$ and $P \ @ 5$ by about $0 . 5 – 1 \%$ on the two datasets. + +Overall, GTPN with beam search and post-processing outperforms both WLDN (Jin et al., 2017) and Seq2Seq (Schwaller et al., 2018) in the top-1 accuracy. For the top-3 and top-5, our model’s performance is comparable to WLDN’s on USPTO- $l 5 k$ and is worse than WLDN’s on USPTO. It is not surprising since our model is trained to accurately predict the top-1 outcomes instead of ranking the candidates directly like WLDN. It is important to emphasize that we did not tune the model hyper-parameters when training on USPTO but reused the optimal settings from $U S P T O – I 5 k$ (which is 25 times smaller than USPTO) so the results may not be optimal (see Appendix A.3 for more training detail). + +# 4 RELATED WORK + +# 4.1 LEARNING TO PREDICT CHEMICAL REACTION + +In chemical reaction prediction, machine learning has replaced rule-based methods (Chen & Baldi, 2009) for better generalizability and scalability. Existing machine learning-based techiques are either template-free (Kayala & Baldi, 2011; Jin et al., 2017; Fooshee et al., 2018) and template-based (Wei et al., 2016; Segler & Waller, 2017; Coley et al., 2017). Both groups share the same mechanism: running multiple stages with the aid of reaction templates or rules. For example, in (Wei et al., 2016) the authors proposed a two-stage model that first classifies reactions into different types based on the neural fingerprint vectors (Duvenaud et al., 2015) of reactant and reagent molecules. Then, it applies pre-designed SMARTS transformation on the reactants with respect to the most suitable predicted reaction type to generate the reaction products. + +The work of (Jin et al., 2017) treats a reaction as a set of bond changes so in the first step, they predict which atom pairs are likely to be reactive using a variant of graph neural networks called Weisfeiler-Lehman Networks (WLNs). In the next step, they do almost the same as (Coley et al., 2017) by modifying the bond type between the selected atom pairs (with chemical rules satisfied) to create product candidates and rank them (with reactant molecules as addition input) using another kind of WLNs called Weifeiler-Lehman Different Networks (WLDNs). + +To the best of our knowledge, (Jin et al., 2017) is the first work that achieves remarkable results (with the Precision $@ 1$ is about $7 9 . 6 \%$ ) on the large USPTO dataset containing more than 480 thousands reactions. Works of (Nam & Kim, 2016) and (Schwaller et al., 2018) avoid multi-stage prediction by building a seq2seq model that generates the (canonical) SMILES string of the single product from the concatenated SMILES strings of the reactants and reagents in an end-to-end manner. However, their methods cannot deal with sets of reactants/reagents/products properly as well as cannot provide concrete reaction mechanism for every reaction. + +The most recent work on this topic is (Bradshaw et al., 2018) which solves the reaction prediction problem by predicting a sequence of bond changes given input reactants and reagents represented as graphs. To handle ordering, they only select reactions with predefined topology. Our method, by contrast, is order-free and can be applied to almost any kind of reactions. + +# 4.2 GRAPH NEURAL NETWORKS FOR MODELING MOLECULES + +In recent years, there has been a fast development of graph neural networks (GNNs) for modeling molecules. These models are proposed to solve different problems in chemistry including toxicity prediction (Duvenaud et al., 2015), drug activity classification (Shervashidze et al., 2011; Dai et al., 2016; Pham et al., 2018), protein interface prediction (Fout et al., 2017) and drug generation (Simonovsky & Komodakis, 2018; Jin et al., 2018). Most of them can be regarded as variants of message-passing graph neural networks (MPGNNs) (Gilmer et al., 2017). + +# 4.3 REINFORCEMENT LEARNING FOR STRUCTURAL REASONING + +Reinforcement learning (RL) has become a standard approach to many structural reasoning problems2 because it allows agents to perform discrete actions. A typical example of using RL for structural reasoning is drug generation (Li et al., 2018; You et al., 2018). Both (Li et al., 2018) and (You et al., 2018) learn the same generative policy whose action set including: i) adding a new atom or a molecular scaffold to the intermediate graph, ii) connecting existing pair of atoms with bonds, and iii) terminating generation. However, (You et al., 2018) uses an adversarial loss to enforce global chemical constraints on the generated molecules as a whole instead of using the common reconstruction loss as in (Li et al., 2018). Other examples are path-based relational reasoning in knowledge graphs (Das et al., 2018) and learning combinatorial optimization over graphs (Khalil et al., 2017). + +# 5 DISCUSSION + +We have introduced a novel method named Graph Transformation Policy Network (GTPN) for predicting products of a chemical reaction. GTPN uses graph neural networks to represent input reactant and reagent molecules, and uses reinforcement learning to find an optimal sequence of bond changes that transforms the reactants into products. We train GTPN using the Advantage Actor-Critic (A2C) method with appropriate constraints to account for notable aspects of chemical reaction. Experiments on real datasets have demonstrated the competitiveness of our model. + +Although the GTPN was proposed to solve the chemical reaction problem, it is indeed generic to solve the graph transformation problem, which can be useful in reasoning about relations (e.g., see (Zambaldi et al., 2018)) and changes in relation. Open rooms include addressing dynamic graphs over time, extending toward full chemical planning and structural reasoning using RL. + +# REFERENCES + +Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. Interaction networks for learning about objects, relations and physics. In Advances in neural information processing systems, pp. 4502–4510, 2016. + +John Bradshaw, Matt J Kusner, Brooks Paige, Marwin HS Segler, and José Miguel Hernández-Lobato. 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Modeling relational data with graph convolutional networks. 15th European Semantic Web Conference (ESWC-18), 2018. + +Philippe Schwaller, Theophile Gaudin, David Lanyi, Costas Bekas, and Teodoro Laino. “found in translation”: Predicting outcome of complex organic chemistry reactions using neural sequence-tosequence models. Chemical Science, 9:6091–6098, 2018. + +Marwin HS Segler and Mark P Waller. Neural-symbolic machine learning for retrosynthesis and reaction prediction. Chemistry–A European Journal, 23(25):5966–5971, 2017. + +Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. Weisfeiler-Lehman graph kernels. Journal of Machine Learning Research, 12(Sep): 2539–2561, 2011. + +Martin Simonovsky and Nikos Komodakis. GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders. arXiv preprint arXiv:1802.03480, 2018. + +Rupesh K Srivastava, Klaus Greff, and Jürgen Schmidhuber. Training very deep networks. In Advances in neural information processing systems, pp. 2377–2385, 2015. + +Ivar Ugi, Johannes Bauer, Josef Brandt, Josef Friedrich, Johann Gasteiger, Clemens Jochum, and Wolfgang Schubert. New applications of computers in chemistry. Angewandte Chemie International Edition in English, 18(2):111–123, 1979. + +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017. + +Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018. + +Jennifer N Wei, David Duvenaud, and Alán Aspuru-Guzik. Neural networks for the prediction of organic chemistry reactions. ACS Central Science, 2(10):725–732, 2016. + +Jiaxuan You, Bowen Liu, Rex Ying, Vijay Pande, and Jure Leskovec. Graph convolutional policy network for goal-directed molecular graph generation. NIPS, 2018. + +Vinicius Zambaldi, David Raposo, Adam Santoro, Victor Bapst, Yujia Li, Igor Babuschkin, Karl Tuyls, David Reichert, Timothy Lillicrap, Edward Lockhart, et al. Relational deep reinforcement learning. arXiv preprint arXiv:1806.01830, 2018. + +# A APPENDIX + +# A.1 GRAPH NEURAL NETWORK + +In this section, we describe our graph neural network (GNN) in detail. Since our GNN does not use the recurrent hidden state $h ^ { \tau }$ , we exclude the time step $\tau$ from our notations for clarity. Instead, we use $t$ to denote a message passing step. + +# GRAPH NOTATIONS + +Input to our GNN is a graph $\mathcal { G } = ( \nu , \mathcal { E } )$ in which each node $i \in \nu$ is represented by a node feature vector ${ \mathbf { } } v _ { i }$ and each edge $( i , j ) \in \mathcal { E }$ is represented by an edge feature vector $e _ { i j }$ . For example of molecular graph, the node feature vector ${ \mathbf { } } v _ { i }$ may include chemical information about the atom $i$ such as its type, charge and degree. Similarly, $e _ { i j }$ captures the bond type between the two atoms $i$ and $j$ We denote by $\bar { \mathcal { N } } ( i )$ the set of all neighbor nodes of node $i$ together with their links to node $i$ : + +$$ +\begin{array} { l l l } { { \mathcal N ( i ) } } & { { \equiv } } & { { \{ ( j , e _ { i j } ) \mid j { \mathrm { ~ i s ~ a ~ n e i g h b o r ~ n o d e ~ o f ~ } } i \} } } \end{array} +$$ + +If we only care about the neighbor nodes of $i$ not their links, we use the notation $\mathcal { N } _ { \mathrm { n } } ( i )$ defined as: + +$$ +\begin{array} { l l l } { { \mathcal N } _ { \mathrm { n } } ( i ) } & { \equiv } & { \left\{ j \ | \ j \ \mathrm { i s ~ a ~ n e i g h b o r ~ n o d e ~ o f } \ i \right\} } \end{array} +$$ + +In addition to ${ \mathbf { } } v _ { i }$ , node $i$ also has a state vector $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ to store information about itself and the surrounding context. This state vector is updated recursively using the neural message passing method (Battaglia et al., 2016; Pham et al., 2017; Hamilton et al., 2017; Gilmer et al., 2017; Schlichtkrull et al., 2018). The initial state $\pmb { x } _ { i } ^ { 0 }$ is the nonlinear mapping of ${ \mathbf { } } v _ { i }$ : + +$$ +\begin{array} { r c l } { { { \pmb x } _ { i } ^ { 0 } } } & { { = } } & { { \sigma \left( W { { \pmb v } _ { i } } + b \right) } } \end{array} +$$ + +# COMPUTING NEIGHBOR MESSAGES + +At the message passing step $t$ , we compute the message $\boldsymbol { m } _ { i j } ^ { t }$ from every neighbor node $j \in \mathcal { N } _ { n } ( i )$ to node $i$ as: + +$$ +\begin{array} { r c l } { { { \pmb m } _ { i j } ^ { t } } } & { { = } } & { { f \left( { \pmb x } _ { i } ^ { t } , { \pmb x } _ { j } ^ { t } , { \pmb e } _ { i j } \right) } } \\ { { } } & { { = } } & { { \sigma \left( W \left[ { \pmb x } _ { i } ^ { t } , { \pmb x } _ { j } ^ { t } , { \pmb e } _ { i j } \right] + b \right) } } \end{array} +$$ + +where $[ \cdot ]$ denotes concatenation; and $\sigma$ is a nonlinear function. + +AGGREGATING NEIGHBOR MESSAGES + +Then, we aggregate all the messages sent to node $i$ into a single message vector by averaging: + +$$ +\begin{array} { r c l } { \pmb { m } _ { i } ^ { t } } & { = } & { \displaystyle \frac { 1 } { \vert \mathcal { N } _ { \mathrm { n } } ( i ) \vert } \sum _ { j \in \mathcal { N } _ { \mathrm { n } } ( i ) } \pmb { m } _ { i j } ^ { t } } \end{array} +$$ + +where $| \mathcal { N } _ { \mathrm { n } } ( i ) |$ is the number of neighbor nodes of node $i$ + +UPDATING NODE STATE + +Finally, we update the state of node $i$ as follows: + +$$ +\begin{array} { r c l } { \pmb { x } _ { i } ^ { t + 1 } } & { = } & { g \left( \pmb { x } _ { i } ^ { t } , \pmb { m } _ { i } ^ { t } , \pmb { v } _ { i } \right) } \end{array} +$$ + +where $g ( . )$ is a Highway Network (Srivastava et al., 2015): + +$$ +\begin{array} { r c l } { { \pmb x _ { i } ^ { t + 1 } } } & { { = } } & { { \mathrm { H i g h w a y } \left( { \pmb x _ { i } ^ { t } , \pmb m _ { i } ^ { t } , \pmb v _ { i } } \right) } } \\ { { } } & { { = } } & { { { \pmb \alpha } * \tilde { \pmb x } _ { i } ^ { t + 1 } + ( 1 - { \pmb \alpha } ) * { \pmb x } _ { i } ^ { t } } } \end{array} +$$ + +where $\tilde { \pmb { x } } _ { i } ^ { t + 1 }$ is the nonlinear part which is computed as: $\bar { \pmb { x } } _ { i } ^ { t + 1 } = \sigma \left( \bar { W _ { 1 } } \left[ \pmb { x } _ { i } ^ { t } , \pmb { m } _ { i } ^ { t } , \pmb { v } _ { i } ^ { t } \right] + b _ { 1 } \right)$ and $_ \alpha$ is the gate controlling the flow of information: + +$$ +\pmb { \alpha } = \mathrm { s i g m o i d } ( W _ { 2 } \left[ \pmb { x } _ { i } ^ { t } , \pmb { m } _ { i } ^ { t } , \pmb { v } _ { i } ^ { t } \right] + b _ { 2 } ) +$$ + +By combining Eqs. (19,20,22) together, one step of message passing update for node $i$ can be written in a generic way as follows: + +$$ +\pmb { x } _ { i } ^ { t + 1 } = \mathrm { M e s s a g e P a s s i n g } \left( \pmb { x } _ { i } ^ { t } , \pmb { v } _ { i } , \mathcal { N } ( i ) \right) +$$ + +# A.2 UPDATING STATES + +UPDATING RNN STATE + +We keep the old representation of the edge that have been modified in the hidden memory of the RNN as follows: + +$$ +\begin{array} { r } { \pmb { h } ^ { \tau } = \mathbf { G } \mathbf { R } \mathbf { U } \left( \pmb { h } ^ { \tau - 1 } , \pmb { z } _ { u v } ^ { \tau } \right) } \end{array} +$$ + +where GRU stands for Gated Recurrent Units (Cho et al., 2014); $\boldsymbol { z } _ { u v } ^ { \tau }$ is the representation vector of the atom pair $( u , v ) ^ { \tau }$ including its old bond (see Eq. 4). Eq. (25) allows the model to keep track of all the changes happening to the graph so far so it can make more accurate prediction later. + +# UPDATING GRAPH STRUCTURE AND NODE STATES + +After predicting a reaction triple $( u , v , b ) ^ { \tau }$ at step $\tau$ , we update the graph structure and node states based on the new bond change. First, to update the graph structure, we simply update the neighbor set of $u$ and $v$ with information from the other atom and the new bond type $b$ as follows: + +$$ +\begin{array} { r l r } { \mathcal { N } ^ { \tau } ( u ) } & { = } & { \left( \mathcal { N } ^ { \tau - 1 } ( u ) \backslash \left( v , b ^ { \mathrm { o l d } } \right) \right) \cup ( v , b ) } \\ { \mathcal { N } ^ { \tau } ( v ) } & { = } & { \left( \mathcal { N } ^ { \tau - 1 } ( v ) \backslash \left( u , b ^ { \mathrm { o l d } } \right) \right) \cup ( u , b ) } \end{array} +$$ + +Next, to update the node states, our model performs one step of message passing for $u$ and $v$ with their new neighbor sets: + +$$ +\begin{array} { r l r } { \pmb { x } _ { u } ^ { \tau } } & { = } & { \mathsf { M e s s a g e P a s s i n g } \left( \pmb { x } _ { u } ^ { \tau - 1 } , \pmb { v } _ { u } , \mathcal { N } ^ { \tau } ( u ) \right) } \\ { \pmb { x } _ { v } ^ { \tau } } & { = } & { \mathsf { M e s s a g e P a s s i n g } \left( \pmb { x } _ { v } ^ { \tau - 1 } , \pmb { v } _ { v } , \mathcal { N } ^ { \tau } ( v ) \right) } \end{array} +$$ + +where the MessagePassing $( . )$ function is defined in Eq. (24). For other nodes in the graph to be aware of the new structures of $u$ and $v$ , we need to perform several message passing steps for all nodes in the graph after Eqs. (28, 29). However, it is very costly to run for every prediction step $\tau$ . Sometimes it is unnecessary since far-away bonds are less likely to be affected by the current bond change (unless the far-way bonds and the new bond are in an aromatic ring). Therefore, in our model, we limit the number of message passing updates for all nodes at step $\tau$ to be 1. + +# A.3 MODEL CONFIGURATIONS + +We optimize our model’s hyper-parameters in two stages: First, we tune the hyper-parameters of the GNN and the NPPN for the reaction atom pair prediction task. Then, we fix the optimal settings of the first two components and optimize the hyper-parameters of the PN for the reaction product prediction task. + +We provide details about the settings that give good results on the USPTO-15k dataset below. With these settings, we trained another model on the USPTO dataset from scratch. Because training on the large dataset such as the USPTO takes time, we did not tune hyper-parameters on the USPTO, eventhough it is possible to increase model sizes for better performance. + +Unless explicitly stated, all neural networks in our model have 2 layers with the same number of hidden units, ReLU activation and residual connections (He et al., 2016). + +Table 4: Data types of atom attributes. + +
Atom attributeData type
Degreenumeric
Explicit valencenumeric
Explicit number of Hsnumeric
Chargenumeric
Part of a ringboolean
+ +Graph Neural Network (GNN) There are 72 different types of atom depending on their atomic numbers and 5 different types of bond including NULL, SINGLE, DOUBLE, TRIPLE and AROMATIC. The size of embedding vectors for atom and bond are 51 and 21, respectively. Apart from atom type, each atom has 5 more attributes listed in Table 4. These attributes are normalized to the range of [0, 1] and are concatenated to the atom embedding vector to form a final atom feature vector of size 56. The state vector and the neighbor message vector for an atom both have the size of 99. The number of message passing steps is 6. + +Node Pair Prediction Network (NPPN) This component consists of two parts. The first part computes the representation vector $z _ { i j }$ of an atom pair $( i , j )$ using a neural network with hidden size of 71. The second part maps $z _ { i j }$ to an unnormalized score $s _ { i j }$ using the function $f ^ { \mathrm { a t o m } } \operatorname { p a i r }$ (see Eqs. (3,5)). This function is also a neural network with hidden size of 51. + +Policy Network (PN) The recurrent network is a GRU (Cho et al., 2014) with 101 hidden units. The value function $V _ { \phi }$ is a neural network with 99 hidden units. The two functions $f ^ { \mathrm { s i g n a l } }$ for computing signal scores (see Eq. (6)) and $f ^ { \mathrm { b o n d } }$ for computing scores over bond types (see Eq. (8)) are neural networks with 81 hidden units. + +Training At each step, we set the reward to be 1.0 for correct prediction of signal/atom pair/bond type and -1.0 for incorrect prediction. After the prediction sequence is terminated (zero signal was emitted), we check whether the entire set of predicted reaction triples is correct or not. If it is correct, we give the model a reward value of 2.0, otherwise $- 2 . 0$ . From the rewards and estimated values for signal, atom pair and bond type, we define the Advantage Actor Critic loss (A2C) as in Eq. (14). The coefficients of components in the final loss $\mathcal { L }$ are set empirically as follows: + +$$ +\mathcal { L } = \mathcal { L } ^ { \mathrm { A 2 C } } + 0 . 5 \times \mathcal { L } ^ { \mathrm { v a l u e } } + \mathcal { L } ^ { \mathrm { a t o m p a i r } } + 0 . 2 \times \mathcal { L } ^ { \mathrm { o v e r l e n g t h } } + 0 . 2 \times \mathcal { L } ^ { \mathrm { i n ~ t o p ~ } K } +$$ + +We trained our model using Adam (Kingma & Ba, 2015) with the initial learning rate of 0.001 for both $U S P T O – I 5 k$ and USPTO. For USPTO-15k, the learning rate will decrease by half if the Precision $@ l$ does not improve on the validation set after 1,000 steps until it reaches the minimum value of $5 \times 1 0 ^ { - 5 }$ . For USPTO, the decay rate is 0.8 after every 500 steps of no improvement until reaching the minimum learning rate is $2 \times 1 0 ^ { - 5 }$ . The maximum number of training iterations is $1 0 ^ { 6 }$ and the batch size is 20. + +# A.4 DECODING WITH BEAM SEARCH + +For decoding, our model generates a sequence of reaction triples (including the stop signal) $( \xi , u , v , b )$ by taking the best $( u , v )$ and $b$ at every step until it outputs a zero signal $\xi = 0$ ). In other words, it computes the argmax of $p \left( ( \xi , u , v , b ) ^ { \tau } \mid \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right)$ at every step $\tau$ . However, this algorithm is not robust for the sequence generation task because just a single error at a step may destroy the entire sequence. To overcome this issue, we employ beam search for decoding. + +During beam search, we keep track of $N > 1$ best subsequences at every step $\tau$ . $N$ is called beam width. Instead of modeling the conditional distribution of generating an output at the current step $\tau$ , we model the joint distribution of the whole subsequence that has been generated from 0 to $\tau$ : + +$$ +\begin{array} { r c l } { \log p \left( ( \xi , u , v , b ) ^ { 0 : \tau } | \mathcal { G } \right) } & { = } & { \log p \left( ( \xi , u , v , b ) ^ { \tau } | \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right) + } \\ & & { \ \log p \left( ( \xi , u , v , b ) ^ { 0 : \tau - 1 } | \mathcal { G } \right) } \end{array} +$$ + +
Beam widthPrecision@k
1235101520
174.49:二二二二-
272.2180.6511---
572.2179.5482.2984.27-二-
1072.1579.5482.1983.9386.01--
1572.1579.5482.1683.9386.1186.981
2074.5680.7282.6284.2386.1487.0487.55
+ +Table 5: Reaction product prediction results using beam search with different values of beam width on $U S P T O – I 5 k$ . + +Computing all configurations of $( \xi , u , v , b ) ^ { \tau }$ jointly is very memory demanding, however. Thus, we decompose the first term as follows: + +$$ +\begin{array} { l l l } { \log p \left( ( \xi , u , v , b ) ^ { \tau } | \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right) } & { = } & { \log p \left( \xi ^ { \tau } | \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right) + \mathstrut } \\ & & { \log p \left( ( u , v ) ^ { \tau } | \xi ^ { \tau } , \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right) + \mathstrut } \\ & & { \log \left( b ^ { \tau } | ( \xi , u , v ) ^ { \tau } , \mathcal { G } , ( \xi , u , v , b ) ^ { 0 : \tau - 1 } \right) } \end{array} +$$ + +At step $\tau$ , we do beam search for the signal $\xi ^ { \tau }$ , then the atom pair $( u , v ) ^ { \tau }$ and finally the bond type $b ^ { \tau }$ . Algorithm 1 describes beam search in detail. Some notable technicalities are: + +• We only do beam search for $( u , v )$ and $b$ if the prediction is ongoing, i.e., when $\xi ^ { \tau } = 1$ . To keep track of this, we use a boolean vector $C$ of length $N$ with $C ^ { 0 }$ is initialized to be all true. +• To avoid beam search favoring short sequences, we normalize the log probability scores over sequence lengths. This is shown in lines 10, 17, 32 and 47 + +# BEAM WIDTH ANALYSIS + +Table 5 reports how beam width affects the decoding performance on the USPTO-15k dataset. +Surprisingly, the top-1 accuracy in case of beam width3 of 1 is higher than the those when beam widths range from 2 to 15. It means that large beam width is not always good in our situation. +However, at beam width of 20, our beam search achieves the best results for different values of $k$ . +Thus, we set the beam width to 20 in subsequent experiments. + +# A.5 USING REAGENT INFORMATION EXPLICITLY + +As can be seen from Table 6, reagent molecules account for about a half of the input molecules on average and $60 \%$ of all reactions containing reagents. It suggests that the proper use of reagent information will lead to better prediction. In our model, before computing the scores for all atom pairs, we append to the representation vector of every atom a binary scalar indicating whether this atom comes from a reagent molecule or not. Then, at the top- $K$ atom pair selection step, we also exclude all atom pairs that have either atoms belong to a reagent molecule. The improvement in prediction accuracy on the validation set of $U S P T O – I 5 k$ is shown in Fig. 4. + +# A.6 COMPARISON WITH ELECTRO + +In method Both GTPN and ELECTRO (Bradshaw et al., 2018) are able to explain the mechanism behind a reaction. ELECTRO regards a reaction as an ordered sequence that alternates between removing and adding a single bond. Our model, on the other hand, assumes no specific order of transformations as well as the amount of valences that a bond can change. Thus, our model is more generic than ELECTRO and can cover a much larger set of reactions. + +
Algorithm 1 Reaction triple prediction using beam search.
Input: A multi-graph G consisting of reactant and reagent molecules, number of bond types E, max prediction steps T,beam width N
1: P0=[(-1,-1,-1,-1),] DThe best N subsequences of(ε,u,v,b)
2: s0 =[0,..] DThe length-normalized log joint probabilities of the best N subsequences 3: Co=[True,.] DThe continuation indicator of the best N subsequences
4: Perform L steps of message passing for all nodes using Eq. (1)
5:x=xii∈V DThe initial states of all nodes before decoding 6: N(i)=N(i) ∀i ∈V DThe initial neighbor set of all nodes before decoding
7: h° is loaded from the saved model DThe initial RNN hidden state before decoding
8: for T from 1 to T do
9: Find the top K atom pairs {(uk, Uk)T |k = 1,K} using Eqs. (4,5)
10: ST-1:0 = ST-1× T-1 Superscript O denotes the sub-step 0
PT-1:0 = PT-1;CT-1;0 =CT-1 11:
12: Beam search for continuation signals
13:
14: Rsignal =@ DStores the log joint probabilities for N × 2 possible signals
15: for n from 1 to N do 16: Computep (|Pn-1:0）usingEq. (6)
17: AddCT-i0× 1 logp(εT =δ| Pπ-10)+ S-1:0 to Rsignal forδ∈ {True,False}
18: end for
19: Sort Rsignal in descending order
20:
21:
22:
Pr-1;1 =extract (-, 23: CT-1;1 = extract (CT-1;0,[-1;1)
24: 25:
26:
27: Beam search for atom pairs 28:
29: Ratom pair = @ DStores the log joint probabilities for N × K possible atom pairs
30: for n from 1 to N do 31: Compute p (u,u)|Sn,P-1;1) using Eq. (7)
Add CT-i1 ×1logp((uU)sn,P-1;1)+S-1:1 to Ratompairk ∈1,K 32:
33: end for
Sort Ratom pair in descending order 34:
35:
36:
37:
Pt-1;2 = extract (P-1;1,[-1;2) 38:
39: CT-1;2 = extract (C-1;1,;2)
40: = extract (,-1;2)
41:
+ +Algorithm 2 Reaction triple prediction using beam search (cont.) + +
45:for n from1 to N do
46:Compute p (bT 1(E,u,o)n,P-1) using Eq. (8)
47:AddCT-i;2× ↓logp (bT =β|(ξ,u,u)，Pπ-1) + S[-1 to Rbond ∀β ∈1,B
48:end for
49: Sort Rbond in descending order
50:ST-1;3 = RO:N Rbond
51:
52:
53:PT-1;3 = extract (Pr-1;2, [T-1;3)
54:CT-1;3 = extract (CT-1;2, [T-1;3)
55:T = extract (T,IT-1;3)
56:(u,u)T = extract (u,u)T,IT-1;3)
57:
58:ST = ST-1;3;CT = CT-1;3
59:P =append(P-1:3，(,u,U,b)n)
60:for n from1 to N do
61:Update the NT(un) and NT(Un) for all n = 1,N using Eqs. (26,27)
62:Update xun and xn using Eq. (1)
63:Perform m steps of message passing for all nodes in the graph
64:Update husing Eq. (25)
65:end for
66: end for Output: PT,ST
+ +Table 6: Proportion of reactions containing reagents and proportion of reagents over input molecules on USPTO-15k and USPTO. + +
Dataset%reactionscontaining reagents%reagents over input molecules
USPTO-15ktrain63.1%41.3%
valid65.3%42.3%
test63.6%40.9%
USPTOtrain79.7%54.0%
valid80.0%54.4%
test79.9%54.2%
+ +![](images/f49e2a5a755feff77e408c1dc8e86781a7fed33932c72027630272324b583ea6.jpg) +Figure 4: Learning curves of our model with and without using reagent information explicitly on USPTO-15k. + +Table 7: Results for the reaction prediction task. $P \ @ k$ is the precision at $k$ . Best results are highlighted in bold. Meanings of markers in our model: $\diamondsuit$ : With beam search (beam width $= 2 0$ ), $\spadesuit$ : Invalid product removal, $\mathbf { \hat { \psi } } _ { \mathbf { * } }$ : Duplicate product removal. + +
ModelProcessed USPTO
P@1P@3P@5
WLDN (Jin et al., 2017)84.091.192.3
ELECTRO (Bradshaw et al., 2018)GTPN**87.094.595.9
87.3590.2290.68
+ +![](images/aebc50a959e34bea76abae1e557fd21e1b1f51ece918963e01707e0931304628.jpg) +Figure 5: Performance with respect to different numbers of bond changes. (a) Top-1 accuracy. (b) Errors grouped by length. In (a), blue: all reactions having that sequence length; orange: correct predicted reactions. In (b), red: the predicted sequence is shorter (than the groundtruth sequence); green: the predicted and the groundtruth have the same length; blue: the predicted sequence is longer; number indidate the average length. + +In performance To do a fair comparison with ELECTRO (Bradshaw et al., 2018), we follow their procedure described in the paper to prepare a new test set that contains only reactions with linear chain topology and single-valence bond changes. It results in 29,808 reactions, close to the reported number of 29,360 in (Bradshaw et al., 2018). We reuse our old model (see Section 3.4) trained on the original USPTO dataset. We also use beam search decoding and post-processing as similar to (Bradshaw et al., 2018). From Table 7, we see that GTPN achieves the highest top-1 accuracy of $8 7 . 3 5 \%$ , outperforming ELECTRO and WLDN by $0 . 3 5 \%$ and $3 \%$ , respectively. For the top-3 and top-5 accuracies, our model, however, does worse than the other two. Especially, while both ELECTRO and WLDN have big jumps from $P \ @ { \cal I }$ to $P ( \omega 3$ with about $7 \%$ improvement, GTPN only has $3 \%$ increase. We conjecture that this problem mainly comes from the fact that GTPN was not optimized on the compatible training and validation sets. + +# A.7 ERROR ANALYSIS + +In this section, we analyze several error types that our model makes during prediction. All the results below are computed on the USPTO-15k dataset by using beam search decoding with the beam width $N = 2 0$ and no post-processing. + +Errors grouped by number of bond changes Fig. 5 shows the top-1 accuracies for reactions with different number of bond changes. Our model performs poorly on reactions with many bond changes. However, those kinds of reactions only accounts for a small proportion in the dataset. From Fig. 5b, we see that the lengths of the error sequences tends to be shorter than the lengths of the groundtruth sequences. + +Errors caused by signal/atom pair/bond type We define a sub-action causing error as the first sub-action that our model makes a wrong decision. In Fig. 6a, we plot the the proportion of errors with respect to the three kinds of sub-actions. Clearly, atom pair prediction causes the most errors + +![](images/4f2ec61d977e14daa78a49521f7d4f16a8bd5f1d149bfd67fb5ba89e7f0dd975.jpg) + +(a) Proportion of the first incorrect sub-action that our model makes. + +![](images/1fa8e0b536b959f903902913cdcd3bc4058776fee20d292f2bb349e260285dcb.jpg) +(b) Proportion of the incorrect top-1 products that have similar structure to the groundtruth products. + +Figure 6: Errors grouped by the first incorrect sub-actions (a), and errors caused by symmetric structures (b). + +(nearly two third). This makes sense because this sub-action is harder than signal prediction and bond type prediction. Therefore, more effort should be put on improving the prediction of atom pairs. + +Errors caused by symmetry There exists cases in which different sequences of bond changes can result in the same products due to symmetric graph structures. Errors caused by symmetry account for $5 . 7 \%$ of the top-1 errors on the USPTO-15k dataset as shown in Fig. 6b. For better understanding, we provide a short list of wrong reaction triple predictions caused by symmetry in Fig. 7. In this list, the top-1 products (along the second column) are incorrect while the top-2 products (along the third column) are correct though both have the same probability. + +![](images/c508e5522ddc1d32c7b23e4219b31cd604fecc99998016a648a66b09069a3155.jpg) +Figure 7: Visualization of some reactions that cause multiple products with symmetric structures. Each row corresponds to a reaction. The columns, from left to right, show: i) reactant and reagent molecules, ii) incorrect top-1 product molecules, iii) correct top-2 product molecules, and iv) major groundtruth product molecules. All atoms in the first three columns are labeled with their atom map numbers. For the top-1 and top-2 products, we highlight the predicted reaction triples in green and provide the probability of the predicted sequence at the bottom. \ No newline at end of file diff --git a/parse/train/r1f78iAcFm/r1f78iAcFm_content_list.json b/parse/train/r1f78iAcFm/r1f78iAcFm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2ce36bdd923d68d45286b30cd5f9d5c5948c716f --- /dev/null +++ b/parse/train/r1f78iAcFm/r1f78iAcFm_content_list.json @@ -0,0 +1,2715 @@ +[ + { + "type": "text", + "text": "GRAPH TRANSFORMATION POLICY NETWORK FOR CHEMICAL REACTION PREDICTION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 732, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We address a fundamental problem in chemistry known as chemical reaction product prediction. Our main insight is that the input reactant and reagent molecules can be jointly represented as a graph, and the process of generating product molecules from reactant molecules can be formulated as a sequence of graph transformations. To this end, we propose Graph Transformation Policy Network (GTPN) − a novel generic method that combines the strengths of graph neural networks and reinforcement learning to learn the reactions directly from data with minimal chemical knowledge. Compared to previous methods, GTPN has some appealing properties such as: end-to-end learning, and making no assumption about the length or the order of graph transformations. In order to guide model search through the complex discrete space of sets of bond changes effectively, we extend the standard policy gradient loss by adding useful constraints. Evaluation results show that GTPN improves the top-1 accuracy over the current state-of-the-art method by about $3 \\%$ on the large USPTO dataset. Our model’s performances and prediction errors are also analyzed carefully in the paper. ", + "bbox": [ + 233, + 263, + 766, + 473 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Chemical reaction product prediction is a fundamental problem in organic chemistry. It paves the way for planning syntheses of new substances (Chen & Baldi, 2009). For decades, huge effort has been spent to solve this problem. However, most methods still depend on the handcrafted reaction rules (Chen & Baldi, 2009; Kayala & Baldi, 2011; Wei et al., 2016) or heuristically extracted reaction templates (Segler & Waller, 2017; Coley et al., 2017), thus are not well generalizable to unseen reactions. ", + "bbox": [ + 174, + 526, + 825, + 611 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A reaction can be regarded as a set (or unordered sequence) of graph transformations in which reactants represented as molecular graphs are transformed into products by modifying the bonds between some atom pairs (Jochum et al., 1980; Ugi et al., 1979). See Fig. 1 for an illustration. We call an atom pair $( u , v )$ that changes its connectivity during reaction and its new bond $b$ a reaction triple $( u , v , b )$ . The reaction product prediction problem now becomes predicting a set of reaction triples given the input reactants and reagents. We argue that in order to solve this problem well, an intelligent system should have two key capabilities: (a) Understanding the molecular graph structure of the input reactants and reagents so that it can identify possible reactivity patterns (i.e., atom pairs with changing connectivity). (b) Knowing how to choose from these reactivity patterns a correct set of reaction triples to generate the desired products. ", + "bbox": [ + 174, + 617, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent state-of-the-art methods (Jin et al., 2017; Bradshaw et al., 2018) have built the first capability by leveraging graph neural networks (Duvenaud et al., 2015; Hamilton et al., 2017; Pham et al., 2017; Gilmer et al., 2017). However, these methods are either unaware of the valid sets of reaction triples (Jin et al., 2017) or limited to sequences of reaction triples with a predefined orders (Bradshaw et al., 2018). The main challenge is that the space of all possible configurations of reaction triples is extremely large and non-differentiable. Moreover, a small change in the predicted set of reaction triples can lead to very different reaction products and a little mistake can produce invalid prediction. ", + "bbox": [ + 174, + 763, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we propose a novel method called Graph Transformation Policy Network (GTPN) that addresses the aforementioned challenges. Our model consists of three main components: a graph neural network (GNN), a node pair prediction network (NPPN) and a policy network (PN). Starting from the initial graph of reactant and reagent molecules, our model iteratively alternates between modeling an input graph using GNN and predicting a reaction triple using NPPN and PN to generate a new intermediate graph as input for the next step until it decides to stop. The final generated graph is considered as the predicted products of the reaction. Importantly, GTPN does not assume any fixed number or any order of bond changes but learn these properties itself. One can view GTPN as a reinforcement learning (RL) agent that operates on a complex and non-differentiable space of sets of reaction triples. To guide our model towards learning a diverse yet robust-to-small-changes policy, we customize our loss function by adding some useful constraints to the standard policy gradient loss (Mnih et al., 2016). ", + "bbox": [ + 176, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/a12f14d83ce2155ad6d7941ebcf1a5e44c5921b556fe473e90e0c85202a223ba.jpg", + "image_caption": [ + "Figure 1: A sample reaction represented as a set of graph transformations from reactants (leftmost) to products (rightmost). Atoms are labeled with their type (Carbon, Oxygen,...) and their index (1, 2,...) in the molecular graph. The atom pairs that change connectivity and their new bonds (if existed) are highlighted in green. There are two bond changes in this case: 1) The double bond between O:1 and C:2 becomes single. 2) A new single bond between C:2 and C:10 is added. " + ], + "image_footnote": [], + "bbox": [ + 215, + 99, + 795, + 218 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 321, + 825, + 434 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To the best of our knowledge, GTPN is the most generic approach for the reaction product prediction problem so far in the sense that: i) It combines graph neural networks and reinforcement learning into a unified framework and trains everything end-to-end; ii) It does not use any handcrafted or heuristically extracted reaction rules/templates to predict the products. Instead, it automatically learns various types of reactions from the training data and can generalize to unseen reactions; iii) It can interpret how the products are formed via the sequence of reaction triples it generates. ", + "bbox": [ + 174, + 440, + 825, + 523 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluate GTPN on two large public datasets named $U S P T O – I 5 k$ and USPTO. Our method significantly outperforms all baselines in the top-1 accuracy, achieving new state-of-the-art results of $8 2 . 3 9 \\%$ and $8 3 . 2 0 \\%$ on $U S P T O – I 5 k$ and USPTO, respectively. In addition, we also provide comprehensive analyses about the performance of GTPN and about different types of errors our model could make. ", + "bbox": [ + 174, + 531, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 625, + 281, + 641 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 CHEMICAL REACTION AS MARKOV DECISION PROCESS OF GRAPH TRANSFORMATIONS ", + "bbox": [ + 176, + 659, + 820, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A reaction occurs when reactant molecules interact with each other in the presence (or absence) of reagent molecules to form new product molecules by breaking or adding some of their bonds. Our main insight is that reaction product prediction can be formulated as predicting a sequence of such bond changes given the reactant and reagent molecules as input. A bond change is characterized by the atom pair (where the change happens) and the new bond type (what is the change). We call this atom pair a reaction atom pair and call this atom pair with the new bond type a reaction triple. ", + "bbox": [ + 174, + 686, + 825, + 770 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "More formally, we represent the entire system of input reactant and reagent molecules as a labeled graph $\\mathcal { G } = ( \\dot { \\mathcal { V } } , \\mathcal { E } )$ with multiple connected components, each of which corresponds to a molecule. Nodes in $\\nu$ are atoms labeled with their atomic numbers and edges in $\\mathcal { E }$ are bonds labeled with their bond types. Given $\\mathcal { G }$ as input, we predict a sequence of reaction triples that transforms $\\mathcal { G }$ into a graph of product molecules $\\mathcal { G } ^ { \\prime }$ . ", + "bbox": [ + 174, + 776, + 825, + 847 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As reactions vary in number of transformation steps, we represent the sequence of reaction triples as $( \\xi , u , v , b ) ^ { 0 } , ( \\dot { \\xi } , u , v , b ) ^ { 1 } , . . . , ( \\xi , u , v , b ) ^ { T - 1 }$ or $( \\dot { \\xi } , \\dot { u } , v , b ) ^ { \\dot { 0 } : T }$ for short. Here $T$ is the maximum number of steps, $( u , v )$ is a pair of nodes, $b$ is the new edge type of $( u , v )$ , and $\\xi$ is a binary signal that indicates the end of the sequence. If the sequence ends at $\\dot { T } _ { \\mathrm { e n d } } < \\dot { T } , \\dot { \\xi } ^ { 0 } , . . . \\xi ^ { \\dot { T } _ { \\mathrm { c n d } } - 1 }$ will be 1 and $\\xi ^ { T _ { \\mathrm { e n d } } } , . . . , \\xi ^ { T - 1 }$ will be 0. At every step $\\tau$ , if $\\xi ^ { \\tau } = 1$ end , we apply the predicted edge change $( u , v , b ) ^ { \\tau }$ on the current graph $\\mathcal { G } ^ { \\tau }$ to create a new intermediate graph $\\mathcal { G } ^ { \\tau + 1 }$ as input for the next step $\\tau + 1$ . This iterative process of graph transformation can be formulated as a Markov Decision Process (MDP) characterized by a tuple $( S , { \\mathcal { A } } , P , R , \\gamma )$ , in which $s$ is a set of states, $\\mathcal { A }$ is a set of actions, $P$ is a state transition function, $R$ is a reward function, and $\\gamma$ is a discount factor. Since the process is finite and contains no loop, we set the discount factor $\\gamma$ to be 1. The rest of the MDP tuple are defined as follows: ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/1228c94e7af5a5048db2eb5fe16475e451da8a13e5b75b55f27945f853ca3bb0.jpg", + "image_caption": [ + "Figure 2: Workflow of a Graph Transformation Policy Network (GTPN). At every step of the forward pass, our model performs 7 major functions: 1) Computing the atom representation vectors, 2) Computing the most possible $K$ reaction atom pairs, 3) Predicting the continuation signal $\\xi$ , 4) Predicting the reaction atom pair $( u , v )$ , 5) Predicting a new bond $b$ of this atom pair, 6) Updating the atom representation vectors, and 7) Updating the recurrent state. " + ], + "image_footnote": [], + "bbox": [ + 279, + 103, + 722, + 323 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 426, + 825, + 510 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• State: A state $s ^ { \\tau } \\in \\boldsymbol { S }$ is an intermediate graph $\\mathcal { G } ^ { \\tau }$ generated at step $\\tau$ $( 0 \\leq \\tau < T )$ ). When $\\tau = 0$ , we denote $s ^ { 0 } = \\mathcal { G } ^ { 0 } = \\mathcal { G }$ . \nAction: An action $a ^ { \\tau } \\in { \\mathcal { A } }$ performed at step $\\tau$ is the tuple $( \\xi , u , v , b ) ^ { \\tau }$ . The action is composed of three consecutive sub-actions: $\\xi ^ { \\tau }$ , $( u , v ) ^ { \\tau }$ , and $b ^ { \\tau }$ . If $\\xi ^ { \\tau } = 0$ , our model will ignore the next sub-actions $( u , v ) ^ { \\tau }$ and $b ^ { \\tau }$ , and all the future actions $( \\xi , u , v , b ) ^ { \\tau + 1 : T }$ . Note that setting $\\xi ^ { \\tau }$ to be the first sub-action is useful in case a reaction does not happen, i.e., $\\xi ^ { 0 } = 0$ \n• State Transition: If $\\xi ^ { \\tau } = 1$ , the current graph $\\mathcal { G } ^ { \\tau }$ is modified based on the reaction triple $( u , v , b ) ^ { \\tau }$ to generate a new intermediate graph $\\mathcal { G } ^ { \\tau + 1 }$ . We do not incorporate chemical rules such as valency check during state transition because the current bond change may result in invalid intermediate molecules $\\mathcal { G } ^ { \\tau }$ , but later, other bond changes may compensate it to create the valid final products $\\mathcal { G } ^ { T _ { \\mathrm { e n d } } }$ . Reward: We use both immediate rewards and delayed rewards to encourage our model to learn the optimal policy faster. At every step $\\tau$ , if the model predicts $\\xi ^ { \\tau }$ , $( u , v ) ^ { \\tau }$ or $b ^ { \\tau }$ correctly, it will receive a positive reward for each correct sub-action. Otherwise, a negative reward is given. After the prediction process has terminated, if the generated products are exactly the same as the groundtruth products, we give the model a positive reward, otherwise a negative reward. The concrete reward values are provided in Appendix A.3. ", + "bbox": [ + 215, + 521, + 825, + 777 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 GRAPH TRANSFORMATION POLICY NETWORK ", + "text_level": 1, + "bbox": [ + 174, + 799, + 537, + 814 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we describe the architecture of our model − a Graph Transformation Policy Network (GTPN). GTPN has three main components namely a Graph Neural Network (GNN), a Node Pair Prediciton Network (NPPN), and a Policy Network (PN). Each component is responsible for one or several key functions shown in Fig. 2: GNN performs functions 1 and 6; NPPN performs function 2; and PN performs functions 3, 4 and 5. Apart from these components, GTPN also has a Recurrent Neural Network (RNN) to keep track of the past transformations. The hidden state $^ { h }$ of this RNN is used by NPPN and PN to make accurate prediction. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2.1 GRAPH NEURAL NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 421, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To model the intermediate graph $\\mathcal { G } ^ { \\tau }$ at step $\\tau$ , we compute the node state vector $\\mathbf { \\boldsymbol { x } } _ { i } ^ { \\tau }$ of every node $i$ in $\\mathcal { G } ^ { \\tau }$ by using a variant of the Message Passing Neural Networks (Gilmer et al., 2017): ", + "bbox": [ + 171, + 127, + 823, + 156 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d77e4320649480dd0a9399b6eb7fd73fe0c25747405e83be9ef3cd63e6ea38ac.jpg", + "text": "$$\n\\begin{array} { r l r } { \\pmb { x } _ { i } ^ { \\tau } } & { = } & { \\mathrm { M e s s a g e P a s s i n g } ^ { m } \\left( \\pmb { x } _ { i } ^ { \\tau - 1 } , \\pmb { v } _ { i } , \\sqrt { \\tau } \\left( i \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 174, + 655, + 191 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $m$ is the number of message passing steps; ${ \\mathbf { } } v _ { i }$ is the feature vector of node $i$ ; $\\mathcal { N } ^ { \\tau } ( i )$ is the set of all neighbor nodes of node $i$ ; and $\\pmb { x } _ { i } ^ { \\tau - 1 }$ is the state vector of node $i$ at the previous step. When $\\tau = 0$ $\\pmb { x } _ { i } ^ { \\tau - 1 }$ is initialized from ${ \\mathbf { } } v _ { i }$ using a neural network. Details about the MessagePassing $( . )$ function are provided in Appendix A.1. ", + "bbox": [ + 174, + 191, + 825, + 250 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2.2 NODE PAIR PREDICTION NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 263, + 477, + 279 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to predict how likely an atom pair $( i , j )$ of the intermediate graph $\\mathcal { G } ^ { \\tau }$ will change its bond, we assign $( i , j )$ with a score $s _ { i j } ^ { \\tau } \\in \\mathbb { R }$ . If $s _ { i j } ^ { \\tau }$ is high, $( i , j )$ is more probably a reaction atom pair, otherwise, less probably. Similar to (Jin et al., 2017), we use two different networks called “local” network and “global” network for this task. In case of the “local” network, $s _ { i j } ^ { \\tau }$ is computed as: ", + "bbox": [ + 174, + 287, + 826, + 344 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1b6390514bd372cd89bdd88a0dde60ab6f47e8860d83bfe9150c214194095545.jpg", + "text": "$$\n\\begin{array} { r c l } { { z _ { i j } ^ { \\tau } } } & { { = } } & { { \\sigma \\left( W _ { 1 } \\left[ { { h ^ { \\tau } } ^ { - 1 } } , ( { x _ { i } ^ { \\tau } } + { { \\bf { x } } _ { j } ^ { \\tau } } ) , { { e _ { i j } } } \\right] + b _ { 1 } \\right) } } \\\\ { { s _ { i j } ^ { \\tau } } } & { { = } } & { { f ^ { \\mathrm { a t o m p a i r } } \\left( { z _ { i j } ^ { \\tau } } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 361, + 653, + 404 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f ^ { \\mathrm { a t o m } } \\operatorname { p a i r }$ is a neural network; $\\sigma$ is a nonlinear activation function (e.g., ReLU); $[ . ]$ denotes vector concatenation; $W _ { 1 }$ and $b _ { 1 }$ are parameters; $\\pmb { h } ^ { \\tau - 1 }$ is the hidden state of the RNN at the previous step; and $e _ { i j }$ is the representation vector of the bond between $( i , j )$ . If there is no bond between $( i , j )$ we assume that its bond type is “NULL”. We consider $z _ { i j }$ as the representation vector for the atom pair $( i , j )$ . ", + "bbox": [ + 173, + 401, + 826, + 473 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The “global” network leverages self-attention (Vaswani et al., 2017; Wang et al., 2018) to detect compatibility between atom $i$ and all other atoms before computing the scores: ", + "bbox": [ + 171, + 479, + 823, + 508 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/933c04373fcfe5ff735c4045bc72b308060f7573839c07c19e20f519e7870a81.jpg", + "text": "$$\n\\begin{array} { r c l } { { r _ { i j } ^ { \\tau } } } & { { = } } & { { \\sigma \\left( V _ { 1 } \\left[ ( { \\bf { x } } _ { i } ^ { \\tau } + { \\bf { x } } _ { j } ^ { \\tau } ) , e _ { i j } \\right] + c _ { 1 } \\right) } } \\\\ { { a _ { i j } ^ { \\tau } } } & { { = } } & { { \\mathrm { s o f t m a x } \\left( V _ { 2 } r _ { i j } ^ { \\tau } + c _ { 2 } \\right) } } \\\\ { { c _ { i } ^ { \\tau } } } & { { = } } & { { \\displaystyle \\sum _ { j \\in \\mathcal { V } } a _ { i j } { \\bf { x } } _ { j } ^ { \\tau } } } \\\\ { { z _ { i j } ^ { \\tau } } } & { { = } } & { { \\sigma \\left( W _ { 1 } \\left[ { \\bf { h } } ^ { \\tau - 1 } , ( { \\bf { x } } _ { i } ^ { \\tau } + { \\bf { x } } _ { j } ^ { \\tau } ) , ( c _ { i } ^ { \\tau } + c _ { j } ^ { \\tau } ) , e _ { i j } \\right] + b _ { 1 } \\right) } } \\\\ { { s _ { i j } ^ { \\tau } } } & { { = } } & { { f _ { \\mathrm { c o m p a i r } } ^ { \\mathrm { a t m } } \\left( z _ { i j } ^ { \\tau } \\right) } } \\\\ { { . } } & { { } } & { { . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 523, + 691, + 643 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $a _ { i j }$ is the attention score from node $\\imath$ to every other node $j$ ; $c _ { i }$ is the context vector of atom $i$ that summarizes the information from all other atoms. ", + "bbox": [ + 176, + 641, + 821, + 667 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "During experiments, we tried both options mentioned above and saw that the “global” network clearly outperforms the “local” network so we set the “global” network as a default module in our model. In addition, since reagents never change their form during a reaction, we explicitly exclude all atom pairs that have either atoms belong to the reagents. This leads to better results than not using reagent information. Detailed analyses are provided in Appendix A.5. ", + "bbox": [ + 173, + 674, + 825, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Top- $K$ atom pairs Because the number of atom pairs that actually participate in a reaction is very small (usually smaller than 10) compared to the total number of atom pairs of the input molecules (usually hundreds or thousands), it is much more efficient to identify reaction triples from a small subset of highly probable reaction atom pairs. For that reason, we extract $K$ $K \\ll | \\mathcal { V } | ^ { 2 } )$ atom pairs with the highest scores. Later, we will predict reaction triples taken from these $K$ atom pairs only. We denote the set of top- $K$ atom pairs, their corresponding scores, and representation vectors as $\\left\\{ ( u _ { k } , v _ { k } ) | k = \\overline { { 1 , K } } \\right\\}$ , $\\left\\{ \\dot { s } _ { u _ { k } v _ { k } } | k = \\overline { { 1 , K } } \\right\\}$ and $Z _ { K } = \\mathsf { \\bar { \\{ } } z _ { u _ { k } v _ { k } } \\mathsf { \\bar { | } } k = \\overline { { 1 , K } } \\mathsf \\}$ , respectively. ", + "bbox": [ + 173, + 758, + 825, + 859 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2.3 POLICY NETWORK ", + "text_level": 1, + "bbox": [ + 174, + 871, + 359, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Predicting continuation signal To account for varying number of transformation steps, PN generates a continuation signal $\\xi ^ { \\tau } \\in \\{ 0 , 1 \\}$ to indicate whether prediction should continue or terminate. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\xi ^ { \\tau }$ is drawn from a Bernoulli distribution: ", + "bbox": [ + 174, + 103, + 447, + 117 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/709d77e53ade9cdb7e160c79b62f567a558ae5c7ccc1590336e5891ff2a1886e.jpg", + "text": "$$\n\\begin{array} { l l l } { p \\left( \\xi ^ { \\tau } = 1 \\right) } & { = } & { \\mathrm { s i g m o i d } \\left( f ^ { \\mathrm { s i g n a l } } \\left( \\left[ { h ^ { \\tau - 1 } } , g \\left( Z _ { K } ^ { \\tau } \\right) \\right] \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 118, + 671, + 137 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\pmb { h } ^ { \\tau - 1 }$ is the previous RNN state; $Z _ { K } ^ { \\tau }$ is the set of representation vectors of the top $K$ atom pairs at the current step; $f ^ { \\mathrm { s i g n a l } }$ is a neural network; $g$ is a function that maps an unordered set of inputs to an output vector. For simplicity, we use a mean function: ", + "bbox": [ + 173, + 140, + 826, + 185 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0e9c7f7c8f57fbffa1573059185dd878f6c44a2b1f423afaa3df4ccddc3b9b61.jpg", + "text": "$$\nz _ { K } ^ { \\tau - 1 } = g \\left( Z _ { K } ^ { \\tau } \\right) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } W z _ { u _ { k } v _ { k } } ^ { \\tau - 1 }\n$$", + "text_format": "latex", + "bbox": [ + 382, + 200, + 614, + 246 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Predicting atom pair At the next sub-step, PN predicts which atom pair changes its bond during the reaction by sampling from the top- $K$ atom pairs with probability: ", + "bbox": [ + 171, + 257, + 823, + 287 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b4fafbedf0de609bab767affb0550c203bd8c905191a98b65789adeeb95db222.jpg", + "text": "$$\np \\left( ( u _ { k } , v _ { k } ) ^ { \\tau } \\right) = \\operatorname { s o f t m a x } _ { K } \\left( s _ { u _ { k } v _ { k } } ^ { \\tau } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 383, + 305, + 614, + 325 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $s _ { u _ { k } v _ { k } } ^ { \\tau }$ is the score of the atom pair $( u _ { k } , v _ { k } ) ^ { \\tau }$ computed in Eq. (5). After predicting the atom pair $( u , v ) ^ { \\tau }$ , we will mask it to ensure that it could not be in the top $K$ again at future steps. ", + "bbox": [ + 171, + 328, + 825, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Predicting bond type Given an atom pair $( u , v ) ^ { \\tau }$ sampled from the previous sub-step, we predict a new bond type $b ^ { \\tau }$ between $u$ and $v$ to get a complete reaction triple $( u , v , b ) ^ { \\tau }$ using the probability: ", + "bbox": [ + 173, + 372, + 826, + 402 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d5b09c39fcdcf3ff65ce1c62f6cbf26fc75e6480a2d2563489e7eaa34c5e72a9.jpg", + "text": "$$\n\\begin{array} { r } { p ( b ^ { \\tau } \\vert ( u , v ) ^ { \\tau } ) = \\mathrm { s o f t m a x } _ { B } ( f ^ { \\mathrm { b o n d } } ( [ h ^ { \\tau - 1 } , z _ { u v } ^ { \\tau } , ( e _ { b } - e _ { b ^ { \\mathrm { o l d } } } ) ] ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 420, + 702, + 439 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $B$ is the total number of bond types; $z _ { u v } ^ { \\tau }$ is the representation vector of $( u , v ) ^ { \\tau }$ computed in Eq. (4); $b ^ { \\mathrm { o l d } }$ is the old bond of $( u , v )$ ; $e _ { b ^ { \\mathrm { o l d } } }$ and $e _ { b }$ are the embedding vectors corresponding to the bond type $b ^ { \\mathrm { o l d } }$ and $b$ , respectively; and $f ^ { \\mathrm { b o n d } }$ is a neural network. ", + "bbox": [ + 173, + 443, + 825, + 486 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 UPDATING STATES ", + "text_level": 1, + "bbox": [ + 174, + 502, + 344, + 517 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After predicting a complete reaction triple $( u , v , b ) ^ { \\tau }$ , our model updates: i) the new recurrent hidden state $\\pmb { h } ^ { \\tau }$ , and ii) the new node representation vectors $\\pmb { x } _ { i } ^ { \\tau + 1 }$ of the new intermediate graph $\\mathcal { G } ^ { \\tau + 1 }$ for $i \\in \\nu$ . These updates are presented in Appendix A.2. ", + "bbox": [ + 173, + 527, + 826, + 573 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 588, + 287, + 603 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Loss function plays a central role in achieving fast training and high performance. We design the following loss: ", + "bbox": [ + 173, + 614, + 823, + 645 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8d862a280a70c08918889b6f10aff6b54ad83d54fe90cded5a7ddb9e6a99e42d.jpg", + "text": "$$\n{ \\mathcal { L } } = \\lambda _ { 1 } { \\mathcal { L } } ^ { \\mathrm { { A 2 C } } } + \\lambda _ { 2 } { \\mathcal { L } } ^ { \\mathrm { { v a l u e } } } + \\lambda _ { 3 } { \\mathcal { L } } ^ { \\mathrm { { a t o m p a i r } } } + \\lambda _ { 4 } { \\mathcal { L } } ^ { \\mathrm { { o v e r l e n g t h } } } + \\lambda _ { 5 } { \\mathcal { L } } ^ { \\mathrm { { i n } } \\mathrm { { t o p } } K }\n$$", + "text_format": "latex", + "bbox": [ + 276, + 661, + 718, + 679 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathcal { L } ^ { \\mathrm { A 2 C } }$ is the Advantage Actor-Critic (A2C) loss (Mnih et al., 2016) to account for the correct sequence of reaction triples; $\\mathcal { L } ^ { \\mathrm { v a l u e } }$ is the loss for estimating the value function used in A2C; $\\mathcal { L } ^ { \\mathrm { a t o m } }$ pair accounts for binary change in the bond of an atom pair; Lover length penalizes long predicted sequences; and $\\mathcal { L } ^ { \\mathrm { i n t o p } ~ K }$ is the rank loss to force a ground-truth reaction atom pair to appear in the top- $K$ ; and $\\lambda _ { 1 } , . . . , \\lambda _ { 5 } > 0$ are tunable coefficients. The component losses are explained in the following. ", + "bbox": [ + 173, + 683, + 825, + 756 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4.1 REACTION TRIPLE LOSS", + "text_level": 1, + "bbox": [ + 174, + 770, + 397, + 785 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The loss follows a policy gradient method known as Advantage Actor-Critic (A2C): ", + "bbox": [ + 173, + 795, + 722, + 810 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a1ec25d303abf6fa1aa773564d1d6f08da8719abd97f464d2e8fc21578e43773.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\mathcal { L } ^ { \\mathrm { A 2 C } } } } & { { = } } & { { \\displaystyle - \\sum _ { \\tau = 0 } ^ { T _ { \\mathrm { e n d } } - 1 } \\left( A _ { \\mathrm { s i g n a l } } ^ { \\tau } \\log p \\left( \\xi ^ { \\tau } \\right) + A _ { \\mathrm { a t o m p a i r } } ^ { \\tau } \\log p \\left( \\left( u , v \\right) ^ { \\tau } \\right) + A _ { \\mathrm { b o n d } } ^ { \\tau } \\log p \\left( b ^ { \\tau } \\right) \\right) } } \\\\ { { } } & { { } } & { { \\displaystyle - A _ { \\mathrm { s i g n a l } } ^ { T _ { \\mathrm { e n d } } } \\log \\pi \\left( \\xi ^ { T _ { \\mathrm { e n d } } } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 829, + 774, + 893 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $T _ { \\mathrm { e n d } }$ is the first step that $\\xi = 0$ ; $A _ { \\mathrm { s i g n a l } }$ , $A _ { \\mathrm { a t o m } \\operatorname { p a i r } }$ and $A _ { \\mathrm { b o n d } }$ are called advantages. To compute these advantages, we use the unbiased estimations called Temporal Different errors, defined as: ", + "bbox": [ + 171, + 896, + 830, + 924 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/64925108e6594163af645890516d8fdff6bcc4d6b0157f23c0675184742e722e.jpg", + "text": "$$\n\\begin{array} { r l r } { A _ { \\mathrm { s i g n a l } } ^ { \\tau } } & { = } & { r _ { \\mathrm { s i g n a l } } ^ { \\tau } + \\gamma V _ { \\phi } \\left( Z _ { K } ^ { \\tau + 1 } \\right) - V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) \\quad } \\\\ { A _ { \\mathrm { a t o m p a i r } } ^ { \\tau } } & { = } & { r _ { \\mathrm { a t o m p a i r } } ^ { \\tau } + \\gamma V _ { \\phi } \\left( Z _ { K } ^ { \\tau + 1 } \\right) - V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) \\quad } \\\\ { A _ { \\mathrm { b o n d } } ^ { \\tau } } & { = } & { r _ { \\mathrm { b o n d } } ^ { \\tau } + \\gamma V _ { \\phi } \\left( Z _ { K } ^ { \\tau + 1 } \\right) - V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) \\quad \\quad } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 117, + 666, + 180 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $r _ { \\mathrm { s i g n a l } } ^ { \\tau }$ , $r _ { \\mathrm { a t o m } \\ p \\mathrm { a i r } } ^ { \\tau }$ , $r _ { \\mathrm { b o n d } } ^ { \\tau }$ are immediate rewards at step $\\tau$ ; at the final step $\\tau = T _ { \\mathrm { e n d } }$ , the model receives additional delayed rewards; $\\gamma$ is the discount factor; and $V _ { \\phi }$ is the parametric value function. We train $V _ { \\phi }$ using the following mean square error loss: ", + "bbox": [ + 176, + 180, + 823, + 223 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7388c44aadaef8815777e0517fa2e9879fa86272ec641118d0016718f9e52823.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { v a l u e } } \\ = \\ \\sum _ { \\tau = 0 } ^ { T _ { \\mathrm { e n d } } } \\left\\| V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) - R ^ { \\tau } \\right\\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 382, + 242, + 616, + 286 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $R ^ { \\tau }$ is the return at step $\\tau$ . ", + "bbox": [ + 174, + 286, + 385, + 301 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Episode termination during training Although the loss defined in Eq. (9) is correct, it is not good to use in practice because: i) If our model selects a wrong sub-action at any sub-step of the step Twrong $\\mathit { T _ { \\mathrm { w r o n g } } } < \\mathit { T _ { \\mathrm { e n d } } } )$ , the whole predicted sequence will be incorrect regardless of what will be predicted from $T _ { \\mathrm { w r o n g } } + 1$ to $T _ { \\mathrm { e n d } }$ . Therefore, computing the loss for actions from $T _ { \\mathrm { w r o n g } } + 1$ to $T _ { \\mathrm { e n d } }$ is redundant. ii) More importantly, the incorrect updates of the graph structure at subsequent steps from $T _ { \\mathrm { w r o n g } } + 1$ to $T _ { \\mathrm { e n d } }$ will lead to cumulative prediction errors which make the training of our model much more difficult. ", + "bbox": [ + 173, + 315, + 825, + 414 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To resolve this issue, during training, we use a binary vector $\\zeta \\in \\{ 0 , 1 \\} ^ { 3 T }$ to keep track of the first wrong sub-action: $\\zeta ^ { t } = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ } } t \\leq t _ { \\mathrm { f i r s t ~ w r o n g } } } \\\\ { 0 } & { { \\mathrm { i f ~ } } t > t _ { \\mathrm { f i r s t ~ w r o n g } } } \\end{array} \\right. }$ if t ≤ tfirst wrong where tfirst wrong denotes the sub-step at which our model chooses a wrong sub-action the first time. The actor-critic loss in Eq. (9) now becomes: ", + "bbox": [ + 173, + 419, + 825, + 483 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c276aa478988f501c20d259a9d7d15da70500b5c3dbfa7a1751ee84793a29485.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { A 2 C } } = - \\sum _ { \\tau = 0 } ^ { T } \\left( \\zeta ^ { \\tau } A _ { \\mathrm { s i g n a l } } ^ { \\tau } \\log p \\left( \\xi ^ { \\tau } \\right) + \\zeta ^ { ( \\tau + 1 ) } A _ { \\mathrm { a t o m p a i r } } ^ { \\tau } \\log p \\left( ( u , v ) ^ { \\tau } \\right) + \\zeta ^ { ( \\tau + 2 ) } A _ { \\mathrm { b o n d } } ^ { \\tau } \\log p \\left( b ^ { \\tau } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 501, + 808, + 545 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $T$ is the maximum number of steps. Similarly, we change the value loss into: ", + "bbox": [ + 171, + 555, + 720, + 571 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a0d280ff73782f38906b641affdabfea7413b7225c2319b82814dfdf08a9acc3.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { v a l u e } } = \\sum _ { \\tau = 0 } ^ { T } \\zeta ^ { \\tau } \\left\\| V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) - R ^ { \\tau } \\right\\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 383, + 587, + 614, + 631 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2.4.2 REACTION ATOM PAIR LOSS ", + "text_level": 1, + "bbox": [ + 176, + 642, + 421, + 657 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To train our model to assign higher scores to reaction atom pairs and lower to non-reaction atom pairs, we use the following cross-entropy loss function: ", + "bbox": [ + 173, + 666, + 825, + 696 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9b7125cc4377f1ad1150a0823cca4d3d2a5db53d1f11fcdc58f88898f9026179.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { a t o m \\ p a i r } } = - \\sum _ { \\tau = 0 } ^ { T _ { \\mathrm { \\bar { m } s t w o u p } } } \\sum _ { i \\in \\mathcal { V } } \\sum _ { j \\in \\mathcal { V } , j \\neq i } \\eta _ { i j \\tau } \\left( y _ { i j } \\log p _ { i j } + ( 1 - y _ { i j } ) \\log ( 1 - p _ { i j } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 251, + 712, + 745, + 758 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { T _ { \\mathrm { f i r s t w r o n g } } = \\biggl \\lfloor \\frac { t _ { \\mathrm { f i r s t w r o n g } } } { 3 } \\biggr \\rfloor ; \\eta _ { i j t } \\in \\{ 0 , 1 \\} } \\end{array}$ is a mask of the atom pair $( i , j )$ at step $\\tau$ ; $y _ { i j } \\in \\{ 0 , 1 \\}$ is the label indicating whether the atom pair $( i , j )$ is a reaction atom pair or not; $p _ { i j } = \\mathrm { s i g m o i d } ( s _ { i j } )$ (see Eq. (5)). ", + "bbox": [ + 174, + 762, + 825, + 815 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2.4.3 CONSTRAINT ON THE SEQUENCE LENGTH ", + "text_level": 1, + "bbox": [ + 178, + 829, + 516, + 844 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "One major difficulty of the chemical reaction prediction problem is to know exactly when to stop prediction so we can make accurate inference. By forcing the model to stop immediately when making wrong prediction, we can prevent cumulative error and significantly reduce variance during training. But it also comes with a cost: The model cannot learn (because it does not have to learn) when to stop. This phenomenon can be visualized easily as the model predicts 1 for the signal at every step $\\tau$ during inference. In order to make the model aware of the correct sequence length during training, we define a loss that punishes the model if it produces a longer sequence than the ground truth sequence: ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/611a5799fc33c79a2e47c58b47efe199ffa1ecfe4b6d8ba884a06c645cb1f3d6.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Statistics of $U S P T O – I 5 k$ and USPTO datasets. “changes” means bond changes, “molecules” means reactants and reagents in a reaction; “atoms” and “bonds” are defined for a molecule. Apart from “#reactions”, other columns are presented in the format “min | max | mean”. " + ], + "table_body": "
Dataset#reactions#changes#molecules#atoms#bonds
USPTO-15ktrain10,500111112.3112013.641100134.931110134.7
valid1,500111112.3112013.67194134.55199134.2
test3,000111112.3111613.67198134.951102134.7
USPTOtrain409,03511612.2212914.891150139.761165138.6
valid30,00011612.2212514.891150139.671158138.5
test40.00011612.2212214.891150139.871162138.7
", + "bbox": [ + 181, + 101, + 816, + 208 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 281, + 825, + 324 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/2159a08fa4939f95bf66528b5af1a9f1a1e1e31c7d272bd3e6f5bad26b48707f.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { o v e r } \\mathrm { l e n g t h } } = - \\sum _ { T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } } \\leq \\tau < T _ { \\mathrm { e n d } } } \\log p \\left( \\xi ^ { \\tau } = 0 \\right)\n$$", + "text_format": "latex", + "bbox": [ + 364, + 342, + 633, + 382 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "wherewhen $T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ end step of the ground-trut. The reason is that forcing nce. with s in Eq. (16) is not applied is not theoretically correct $T _ { \\mathrm { e n d } } \\leq T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ $\\xi ^ { \\tau } = 1$ $T _ { \\mathrm { e n d } } \\leq \\tau < T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ because all the signals after $T _ { \\mathrm { e n d } }$ are assumed to be 0. The incentive to force $T _ { \\mathrm { e n d } }$ close to $T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ when it is smaller than $T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ has already been included in the advantages in Eq. (14). ", + "bbox": [ + 173, + 387, + 825, + 453 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2.4.4 CONSTRAINT ON THE TOP- $K$ ATOM PAIRS ", + "text_level": 1, + "bbox": [ + 174, + 468, + 519, + 483 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ideally, the loss from Eq. (15) pushes a reaction atom pair $( \\tilde { u } , \\tilde { v } ) ^ { \\tau }$ into the top- $K$ atom pairs at each step $\\tau < T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ . However, this is not guaranteed, especially when τ $\\tau$ comes close to $T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }$ . To encourage the ground-truth reaction atom pair with the highest score to appear in the top $K$ , we introduce an additional rank-based loss: ", + "bbox": [ + 173, + 493, + 825, + 553 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/368df5c4c2a08352221c09d60579e92caa32a13ab44d97b09135ea3f5959e988.jpg", + "text": "$$\n{ \\mathcal { L } } ^ { \\mathrm { i n } \\mathrm { t o p } K } = - \\sum _ { \\tau = 0 } ^ { T _ { \\mathrm { f i r s t w r o n g } } } \\log p \\left( ( { \\tilde { u } } , { \\tilde { v } } ) ^ { \\tau } \\operatorname { i n } \\mathrm { t o p } K \\right)\n$$", + "text_format": "latex", + "bbox": [ + 351, + 570, + 647, + 616 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $p \\left( ( \\tilde { u } , \\tilde { v } ) ^ { \\tau } \\right.$ in top $K$ ) is computed as: ", + "bbox": [ + 174, + 619, + 455, + 637 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3ce54945db588ad83ae194eb3f376e872d5538a5caacec940fe8ff05093f1361.jpg", + "text": "$$\n\\begin{array} { r l r } { p \\left( ( \\tilde { u } , \\tilde { v } ) ^ { \\tau } \\mathrm { i n } \\mathrm { t o p } K \\right) } & { = } & { \\frac { \\exp \\left( s _ { \\tilde { u } \\tilde { v } } ^ { \\tau } \\right) } { \\exp \\left( s _ { \\tilde { u } \\tilde { v } } ^ { \\tau } \\right) + \\sum _ { k = 1 } ^ { K } \\exp \\left( s _ { u _ { k } v _ { k } } ^ { \\tau } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 656, + 694, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 712, + 328, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.1 DATASET ", + "text_level": 1, + "bbox": [ + 174, + 744, + 281, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate our model on two standard datasets USPTO-15k (15K reactions) and USPTO (480K reactions) which have been used in previous works (Jin et al., 2017; Schwaller et al., 2018; Bradshaw et al., 2018). Details about these datasets are given in Table 1. The USPTO dataset contains reactant, reagent and product molecules represented as SMILES strings. Using RDKit1, we convert the SMILES strings into molecule objects and store them as graphs. For each reaction, every atom in the reactant and reagent molecules is identified with a unique “atom map number”. This identity is the same in the products. Using this knowledge, we compare every atom pair in the input molecules with the correspondent in the product molecules to obtain a ground-truth set of reaction triples for training. In USPTO-15k, the ground-truth sets of reaction triples was precomputed by (Jin et al., 2017). ", + "bbox": [ + 173, + 770, + 826, + 897 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/40db2a4b404abab781bb4f78d3eb61e5842fa6023ada8fe496557badbf7ff8fe.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelUSPTO-15kUSPTO
C@6C@8C@10C@6C@8C@10
WLN* (Jin et al., 2017)81.686.189.189.892.093.3
WLN (Jin et al., 2017)88.4591.6593.3490.9793.9895.26
CLN (Pham et al., 2017)88.6891.6393.0790.7293.5794.80
Our GNN88.9292.0093.5791.2494.1795.33
", + "bbox": [ + 228, + 101, + 769, + 193 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2: Results for reaction atom pair prediction. $C @ k$ is coverage at $k$ . Best results are highlighted in bold. $\\mathrm { W L N ^ { \\star } }$ is the original model from (Jin et al., 2017) while WLN is our re-implemented version. Except for $\\mathrm { W L N ^ { \\star } }$ , other models explicitly use reagent information. ", + "bbox": [ + 178, + 199, + 825, + 241 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/14668a9481c915b9fba00c8d77e2952ac553712c8edb6065fb0fd1071dd67232.jpg", + "image_caption": [ + "Figure 3: Coverage $@ k$ and Recall $@ k$ with respect to $k$ for the USPTO dataset. " + ], + "image_footnote": [], + "bbox": [ + 348, + 275, + 647, + 443 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.2 REACTION ATOM PAIR PREDICTION ", + "text_level": 1, + "bbox": [ + 178, + 513, + 464, + 529 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we test our model’s ability to identify reaction atom pairs by formulating it as a ranking problem with the scores computed in Eq. (5). Similar to (Jin et al., 2017), we use Coverage $@ k$ as the evaluation metric, which is the proportion of reactions that have all groundtruth reaction atom pairs appear in the top $k$ predicted atom pairs. ", + "bbox": [ + 174, + 545, + 825, + 602 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare our proposed graph neural network (GNN) with Weisfeiler-Lehman Network (WLN) (Jin et al., 2017) and Column Network (CLN) (Pham et al., 2017). Since our GNN explicitly uses reagent information to compute the scores of atom pairs, we modify the implementation of WLN and CLN accordingly for fair comparison. From Table 2, we observe that our GNN clearly outperforms WLN and CLN in all cases. We attribute this improvement to the use of a separate node state vector $\\boldsymbol { x } _ { i } ^ { t }$ (different from the node feature vector ${ \\mathbf { } } v _ { i }$ ) for updating the structural information of a node (see Eq. (21)). The other two models, on the other hand, only use a single vector to store both the node features and structure, hence, some information may be lost. In addition, using explicit reagent information boosts the prediction accuracy, which improves the WLN by $1 - 7 \\%$ depending on the metrics. The presence of reagent information reduces the number of atom pairs to be searched on and contributes to the likelihood of reaction atom pairs. Further results are presented in Appendix A.5. ", + "bbox": [ + 173, + 609, + 825, + 762 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 TOP- $K$ ATOM PAIR EXTRACTION ", + "text_level": 1, + "bbox": [ + 176, + 794, + 444, + 808 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The performance of our model depends on the number of selected top atom pairs $K$ . The value of $K$ presents a trade-off between coverage and efficiency. In addition to the metric Coverage $@ k$ in Sec. 3.2, we use Recall $@ k$ which is the proportion of correct atom pairs that appear in top $k$ to find the good $K$ . Fig. 3 shows Coverage $@ k$ and Recall $@ k$ for the USPTO dataset with respect to $k$ . We see that both curves increase rapidly when $k < 1 0$ and stablize when $k > 1 0$ . We also ran experiments with $k = 1 0$ , 15, 20 and observed that their prediction results are quite similar. Hence, in what follows we select $K = 1 0$ for efficiency. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/078df044119ba38679194a3cf761c77c41fb92600340ab9dee45aec238a516fa.jpg", + "table_caption": [ + "Table 3: Results for reaction prediction. $P \\ @ k$ is precision at $k$ . State-of-the-art results from (Jin et al., 2017) are written in italic. Results from (Schwaller et al., 2018) are marked with ? and they are computed on a slightly different version of USPTO that contains only single-product reactions. Best results are highlighted in bold. $\\diamondsuit$ : With beam search (beam width $= 2 0$ ), $\\lessgtr$ : Invalid product removal, ♣: Duplicated product removal. " + ], + "table_footnote": [], + "table_body": "
ModelUSPTO-15kUSPTO
P@1P@3P@5P@1P@3P@5
WLDN (Jin et al., 2017)76.785.686.879.687.789.2
Seq2Seq (Schwaller et al., 2018)---80.3*86.2*87.5*
GTPN72.311171.26--
GTPN74.5682.6284.2373.2580.5683.53
GTPN474.5683.1984.9773.2584.3185.76
GTPN82.3985.6086.6883.2084.9785.90
GTPN**82.3985.7386.7883.2086.0386.48
", + "bbox": [ + 207, + 101, + 790, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.4 REACTION PRODUCT PREDICTION ", + "text_level": 1, + "bbox": [ + 178, + 347, + 450, + 361 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This experiment validates GTPN on full reaction product prediction against the recent state-of-the-art methods (Jin et al., 2017; Schwaller et al., 2018) using the accuracy metric. The recent method ELECTRO (Bradshaw et al., 2018) is not compatible here because it was only evaluated on a subset of USPTO limited to linear chain topology. Comparison against ELECTRO is reported separately in Appendix A.6. Table 3 shows the prediction results. We produce multiple reaction product candidates by using beam search decoding with beam width $N = 2 0$ . Details about beam search and its behaviors are presented in Appendix A.4. ", + "bbox": [ + 173, + 375, + 825, + 473 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In brief, we compute the normalized-over-length log probabilities of $N$ predicted sequences of reaction triples and sort these values in descending order to get a rank list of $N$ possible reaction outcomes. Given a predicted sequence of reaction triples $( \\overline { { u } } , v , b ) ^ { 0 : T }$ , we can generate reaction products from input reactants simply by replacing the old bond of $( u , v ) ^ { \\tau }$ with $b ^ { \\tau }$ . However, these products are not guaranteed to be valid (e.g., maximum valence constraint violation or aromatic molecules cannot be kekulized) so we post-process the outputs by removing all invalid products. The removal increases the top-1 accuracy by about $8 \\%$ and $10 \\%$ on $U S P T O – I 5 k$ and USPTO, respectively. Due to the permutation invariance of the predicted sequence of reaction triples, some product candidates are duplicate and will also be removed. This does not lead to any change in $P \\ @ { \\cal I }$ but slightly improves $P \\ @ 3$ and $P \\ @ 5$ by about $0 . 5 – 1 \\%$ on the two datasets. ", + "bbox": [ + 173, + 479, + 825, + 618 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Overall, GTPN with beam search and post-processing outperforms both WLDN (Jin et al., 2017) and Seq2Seq (Schwaller et al., 2018) in the top-1 accuracy. For the top-3 and top-5, our model’s performance is comparable to WLDN’s on USPTO- $l 5 k$ and is worse than WLDN’s on USPTO. It is not surprising since our model is trained to accurately predict the top-1 outcomes instead of ranking the candidates directly like WLDN. It is important to emphasize that we did not tune the model hyper-parameters when training on USPTO but reused the optimal settings from $U S P T O – I 5 k$ (which is 25 times smaller than USPTO) so the results may not be optimal (see Appendix A.3 for more training detail). ", + "bbox": [ + 174, + 626, + 825, + 738 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 762, + 344, + 780 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.1 LEARNING TO PREDICT CHEMICAL REACTION ", + "text_level": 1, + "bbox": [ + 174, + 797, + 537, + 813 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In chemical reaction prediction, machine learning has replaced rule-based methods (Chen & Baldi, 2009) for better generalizability and scalability. Existing machine learning-based techiques are either template-free (Kayala & Baldi, 2011; Jin et al., 2017; Fooshee et al., 2018) and template-based (Wei et al., 2016; Segler & Waller, 2017; Coley et al., 2017). Both groups share the same mechanism: running multiple stages with the aid of reaction templates or rules. For example, in (Wei et al., 2016) the authors proposed a two-stage model that first classifies reactions into different types based on the neural fingerprint vectors (Duvenaud et al., 2015) of reactant and reagent molecules. Then, it applies pre-designed SMARTS transformation on the reactants with respect to the most suitable predicted reaction type to generate the reaction products. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The work of (Jin et al., 2017) treats a reaction as a set of bond changes so in the first step, they predict which atom pairs are likely to be reactive using a variant of graph neural networks called Weisfeiler-Lehman Networks (WLNs). In the next step, they do almost the same as (Coley et al., 2017) by modifying the bond type between the selected atom pairs (with chemical rules satisfied) to create product candidates and rank them (with reactant molecules as addition input) using another kind of WLNs called Weifeiler-Lehman Different Networks (WLDNs). ", + "bbox": [ + 174, + 138, + 825, + 222 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To the best of our knowledge, (Jin et al., 2017) is the first work that achieves remarkable results (with the Precision $@ 1$ is about $7 9 . 6 \\%$ ) on the large USPTO dataset containing more than 480 thousands reactions. Works of (Nam & Kim, 2016) and (Schwaller et al., 2018) avoid multi-stage prediction by building a seq2seq model that generates the (canonical) SMILES string of the single product from the concatenated SMILES strings of the reactants and reagents in an end-to-end manner. However, their methods cannot deal with sets of reactants/reagents/products properly as well as cannot provide concrete reaction mechanism for every reaction. ", + "bbox": [ + 174, + 229, + 825, + 327 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The most recent work on this topic is (Bradshaw et al., 2018) which solves the reaction prediction problem by predicting a sequence of bond changes given input reactants and reagents represented as graphs. To handle ordering, they only select reactions with predefined topology. Our method, by contrast, is order-free and can be applied to almost any kind of reactions. ", + "bbox": [ + 176, + 334, + 825, + 390 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.2 GRAPH NEURAL NETWORKS FOR MODELING MOLECULES ", + "text_level": 1, + "bbox": [ + 174, + 407, + 620, + 421 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In recent years, there has been a fast development of graph neural networks (GNNs) for modeling molecules. These models are proposed to solve different problems in chemistry including toxicity prediction (Duvenaud et al., 2015), drug activity classification (Shervashidze et al., 2011; Dai et al., 2016; Pham et al., 2018), protein interface prediction (Fout et al., 2017) and drug generation (Simonovsky & Komodakis, 2018; Jin et al., 2018). Most of them can be regarded as variants of message-passing graph neural networks (MPGNNs) (Gilmer et al., 2017). ", + "bbox": [ + 174, + 434, + 825, + 517 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4.3 REINFORCEMENT LEARNING FOR STRUCTURAL REASONING ", + "text_level": 1, + "bbox": [ + 176, + 534, + 635, + 549 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Reinforcement learning (RL) has become a standard approach to many structural reasoning problems2 because it allows agents to perform discrete actions. A typical example of using RL for structural reasoning is drug generation (Li et al., 2018; You et al., 2018). Both (Li et al., 2018) and (You et al., 2018) learn the same generative policy whose action set including: i) adding a new atom or a molecular scaffold to the intermediate graph, ii) connecting existing pair of atoms with bonds, and iii) terminating generation. However, (You et al., 2018) uses an adversarial loss to enforce global chemical constraints on the generated molecules as a whole instead of using the common reconstruction loss as in (Li et al., 2018). Other examples are path-based relational reasoning in knowledge graphs (Das et al., 2018) and learning combinatorial optimization over graphs (Khalil et al., 2017). ", + "bbox": [ + 174, + 560, + 825, + 700 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 719, + 310, + 736 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have introduced a novel method named Graph Transformation Policy Network (GTPN) for predicting products of a chemical reaction. GTPN uses graph neural networks to represent input reactant and reagent molecules, and uses reinforcement learning to find an optimal sequence of bond changes that transforms the reactants into products. We train GTPN using the Advantage Actor-Critic (A2C) method with appropriate constraints to account for notable aspects of chemical reaction. Experiments on real datasets have demonstrated the competitiveness of our model. ", + "bbox": [ + 174, + 752, + 825, + 837 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Although the GTPN was proposed to solve the chemical reaction problem, it is indeed generic to solve the graph transformation problem, which can be useful in reasoning about relations (e.g., see (Zambaldi et al., 2018)) and changes in relation. Open rooms include addressing dynamic graphs over time, extending toward full chemical planning and structural reasoning using RL. 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", + "bbox": [ + 178, + 880, + 823, + 921 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 GRAPH NEURAL NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 133, + 410, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we describe our graph neural network (GNN) in detail. Since our GNN does not use the recurrent hidden state $h ^ { \\tau }$ , we exclude the time step $\\tau$ from our notations for clarity. Instead, we use $t$ to denote a message passing step. ", + "bbox": [ + 174, + 159, + 825, + 203 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "GRAPH NOTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 217, + 310, + 231 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Input to our GNN is a graph $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ in which each node $i \\in \\nu$ is represented by a node feature vector ${ \\mathbf { } } v _ { i }$ and each edge $( i , j ) \\in \\mathcal { E }$ is represented by an edge feature vector $e _ { i j }$ . For example of molecular graph, the node feature vector ${ \\mathbf { } } v _ { i }$ may include chemical information about the atom $i$ such as its type, charge and degree. Similarly, $e _ { i j }$ captures the bond type between the two atoms $i$ and $j$ We denote by $\\bar { \\mathcal { N } } ( i )$ the set of all neighbor nodes of node $i$ together with their links to node $i$ : ", + "bbox": [ + 173, + 241, + 825, + 311 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/69ebfd8ea3f9e275c82afa1a65cd66a3cd2e14e1d097d3bba631a59f12ffe28f.jpg", + "text": "$$\n\\begin{array} { l l l } { { \\mathcal N ( i ) } } & { { \\equiv } } & { { \\{ ( j , e _ { i j } ) \\mid j { \\mathrm { ~ i s ~ a ~ n e i g h b o r ~ n o d e ~ o f ~ } } i \\} } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 334, + 658, + 351 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "If we only care about the neighbor nodes of $i$ not their links, we use the notation $\\mathcal { N } _ { \\mathrm { n } } ( i )$ defined as: ", + "bbox": [ + 179, + 352, + 808, + 367 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/3b4e792e0828d25cd8a2c8175b32818911159eb39fce10e274d3d5c0f86aaa1f.jpg", + "text": "$$\n\\begin{array} { l l l } { { \\mathcal N } _ { \\mathrm { n } } ( i ) } & { \\equiv } & { \\left\\{ j \\ | \\ j \\ \\mathrm { i s ~ a ~ n e i g h b o r ~ n o d e ~ o f } \\ i \\right\\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 388, + 642, + 405 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In addition to ${ \\mathbf { } } v _ { i }$ , node $i$ also has a state vector $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ to store information about itself and the surrounding context. This state vector is updated recursively using the neural message passing method (Battaglia et al., 2016; Pham et al., 2017; Hamilton et al., 2017; Gilmer et al., 2017; Schlichtkrull et al., 2018). The initial state $\\pmb { x } _ { i } ^ { 0 }$ is the nonlinear mapping of ${ \\mathbf { } } v _ { i }$ : ", + "bbox": [ + 174, + 406, + 825, + 462 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/74badc6d50554f991ec2e97e0edb03943f79ba23173d8ea9495d82c77fba82ba.jpg", + "text": "$$\n\\begin{array} { r c l } { { { \\pmb x } _ { i } ^ { 0 } } } & { { = } } & { { \\sigma \\left( W { { \\pmb v } _ { i } } + b \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 421, + 464, + 575, + 483 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "COMPUTING NEIGHBOR MESSAGES ", + "text_level": 1, + "bbox": [ + 176, + 496, + 419, + 511 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "At the message passing step $t$ , we compute the message $\\boldsymbol { m } _ { i j } ^ { t }$ from every neighbor node $j \\in \\mathcal { N } _ { n } ( i )$ to node $i$ as: ", + "bbox": [ + 173, + 520, + 826, + 549 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/fcbc5ee59354814fadb7dc25bde72ddb0770ec5d984d7839eda459869421e243.jpg", + "text": "$$\n\\begin{array} { r c l } { { { \\pmb m } _ { i j } ^ { t } } } & { { = } } & { { f \\left( { \\pmb x } _ { i } ^ { t } , { \\pmb x } _ { j } ^ { t } , { \\pmb e } _ { i j } \\right) } } \\\\ { { } } & { { = } } & { { \\sigma \\left( W \\left[ { \\pmb x } _ { i } ^ { t } , { \\pmb x } _ { j } ^ { t } , { \\pmb e } _ { i j } \\right] + b \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 380, + 569, + 616, + 611 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $[ \\cdot ]$ denotes concatenation; and $\\sigma$ is a nonlinear function. ", + "bbox": [ + 173, + 611, + 581, + 626 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "AGGREGATING NEIGHBOR MESSAGES ", + "bbox": [ + 174, + 642, + 437, + 656 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then, we aggregate all the messages sent to node $i$ into a single message vector by averaging: ", + "bbox": [ + 166, + 665, + 787, + 681 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/8b6a58bc215847c6bf156cde2dee8316138872290d852876df15b5cc521d33ca.jpg", + "text": "$$\n\\begin{array} { r c l } { \\pmb { m } _ { i } ^ { t } } & { = } & { \\displaystyle \\frac { 1 } { \\vert \\mathcal { N } _ { \\mathrm { n } } ( i ) \\vert } \\sum _ { j \\in \\mathcal { N } _ { \\mathrm { n } } ( i ) } \\pmb { m } _ { i j } ^ { t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 398, + 699, + 599, + 738 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $| \\mathcal { N } _ { \\mathrm { n } } ( i ) |$ is the number of neighbor nodes of node $i$ ", + "bbox": [ + 173, + 739, + 547, + 756 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "UPDATING NODE STATE ", + "bbox": [ + 174, + 770, + 339, + 785 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Finally, we update the state of node $i$ as follows: ", + "bbox": [ + 173, + 795, + 493, + 809 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f1b3fe14e64e67baed751439836380134bb5b3531f0432270f0fab6adeec114d.jpg", + "text": "$$\n\\begin{array} { r c l } { \\pmb { x } _ { i } ^ { t + 1 } } & { = } & { g \\left( \\pmb { x } _ { i } ^ { t } , \\pmb { m } _ { i } ^ { t } , \\pmb { v } _ { i } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 410, + 829, + 586, + 849 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $g ( . )$ is a Highway Network (Srivastava et al., 2015): ", + "bbox": [ + 173, + 849, + 558, + 866 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/611c5ca80543c8a75c2d7a7062d8f56c9f25bae302fb6d770d1d329b50b0f996.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\pmb x _ { i } ^ { t + 1 } } } & { { = } } & { { \\mathrm { H i g h w a y } \\left( { \\pmb x _ { i } ^ { t } , \\pmb m _ { i } ^ { t } , \\pmb v _ { i } } \\right) } } \\\\ { { } } & { { = } } & { { { \\pmb \\alpha } * \\tilde { \\pmb x } _ { i } ^ { t + 1 } + ( 1 - { \\pmb \\alpha } ) * { \\pmb x } _ { i } ^ { t } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 885, + 622, + 926 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $\\tilde { \\pmb { x } } _ { i } ^ { t + 1 }$ is the nonlinear part which is computed as: $\\bar { \\pmb { x } } _ { i } ^ { t + 1 } = \\sigma \\left( \\bar { W _ { 1 } } \\left[ \\pmb { x } _ { i } ^ { t } , \\pmb { m } _ { i } ^ { t } , \\pmb { v } _ { i } ^ { t } \\right] + b _ { 1 } \\right)$ and $_ \\alpha$ is the gate controlling the flow of information: ", + "bbox": [ + 173, + 102, + 614, + 148 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/53e933465545dce9b796151fa0986bd5939f3a92542be6e727f34e32b07e067f.jpg", + "text": "$$\n\\pmb { \\alpha } = \\mathrm { s i g m o i d } ( W _ { 2 } \\left[ \\pmb { x } _ { i } ^ { t } , \\pmb { m } _ { i } ^ { t } , \\pmb { v } _ { i } ^ { t } \\right] + b _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 375, + 166, + 620, + 185 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By combining Eqs. (19,20,22) together, one step of message passing update for node $i$ can be written in a generic way as follows: ", + "bbox": [ + 173, + 185, + 826, + 214 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b4e04240471ea34e85e0fd90a18a1b3ce676b1d7f72e23c67c9356d0331e60d4.jpg", + "text": "$$\n\\pmb { x } _ { i } ^ { t + 1 } = \\mathrm { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { i } ^ { t } , \\pmb { v } _ { i } , \\mathcal { N } ( i ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 367, + 231, + 630, + 251 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 UPDATING STATES ", + "text_level": 1, + "bbox": [ + 176, + 265, + 348, + 279 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "UPDATING RNN STATE ", + "bbox": [ + 176, + 290, + 338, + 305 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We keep the old representation of the edge that have been modified in the hidden memory of the RNN as follows: ", + "bbox": [ + 173, + 315, + 823, + 343 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e8266b890dd6cec36d096d56b55087b4dc1abae35f9a0f05acfffe6b3708f8dd.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { h } ^ { \\tau } = \\mathbf { G } \\mathbf { R } \\mathbf { U } \\left( \\pmb { h } ^ { \\tau - 1 } , \\pmb { z } _ { u v } ^ { \\tau } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 415, + 361, + 580, + 381 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where GRU stands for Gated Recurrent Units (Cho et al., 2014); $\\boldsymbol { z } _ { u v } ^ { \\tau }$ is the representation vector of the atom pair $( u , v ) ^ { \\tau }$ including its old bond (see Eq. 4). Eq. (25) allows the model to keep track of all the changes happening to the graph so far so it can make more accurate prediction later. ", + "bbox": [ + 174, + 381, + 825, + 424 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "UPDATING GRAPH STRUCTURE AND NODE STATES ", + "text_level": 1, + "bbox": [ + 174, + 439, + 521, + 452 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "After predicting a reaction triple $( u , v , b ) ^ { \\tau }$ at step $\\tau$ , we update the graph structure and node states based on the new bond change. First, to update the graph structure, we simply update the neighbor set of $u$ and $v$ with information from the other atom and the new bond type $b$ as follows: ", + "bbox": [ + 174, + 462, + 825, + 505 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/711fda8a1915bd0c770d210262fb5bf19078bfe031660032107382cf0a3b1810.jpg", + "text": "$$\n\\begin{array} { r l r } { \\mathcal { N } ^ { \\tau } ( u ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( u ) \\backslash \\left( v , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( v , b ) } \\\\ { \\mathcal { N } ^ { \\tau } ( v ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( v ) \\backslash \\left( u , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( u , b ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 522, + 642, + 563 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Next, to update the node states, our model performs one step of message passing for $u$ and $v$ with their new neighbor sets: ", + "bbox": [ + 174, + 561, + 826, + 588 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1037ecb9d4b964ec79c198c33b5dbb951a46c54f5a12e1d58afc79a824035a6f.jpg", + "text": "$$\n\\begin{array} { r l r } { \\pmb { x } _ { u } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { u } ^ { \\tau - 1 } , \\pmb { v } _ { u } , \\mathcal { N } ^ { \\tau } ( u ) \\right) } \\\\ { \\pmb { x } _ { v } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { v } ^ { \\tau - 1 } , \\pmb { v } _ { v } , \\mathcal { N } ^ { \\tau } ( v ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 604, + 650, + 646 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the MessagePassing $( . )$ function is defined in Eq. (24). For other nodes in the graph to be aware of the new structures of $u$ and $v$ , we need to perform several message passing steps for all nodes in the graph after Eqs. (28, 29). However, it is very costly to run for every prediction step $\\tau$ . Sometimes it is unnecessary since far-away bonds are less likely to be affected by the current bond change (unless the far-way bonds and the new bond are in an aromatic ring). Therefore, in our model, we limit the number of message passing updates for all nodes at step $\\tau$ to be 1. ", + "bbox": [ + 174, + 643, + 826, + 728 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 MODEL CONFIGURATIONS ", + "text_level": 1, + "bbox": [ + 176, + 743, + 401, + 758 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We optimize our model’s hyper-parameters in two stages: First, we tune the hyper-parameters of the GNN and the NPPN for the reaction atom pair prediction task. Then, we fix the optimal settings of the first two components and optimize the hyper-parameters of the PN for the reaction product prediction task. ", + "bbox": [ + 174, + 768, + 825, + 825 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We provide details about the settings that give good results on the USPTO-15k dataset below. With these settings, we trained another model on the USPTO dataset from scratch. Because training on the large dataset such as the USPTO takes time, we did not tune hyper-parameters on the USPTO, eventhough it is possible to increase model sizes for better performance. ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Unless explicitly stated, all neural networks in our model have 2 layers with the same number of hidden units, ReLU activation and residual connections (He et al., 2016). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/af15e01011e2f02b1e6ac95f4aac3c5dd0520d54e5ba2400c7a03855a3d03f1d.jpg", + "table_caption": [ + "Table 4: Data types of atom attributes. " + ], + "table_footnote": [], + "table_body": "
Atom attributeData type
Degreenumeric
Explicit valencenumeric
Explicit number of Hsnumeric
Chargenumeric
Part of a ringboolean
", + "bbox": [ + 372, + 101, + 624, + 193 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Graph Neural Network (GNN) There are 72 different types of atom depending on their atomic numbers and 5 different types of bond including NULL, SINGLE, DOUBLE, TRIPLE and AROMATIC. The size of embedding vectors for atom and bond are 51 and 21, respectively. Apart from atom type, each atom has 5 more attributes listed in Table 4. These attributes are normalized to the range of [0, 1] and are concatenated to the atom embedding vector to form a final atom feature vector of size 56. The state vector and the neighbor message vector for an atom both have the size of 99. The number of message passing steps is 6. ", + "bbox": [ + 173, + 236, + 826, + 335 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Node Pair Prediction Network (NPPN) This component consists of two parts. The first part computes the representation vector $z _ { i j }$ of an atom pair $( i , j )$ using a neural network with hidden size of 71. The second part maps $z _ { i j }$ to an unnormalized score $s _ { i j }$ using the function $f ^ { \\mathrm { a t o m } } \\operatorname { p a i r }$ (see Eqs. (3,5)). This function is also a neural network with hidden size of 51. ", + "bbox": [ + 173, + 349, + 825, + 409 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Policy Network (PN) The recurrent network is a GRU (Cho et al., 2014) with 101 hidden units. The value function $V _ { \\phi }$ is a neural network with 99 hidden units. The two functions $f ^ { \\mathrm { s i g n a l } }$ for computing signal scores (see Eq. (6)) and $f ^ { \\mathrm { b o n d } }$ for computing scores over bond types (see Eq. (8)) are neural networks with 81 hidden units. ", + "bbox": [ + 173, + 422, + 825, + 481 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Training At each step, we set the reward to be 1.0 for correct prediction of signal/atom pair/bond type and -1.0 for incorrect prediction. After the prediction sequence is terminated (zero signal was emitted), we check whether the entire set of predicted reaction triples is correct or not. If it is correct, we give the model a reward value of 2.0, otherwise $- 2 . 0$ . From the rewards and estimated values for signal, atom pair and bond type, we define the Advantage Actor Critic loss (A2C) as in Eq. (14). The coefficients of components in the final loss $\\mathcal { L }$ are set empirically as follows: ", + "bbox": [ + 173, + 494, + 826, + 580 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d06fb9511efe0c12dce034dae155e1aeb45209904aa758e768521da1a5741b6d.jpg", + "text": "$$\n\\mathcal { L } = \\mathcal { L } ^ { \\mathrm { A 2 C } } + 0 . 5 \\times \\mathcal { L } ^ { \\mathrm { v a l u e } } + \\mathcal { L } ^ { \\mathrm { a t o m p a i r } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { o v e r l e n g t h } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { i n ~ t o p ~ } K }\n$$", + "text_format": "latex", + "bbox": [ + 258, + 597, + 738, + 614 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We trained our model using Adam (Kingma & Ba, 2015) with the initial learning rate of 0.001 for both $U S P T O – I 5 k$ and USPTO. For USPTO-15k, the learning rate will decrease by half if the Precision $@ l$ does not improve on the validation set after 1,000 steps until it reaches the minimum value of $5 \\times 1 0 ^ { - 5 }$ . For USPTO, the decay rate is 0.8 after every 500 steps of no improvement until reaching the minimum learning rate is $2 \\times 1 0 ^ { - 5 }$ . The maximum number of training iterations is $1 0 ^ { 6 }$ and the batch size is 20. ", + "bbox": [ + 173, + 618, + 825, + 703 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.4 DECODING WITH BEAM SEARCH ", + "text_level": 1, + "bbox": [ + 176, + 719, + 446, + 734 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For decoding, our model generates a sequence of reaction triples (including the stop signal) $( \\xi , u , v , b )$ by taking the best $( u , v )$ and $b$ at every step until it outputs a zero signal $\\xi = 0$ ). In other words, it computes the argmax of $p \\left( ( \\xi , u , v , b ) ^ { \\tau } \\mid \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right)$ at every step $\\tau$ . However, this algorithm is not robust for the sequence generation task because just a single error at a step may destroy the entire sequence. To overcome this issue, we employ beam search for decoding. ", + "bbox": [ + 174, + 744, + 825, + 818 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "During beam search, we keep track of $N > 1$ best subsequences at every step $\\tau$ . $N$ is called beam width. Instead of modeling the conditional distribution of generating an output at the current step $\\tau$ , we model the joint distribution of the whole subsequence that has been generated from 0 to $\\tau$ : ", + "bbox": [ + 174, + 824, + 826, + 866 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ce5aba8973b307de115ccbafd60884e53cfccef894aa2e8221397b7401f323d4.jpg", + "text": "$$\n\\begin{array} { r c l } { \\log p \\left( ( \\xi , u , v , b ) ^ { 0 : \\tau } | \\mathcal { G } \\right) } & { = } & { \\log p \\left( ( \\xi , u , v , b ) ^ { \\tau } | \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) + } \\\\ & & { \\ \\log p \\left( ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } | \\mathcal { G } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 885, + 730, + 928 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/89043796ab63f2d5448c2d1672e4df6449ebba24bbeeb93de070420b3cde1666.jpg", + "table_caption": [], + "table_footnote": [ + "Table 5: Reaction product prediction results using beam search with different values of beam width on $U S P T O – I 5 k$ . " + ], + "table_body": "
Beam widthPrecision@k
1235101520
174.49:二二二二-
272.2180.6511---
572.2179.5482.2984.27-二-
1072.1579.5482.1983.9386.01--
1572.1579.5482.1683.9386.1186.981
2074.5680.7282.6284.2386.1487.0487.55
", + "bbox": [ + 248, + 101, + 750, + 220 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Computing all configurations of $( \\xi , u , v , b ) ^ { \\tau }$ jointly is very memory demanding, however. Thus, we decompose the first term as follows: ", + "bbox": [ + 174, + 280, + 823, + 309 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3102af7b53b8fc38e42d9ff2d6978fe328c3fc7aa6f92308b75b606cab663d15.jpg", + "text": "$$\n\\begin{array} { l l l } { \\log p \\left( ( \\xi , u , v , b ) ^ { \\tau } | \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) } & { = } & { \\log p \\left( \\xi ^ { \\tau } | \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) + \\mathstrut } \\\\ & & { \\log p \\left( ( u , v ) ^ { \\tau } | \\xi ^ { \\tau } , \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) + \\mathstrut } \\\\ & & { \\log \\left( b ^ { \\tau } | ( \\xi , u , v ) ^ { \\tau } , \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 330, + 777, + 393 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "At step $\\tau$ , we do beam search for the signal $\\xi ^ { \\tau }$ , then the atom pair $( u , v ) ^ { \\tau }$ and finally the bond type $b ^ { \\tau }$ . Algorithm 1 describes beam search in detail. Some notable technicalities are: ", + "bbox": [ + 174, + 395, + 823, + 422 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• We only do beam search for $( u , v )$ and $b$ if the prediction is ongoing, i.e., when $\\xi ^ { \\tau } = 1$ . To keep track of this, we use a boolean vector $C$ of length $N$ with $C ^ { 0 }$ is initialized to be all true. \n• To avoid beam search favoring short sequences, we normalize the log probability scores over sequence lengths. This is shown in lines 10, 17, 32 and 47 ", + "bbox": [ + 214, + 435, + 825, + 508 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "BEAM WIDTH ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 531, + 343, + 545 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Table 5 reports how beam width affects the decoding performance on the USPTO-15k dataset. \nSurprisingly, the top-1 accuracy in case of beam width3 of 1 is higher than the those when beam widths range from 2 to 15. It means that large beam width is not always good in our situation. \nHowever, at beam width of 20, our beam search achieves the best results for different values of $k$ . \nThus, we set the beam width to 20 in subsequent experiments. ", + "bbox": [ + 174, + 556, + 826, + 626 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.5 USING REAGENT INFORMATION EXPLICITLY ", + "text_level": 1, + "bbox": [ + 176, + 645, + 529, + 659 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "As can be seen from Table 6, reagent molecules account for about a half of the input molecules on average and $60 \\%$ of all reactions containing reagents. It suggests that the proper use of reagent information will lead to better prediction. In our model, before computing the scores for all atom pairs, we append to the representation vector of every atom a binary scalar indicating whether this atom comes from a reagent molecule or not. Then, at the top- $K$ atom pair selection step, we also exclude all atom pairs that have either atoms belong to a reagent molecule. The improvement in prediction accuracy on the validation set of $U S P T O – I 5 k$ is shown in Fig. 4. ", + "bbox": [ + 174, + 671, + 825, + 768 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.6 COMPARISON WITH ELECTRO ", + "text_level": 1, + "bbox": [ + 176, + 787, + 436, + 801 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In method Both GTPN and ELECTRO (Bradshaw et al., 2018) are able to explain the mechanism behind a reaction. ELECTRO regards a reaction as an ordered sequence that alternates between removing and adding a single bond. Our model, on the other hand, assumes no specific order of transformations as well as the amount of valences that a bond can change. Thus, our model is more generic than ELECTRO and can cover a much larger set of reactions. ", + "bbox": [ + 174, + 814, + 825, + 885 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/c27d8f01744a2cb3d9752593a74556a8b3a40c8eedef9c8d7e9a53c5a6281d86.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm 1 Reaction triple prediction using beam search.
Input: A multi-graph G consisting of reactant and reagent molecules, number of bond types E, max prediction steps T,beam width N
1: P0=[(-1,-1,-1,-1),] DThe best N subsequences of(ε,u,v,b)
2: s0 =[0,..] DThe length-normalized log joint probabilities of the best N subsequences 3: Co=[True,.] DThe continuation indicator of the best N subsequences
4: Perform L steps of message passing for all nodes using Eq. (1)
5:x=xii∈V DThe initial states of all nodes before decoding 6: N(i)=N(i) ∀i ∈V DThe initial neighbor set of all nodes before decoding
7: h° is loaded from the saved model DThe initial RNN hidden state before decoding
8: for T from 1 to T do
9: Find the top K atom pairs {(uk, Uk)T |k = 1,K} using Eqs. (4,5)
10: ST-1:0 = ST-1× T-1 Superscript O denotes the sub-step 0
PT-1:0 = PT-1;CT-1;0 =CT-1 11:
12: Beam search for continuation signals
13:
14: Rsignal =@ DStores the log joint probabilities for N × 2 possible signals
15: for n from 1 to N do 16: Computep (|Pn-1:0）usingEq. (6)
17: AddCT-i0× 1 logp(εT =δ| Pπ-10)+ S-1:0 to Rsignal forδ∈ {True,False}
18: end for
19: Sort Rsignal in descending order
20:
21:
22:
Pr-1;1 =extract (-, 23: CT-1;1 = extract (CT-1;0,[-1;1)
24: 25:
26:
27: Beam search for atom pairs 28:
29: Ratom pair = @ DStores the log joint probabilities for N × K possible atom pairs
30: for n from 1 to N do 31: Compute p (u,u)|Sn,P-1;1) using Eq. (7)
Add CT-i1 ×1logp((uU)sn,P-1;1)+S-1:1 to Ratompairk ∈1,K 32:
33: end for
Sort Ratom pair in descending order 34:
35:
36:
37:
Pt-1;2 = extract (P-1;1,[-1;2) 38:
39: CT-1;2 = extract (C-1;1,;2)
40: = extract (,-1;2)
41:
", + "bbox": [ + 171, + 56, + 828, + 912 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/4b2872855af5344e9e935cdf2130f461a8edacd688102567396757d68178b8c5.jpg", + "table_caption": [ + "Algorithm 2 Reaction triple prediction using beam search (cont.) " + ], + "table_footnote": [], + "table_body": "
45:for n from1 to N do
46:Compute p (bT 1(E,u,o)n,P-1) using Eq. (8)
47:AddCT-i;2× ↓logp (bT =β|(ξ,u,u)，Pπ-1) + S[-1 to Rbond ∀β ∈1,B
48:end for
49: Sort Rbond in descending order
50:ST-1;3 = RO:N Rbond
51:
52:
53:PT-1;3 = extract (Pr-1;2, [T-1;3)
54:CT-1;3 = extract (CT-1;2, [T-1;3)
55:T = extract (T,IT-1;3)
56:(u,u)T = extract (u,u)T,IT-1;3)
57:
58:ST = ST-1;3;CT = CT-1;3
59:P =append(P-1:3，(,u,U,b)n)
60:for n from1 to N do
61:Update the NT(un) and NT(Un) for all n = 1,N using Eqs. (26,27)
62:Update xun and xn using Eq. (1)
63:Perform m steps of message passing for all nodes in the graph
64:Update husing Eq. (25)
65:end for
66: end for Output: PT,ST
", + "bbox": [ + 171, + 127, + 767, + 493 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/9f65838cdb1528ab2e922862a26db57939f9dbdd1a11736b34d5809570c9c9bc.jpg", + "table_caption": [ + "Table 6: Proportion of reactions containing reagents and proportion of reagents over input molecules on USPTO-15k and USPTO. " + ], + "table_footnote": [], + "table_body": "
Dataset%reactionscontaining reagents%reagents over input molecules
USPTO-15ktrain63.1%41.3%
valid65.3%42.3%
test63.6%40.9%
USPTOtrain79.7%54.0%
valid80.0%54.4%
test79.9%54.2%
", + "bbox": [ + 271, + 516, + 725, + 637 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/f49e2a5a755feff77e408c1dc8e86781a7fed33932c72027630272324b583ea6.jpg", + "image_caption": [ + "Figure 4: Learning curves of our model with and without using reagent information explicitly on USPTO-15k. " + ], + "image_footnote": [], + "bbox": [ + 349, + 702, + 648, + 871 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/40f884d517ce4b417d6e71bd536d894e28c9e8354fad66cbe5d1db0dabd5133a.jpg", + "table_caption": [ + "Table 7: Results for the reaction prediction task. $P \\ @ k$ is the precision at $k$ . Best results are highlighted in bold. Meanings of markers in our model: $\\diamondsuit$ : With beam search (beam width $= 2 0$ ), $\\spadesuit$ : Invalid product removal, $\\mathbf { \\hat { \\psi } } _ { \\mathbf { * } }$ : Duplicate product removal. " + ], + "table_footnote": [], + "table_body": "
ModelProcessed USPTO
P@1P@3P@5
WLDN (Jin et al., 2017)84.091.192.3
ELECTRO (Bradshaw et al., 2018)GTPN**87.094.595.9
87.3590.2290.68
", + "bbox": [ + 287, + 101, + 710, + 183 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/aebc50a959e34bea76abae1e557fd21e1b1f51ece918963e01707e0931304628.jpg", + "image_caption": [ + "Figure 5: Performance with respect to different numbers of bond changes. (a) Top-1 accuracy. (b) Errors grouped by length. In (a), blue: all reactions having that sequence length; orange: correct predicted reactions. In (b), red: the predicted sequence is shorter (than the groundtruth sequence); green: the predicted and the groundtruth have the same length; blue: the predicted sequence is longer; number indidate the average length. " + ], + "image_footnote": [], + "bbox": [ + 187, + 262, + 808, + 425 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In performance To do a fair comparison with ELECTRO (Bradshaw et al., 2018), we follow their procedure described in the paper to prepare a new test set that contains only reactions with linear chain topology and single-valence bond changes. It results in 29,808 reactions, close to the reported number of 29,360 in (Bradshaw et al., 2018). We reuse our old model (see Section 3.4) trained on the original USPTO dataset. We also use beam search decoding and post-processing as similar to (Bradshaw et al., 2018). From Table 7, we see that GTPN achieves the highest top-1 accuracy of $8 7 . 3 5 \\%$ , outperforming ELECTRO and WLDN by $0 . 3 5 \\%$ and $3 \\%$ , respectively. For the top-3 and top-5 accuracies, our model, however, does worse than the other two. Especially, while both ELECTRO and WLDN have big jumps from $P \\ @ { \\cal I }$ to $P ( \\omega 3$ with about $7 \\%$ improvement, GTPN only has $3 \\%$ increase. We conjecture that this problem mainly comes from the fact that GTPN was not optimized on the compatible training and validation sets. ", + "bbox": [ + 173, + 530, + 825, + 683 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.7 ERROR ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 704, + 343, + 718 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this section, we analyze several error types that our model makes during prediction. All the results below are computed on the USPTO-15k dataset by using beam search decoding with the beam width $N = 2 0$ and no post-processing. ", + "bbox": [ + 174, + 732, + 825, + 773 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Errors grouped by number of bond changes Fig. 5 shows the top-1 accuracies for reactions with different number of bond changes. Our model performs poorly on reactions with many bond changes. However, those kinds of reactions only accounts for a small proportion in the dataset. From Fig. 5b, we see that the lengths of the error sequences tends to be shorter than the lengths of the groundtruth sequences. ", + "bbox": [ + 174, + 792, + 825, + 862 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Errors caused by signal/atom pair/bond type We define a sub-action causing error as the first sub-action that our model makes a wrong decision. In Fig. 6a, we plot the the proportion of errors with respect to the three kinds of sub-actions. Clearly, atom pair prediction causes the most errors ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/4f2ec61d977e14daa78a49521f7d4f16a8bd5f1d149bfd67fb5ba89e7f0dd975.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 209, + 125, + 441, + 308 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(a) Proportion of the first incorrect sub-action that our model makes. ", + "bbox": [ + 179, + 318, + 457, + 344 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/1fa8e0b536b959f903902913cdcd3bc4058776fee20d292f2bb349e260285dcb.jpg", + "image_caption": [ + "(b) Proportion of the incorrect top-1 products that have similar structure to the groundtruth products. " + ], + "image_footnote": [], + "bbox": [ + 570, + 118, + 795, + 309 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Figure 6: Errors grouped by the first incorrect sub-actions (a), and errors caused by symmetric structures (b). ", + "bbox": [ + 173, + 364, + 823, + 393 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(nearly two third). This makes sense because this sub-action is harder than signal prediction and bond type prediction. Therefore, more effort should be put on improving the prediction of atom pairs. ", + "bbox": [ + 173, + 417, + 823, + 445 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Errors caused by symmetry There exists cases in which different sequences of bond changes can result in the same products due to symmetric graph structures. Errors caused by symmetry account for $5 . 7 \\%$ of the top-1 errors on the USPTO-15k dataset as shown in Fig. 6b. For better understanding, we provide a short list of wrong reaction triple predictions caused by symmetry in Fig. 7. In this list, the top-1 products (along the second column) are incorrect while the top-2 products (along the third column) are correct though both have the same probability. ", + "bbox": [ + 173, + 460, + 825, + 545 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/c508e5522ddc1d32c7b23e4219b31cd604fecc99998016a648a66b09069a3155.jpg", + "image_caption": [ + "Figure 7: Visualization of some reactions that cause multiple products with symmetric structures. Each row corresponds to a reaction. The columns, from left to right, show: i) reactant and reagent molecules, ii) incorrect top-1 product molecules, iii) correct top-2 product molecules, and iv) major groundtruth product molecules. All atoms in the first three columns are labeled with their atom map numbers. For the top-1 and top-2 products, we highlight the predicted reaction triples in green and provide the probability of the predicted sequence at the bottom. 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Our model’s performances and prediction errors are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 288, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 288, + 377 + ], + "score": 1.0, + "content": "also analyzed carefully in the paper.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Chemical reaction product prediction is a fundamental problem in organic chemistry. It paves the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "way for planning syntheses of new substances (Chen & Baldi, 2009). For decades, huge effort has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "been spent to solve this problem. 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See Fig. 1 for an illustration. We", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 177, + 534 + ], + "score": 1.0, + "content": "call an atom pair", + "type": "text" + }, + { + "bbox": [ + 177, + 523, + 201, + 534 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 523, + 455, + 534 + ], + "score": 1.0, + "content": "that changes its connectivity during reaction and its new bond", + "type": "text" + }, + { + "bbox": [ + 455, + 523, + 461, + 533 + ], + "score": 0.66, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "a reaction", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 131, + 545 + ], + "score": 1.0, + "content": "triple", + "type": "text" + }, + { + "bbox": [ + 132, + 533, + 164, + 545 + ], + "score": 0.92, + "content": "( u , v , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 533, + 504, + 545 + ], + "score": 1.0, + "content": ". The reaction product prediction problem now becomes predicting a set of reaction", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "triples given the input reactants and reagents. 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(b) Knowing how to choose from these reactivity patterns a correct set", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 589, + 311, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 311, + 600 + ], + "score": 1.0, + "content": "of reaction triples to generate the desired products.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Recent state-of-the-art methods (Jin et al., 2017; Bradshaw et al., 2018) have built the first capability", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "by leveraging graph neural networks (Duvenaud et al., 2015; Hamilton et al., 2017; Pham et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "2017; Gilmer et al., 2017). 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In order to guide model search through the complex", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "score": 1.0, + "content": "discrete space of sets of bond changes effectively, we extend the standard policy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 469, + 342 + ], + "score": 1.0, + "content": "gradient loss by adding useful constraints. Evaluation results show that GTPN", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 454, + 353 + ], + "score": 1.0, + "content": "improves the top-1 accuracy over the current state-of-the-art method by about", + "type": "text" + }, + { + "bbox": [ + 454, + 341, + 469, + 352 + ], + "score": 0.83, + "content": "3 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 365 + ], + "score": 1.0, + "content": "on the large USPTO dataset. Our model’s performances and prediction errors are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 288, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 288, + 377 + ], + "score": 1.0, + "content": "also analyzed carefully in the paper.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 209, + 471, + 377 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Chemical reaction product prediction is a fundamental problem in organic chemistry. It paves the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "way for planning syntheses of new substances (Chen & Baldi, 2009). For decades, huge effort has", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "been spent to solve this problem. However, most methods still depend on the handcrafted reaction", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "rules (Chen & Baldi, 2009; Kayala & Baldi, 2011; Wei et al., 2016) or heuristically extracted reaction", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "templates (Segler & Waller, 2017; Coley et al., 2017), thus are not well generalizable to unseen", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 148, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 148, + 485 + ], + "score": 1.0, + "content": "reactions.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 417, + 506, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "A reaction can be regarded as a set (or unordered sequence) of graph transformations in which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "reactants represented as molecular graphs are transformed into products by modifying the bonds", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 523 + ], + "score": 1.0, + "content": "between some atom pairs (Jochum et al., 1980; Ugi et al., 1979). See Fig. 1 for an illustration. We", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 177, + 534 + ], + "score": 1.0, + "content": "call an atom pair", + "type": "text" + }, + { + "bbox": [ + 177, + 523, + 201, + 534 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 523, + 455, + 534 + ], + "score": 1.0, + "content": "that changes its connectivity during reaction and its new bond", + "type": "text" + }, + { + "bbox": [ + 455, + 523, + 461, + 533 + ], + "score": 0.66, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "a reaction", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 131, + 545 + ], + "score": 1.0, + "content": "triple", + "type": "text" + }, + { + "bbox": [ + 132, + 533, + 164, + 545 + ], + "score": 0.92, + "content": "( u , v , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 533, + 504, + 545 + ], + "score": 1.0, + "content": ". The reaction product prediction problem now becomes predicting a set of reaction", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "triples given the input reactants and reagents. We argue that in order to solve this problem well, an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 556, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 567 + ], + "score": 1.0, + "content": "intelligent system should have two key capabilities: (a) Understanding the molecular graph structure", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "of the input reactants and reagents so that it can identify possible reactivity patterns (i.e., atom pairs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "with changing connectivity). (b) Knowing how to choose from these reactivity patterns a correct set", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 589, + 311, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 311, + 600 + ], + "score": 1.0, + "content": "of reaction triples to generate the desired products.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 489, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Recent state-of-the-art methods (Jin et al., 2017; Bradshaw et al., 2018) have built the first capability", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "by leveraging graph neural networks (Duvenaud et al., 2015; Hamilton et al., 2017; Pham et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "2017; Gilmer et al., 2017). However, these methods are either unaware of the valid sets of reaction", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "triples (Jin et al., 2017) or limited to sequences of reaction triples with a predefined orders (Bradshaw", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "et al., 2018). The main challenge is that the space of all possible configurations of reaction triples", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "is extremely large and non-differentiable. Moreover, a small change in the predicted set of reaction", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 672, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 683 + ], + "score": 1.0, + "content": "triples can lead to very different reaction products and a little mistake can produce invalid prediction.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 604, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "In this paper, we propose a novel method called Graph Transformation Policy Network (GTPN) that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "addresses the aforementioned challenges. Our model consists of three main components: a graph", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "neural network (GNN), a node pair prediction network (NPPN) and a policy network (PN). Starting", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "from the initial graph of reactant and reagent molecules, our model iteratively alternates between", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "modeling an input graph using GNN and predicting a reaction triple using NPPN and PN to generate", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "a new intermediate graph as input for the next step until it decides to stop. The final generated graph", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "is considered as the predicted products of the reaction. Importantly, GTPN does not assume any", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "score": 1.0, + "content": "fixed number or any order of bond changes but learn these properties itself. One can view GTPN as", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "a reinforcement learning (RL) agent that operates on a complex and non-differentiable space of sets", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "score": 1.0, + "content": "of reaction triples. To guide our model towards learning a diverse yet robust-to-small-changes policy,", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "we customize our loss function by adding some useful constraints to the standard policy gradient loss", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 333, + 186, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 186, + 345 + ], + "score": 1.0, + "content": "(Mnih et al., 2016).", + "type": "text", + "cross_page": true + } + ], + "index": 15 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 79, + 487, + 173 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 79, + 487, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 79, + 487, + 173 + ], + "spans": [ + { + "bbox": [ + 132, + 79, + 487, + 173 + ], + "score": 0.963, + "type": "image", + "image_path": "a12f14d83ce2155ad6d7941ebcf1a5e44c5921b556fe473e90e0c85202a223ba.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 79, + 487, + 110.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 110.33333333333333, + 487, + 141.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 141.66666666666666, + 487, + 173.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 179, + 506, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "score": 1.0, + "content": "Figure 1: A sample reaction represented as a set of graph transformations from reactants (leftmost) to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "products (rightmost). Atoms are labeled with their type (Carbon, Oxygen,...) and their index (1, 2,...)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "score": 1.0, + "content": "in the molecular graph. The atom pairs that change connectivity and their new bonds (if existed) are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "highlighted in green. There are two bond changes in this case: 1) The double bond between O:1 and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 408, + 236 + ], + "score": 1.0, + "content": "C:2 becomes single. 2) A new single bond between C:2 and C:10 is added.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "modeling an input graph using GNN and predicting a reaction triple using NPPN and PN to generate", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "a new intermediate graph as input for the next step until it decides to stop. The final generated graph", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "is considered as the predicted products of the reaction. Importantly, GTPN does not assume any", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "score": 1.0, + "content": "fixed number or any order of bond changes but learn these properties itself. One can view GTPN as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "a reinforcement learning (RL) agent that operates on a complex and non-differentiable space of sets", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "score": 1.0, + "content": "of reaction triples. To guide our model towards learning a diverse yet robust-to-small-changes policy,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "we customize our loss function by adding some useful constraints to the standard policy gradient loss", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 333, + 186, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 186, + 345 + ], + "score": 1.0, + "content": "(Mnih et al., 2016).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "To the best of our knowledge, GTPN is the most generic approach for the reaction product prediction", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "problem so far in the sense that: i) It combines graph neural networks and reinforcement learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "into a unified framework and trains everything end-to-end; ii) It does not use any handcrafted or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "heuristically extracted reaction rules/templates to predict the products. Instead, it automatically learns", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "various types of reactions from the training data and can generalize to unseen reactions; iii) It can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 405, + 451, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 451, + 417 + ], + "score": 1.0, + "content": "interpret how the products are formed via the sequence of reaction triples it generates.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 344, + 434 + ], + "score": 1.0, + "content": "We evaluate GTPN on two large public datasets named", + "type": "text" + }, + { + "bbox": [ + 344, + 421, + 395, + 432 + ], + "score": 0.51, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "and USPTO. Our method", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "significantly outperforms all baselines in the top-1 accuracy, achieving new state-of-the-art results", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 118, + 456 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 443, + 151, + 454 + ], + "score": 0.87, + "content": "8 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 442, + 171, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 443, + 204, + 454 + ], + "score": 0.87, + "content": "8 3 . 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 442, + 220, + 456 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 220, + 443, + 271, + 454 + ], + "score": 0.63, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "and USPTO, respectively. In addition, we also provide", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "comprehensive analyses about the performance of GTPN and about different types of errors our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 465, + 184, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 184, + 477 + ], + "score": 1.0, + "content": "model could make.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 495, + 172, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 493, + 174, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 174, + 511 + ], + "score": 1.0, + "content": "2 METHOD", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 522, + 502, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "2.1 CHEMICAL REACTION AS MARKOV DECISION PROCESS OF GRAPH TRANSFORMATIONS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "A reaction occurs when reactant molecules interact with each other in the presence (or absence) of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "reagent molecules to form new product molecules by breaking or adding some of their bonds. Our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "main insight is that reaction product prediction can be formulated as predicting a sequence of such", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "bond changes given the reactant and reagent molecules as input. A bond change is characterized by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "the atom pair (where the change happens) and the new bond type (what is the change). We call this", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 599, + 486, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 486, + 612 + ], + "score": 1.0, + "content": "atom pair a reaction atom pair and call this atom pair with the new bond type a reaction triple.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "More formally, we represent the entire system of input reactant and reagent molecules as a labeled", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 132, + 639 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 133, + 627, + 179, + 639 + ], + "score": 0.94, + "content": "\\mathcal { G } = ( \\dot { \\mathcal { V } } , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "with multiple connected components, each of which corresponds to a molecule.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 144, + 650 + ], + "score": 1.0, + "content": "Nodes in", + "type": "text" + }, + { + "bbox": [ + 145, + 639, + 153, + 648 + ], + "score": 0.75, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 638, + 384, + 650 + ], + "score": 1.0, + "content": "are atoms labeled with their atomic numbers and edges in", + "type": "text" + }, + { + "bbox": [ + 384, + 639, + 392, + 648 + ], + "score": 0.77, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "are bonds labeled with their", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 180, + 662 + ], + "score": 1.0, + "content": "bond types. 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Atoms are labeled with their type (Carbon, Oxygen,...) and their index (1, 2,...)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "score": 1.0, + "content": "in the molecular graph. The atom pairs that change connectivity and their new bonds (if existed) are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "highlighted in green. There are two bond changes in this case: 1) The double bond between O:1 and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 408, + 236 + ], + "score": 1.0, + "content": "C:2 becomes single. 2) A new single bond between C:2 and C:10 is added.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 344 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 255, + 506, + 345 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "To the best of our knowledge, GTPN is the most generic approach for the reaction product prediction", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "problem so far in the sense that: i) It combines graph neural networks and reinforcement learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "into a unified framework and trains everything end-to-end; ii) It does not use any handcrafted or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "heuristically extracted reaction rules/templates to predict the products. Instead, it automatically learns", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "various types of reactions from the training data and can generalize to unseen reactions; iii) It can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 405, + 451, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 451, + 417 + ], + "score": 1.0, + "content": "interpret how the products are formed via the sequence of reaction triples it generates.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 349, + 506, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 344, + 434 + ], + "score": 1.0, + "content": "We evaluate GTPN on two large public datasets named", + "type": "text" + }, + { + "bbox": [ + 344, + 421, + 395, + 432 + ], + "score": 0.51, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "and USPTO. Our method", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "significantly outperforms all baselines in the top-1 accuracy, achieving new state-of-the-art results", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 118, + 456 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 443, + 151, + 454 + ], + "score": 0.87, + "content": "8 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 442, + 171, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 443, + 204, + 454 + ], + "score": 0.87, + "content": "8 3 . 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 442, + 220, + 456 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 220, + 443, + 271, + 454 + ], + "score": 0.63, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "and USPTO, respectively. In addition, we also provide", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "score": 1.0, + "content": "comprehensive analyses about the performance of GTPN and about different types of errors our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 465, + 184, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 184, + 477 + ], + "score": 1.0, + "content": "model could make.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 421, + 506, + 477 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 495, + 172, + 508 + ], + "lines": [ + { + "bbox": [ + 104, + 493, + 174, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 174, + 511 + ], + "score": 1.0, + "content": "2 METHOD", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 522, + 502, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "2.1 CHEMICAL REACTION AS MARKOV DECISION PROCESS OF GRAPH TRANSFORMATIONS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 521, + 505, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "A reaction occurs when reactant molecules interact with each other in the presence (or absence) of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "reagent molecules to form new product molecules by breaking or adding some of their bonds. Our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "main insight is that reaction product prediction can be formulated as predicting a sequence of such", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "bond changes given the reactant and reagent molecules as input. A bond change is characterized by", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "the atom pair (where the change happens) and the new bond type (what is the change). We call this", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 599, + 486, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 486, + 612 + ], + "score": 1.0, + "content": "atom pair a reaction atom pair and call this atom pair with the new bond type a reaction triple.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 545, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "More formally, we represent the entire system of input reactant and reagent molecules as a labeled", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 132, + 639 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 133, + 627, + 179, + 639 + ], + "score": 0.94, + "content": "\\mathcal { G } = ( \\dot { \\mathcal { V } } , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "with multiple connected components, each of which corresponds to a molecule.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 144, + 650 + ], + "score": 1.0, + "content": "Nodes in", + "type": "text" + }, + { + "bbox": [ + 145, + 639, + 153, + 648 + ], + "score": 0.75, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 638, + 384, + 650 + ], + "score": 1.0, + "content": "are atoms labeled with their atomic numbers and edges in", + "type": "text" + }, + { + "bbox": [ + 384, + 639, + 392, + 648 + ], + "score": 0.77, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "are bonds labeled with their", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 647, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 180, + 662 + ], + "score": 1.0, + "content": "bond types. Given", + "type": "text" + }, + { + "bbox": [ + 181, + 649, + 189, + 659 + ], + "score": 0.83, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 647, + 446, + 662 + ], + "score": 1.0, + "content": "as input, we predict a sequence of reaction triples that transforms", + "type": "text" + }, + { + "bbox": [ + 447, + 649, + 455, + 659 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 647, + 506, + 662 + ], + "score": 1.0, + "content": "into a graph", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 659, + 207, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 193, + 673 + ], + "score": 1.0, + "content": "of product molecules", + "type": "text" + }, + { + "bbox": [ + 193, + 660, + 203, + 671 + ], + "score": 0.87, + "content": "\\mathcal { G } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 659, + 207, + 673 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 616, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "As reactions vary in number of transformation steps, we represent the sequence of reaction triples", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 118, + 700 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 687, + 289, + 700 + ], + "score": 0.91, + "content": "( \\xi , u , v , b ) ^ { 0 } , ( \\dot { \\xi } , u , v , b ) ^ { 1 } , . . . , ( \\xi , u , v , b ) ^ { T - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 686, + 303, + 700 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 303, + 687, + 357, + 700 + ], + "score": 0.92, + "content": "( \\dot { \\xi } , \\dot { u } , v , b ) ^ { \\dot { 0 } : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 686, + 425, + 700 + ], + "score": 1.0, + "content": "for short. Here", + "type": "text" + }, + { + "bbox": [ + 425, + 688, + 434, + 698 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "is the maximum", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 177, + 711 + ], + "score": 1.0, + "content": "number of steps,", + "type": "text" + }, + { + "bbox": [ + 177, + 700, + 201, + 711 + ], + "score": 0.9, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 698, + 277, + 711 + ], + "score": 1.0, + "content": "is a pair of nodes,", + "type": "text" + }, + { + "bbox": [ + 277, + 700, + 282, + 708 + ], + "score": 0.8, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 698, + 381, + 711 + ], + "score": 1.0, + "content": "is the new edge type of", + "type": "text" + }, + { + "bbox": [ + 381, + 699, + 405, + 710 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 698, + 425, + 711 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 426, + 699, + 432, + 709 + ], + "score": 0.75, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "is a binary signal", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 356, + 723 + ], + "score": 1.0, + "content": "that indicates the end of the sequence. 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This leads to better results than not using reagent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 356, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 356, + 591 + ], + "score": 1.0, + "content": "information. Detailed analyses are provided in Appendix A.5.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 535, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 126, + 615 + ], + "score": 1.0, + "content": "Top-", + "type": "text" + }, + { + "bbox": [ + 127, + 602, + 137, + 612 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "atom pairs Because the number of atom pairs that actually participate in a reaction is very", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "small (usually smaller than 10) compared to the total number of atom pairs of the input molecules", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "(usually hundreds or thousands), it is much more efficient to identify reaction triples from a small", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 397, + 647 + ], + "score": 1.0, + "content": "subset of highly probable reaction atom pairs. For that reason, we extract", + "type": "text" + }, + { + "bbox": [ + 398, + 635, + 408, + 645 + ], + "score": 0.68, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 634, + 460, + 646 + ], + "score": 0.9, + "content": "K \\ll | \\mathcal { V } | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "atom pairs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 427, + 659 + ], + "score": 1.0, + "content": "with the highest scores. Later, we will predict reaction triples taken from these", + "type": "text" + }, + { + "bbox": [ + 428, + 646, + 438, + 655 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "atom pairs only.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 210, + 670 + ], + "score": 1.0, + "content": "We denote the set of top-", + "type": "text" + }, + { + "bbox": [ + 210, + 657, + 220, + 666 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "atom pairs, their corresponding scores, and representation vectors as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 666, + 450, + 682 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 191, + 680 + ], + "score": 0.92, + "content": "\\left\\{ ( u _ { k } , v _ { k } ) | k = \\overline { { 1 , K } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 666, + 196, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 196, + 667, + 271, + 680 + ], + "score": 0.91, + "content": "\\left\\{ \\dot { s } _ { u _ { k } v _ { k } } | k = \\overline { { 1 , K } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 666, + 289, + 682 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 290, + 667, + 394, + 680 + ], + "score": 0.94, + "content": "Z _ { K } = \\mathsf { \\bar { \\{ } } z _ { u _ { k } v _ { k } } \\mathsf { \\bar { | } } k = \\overline { { 1 , K } } \\mathsf \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 666, + 450, + 682 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 601, + 506, + 682 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 220, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 222, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 222, + 703 + ], + "score": 1.0, + "content": "2.2.3 POLICY NETWORK", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Predicting continuation signal To account for varying number of transformation steps, PN gener-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 212, + 733 + ], + "score": 1.0, + "content": "ates a continuation signal", + "type": "text" + }, + { + "bbox": [ + 212, + 721, + 259, + 732 + ], + "score": 0.93, + "content": "\\xi ^ { \\tau } \\in \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "to indicate whether prediction should continue or terminate.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 709, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 274, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 275, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 117, + 94 + ], + "score": 0.86, + "content": "\\xi ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 81, + 275, + 96 + ], + "score": 1.0, + "content": "is drawn from a Bernoulli distribution:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 94, + 411, + 109 + ], + "lines": [ + { + "bbox": [ + 199, + 94, + 411, + 109 + ], + "spans": [ + { + "bbox": [ + 199, + 94, + 411, + 109 + ], + "score": 0.88, + "content": "\\begin{array} { l l l } { p \\left( \\xi ^ { \\tau } = 1 \\right) } & { = } & { \\mathrm { s i g m o i d } \\left( f ^ { \\mathrm { s i g n a l } } \\left( \\left[ { h ^ { \\tau - 1 } } , g \\left( Z _ { K } ^ { \\tau } \\right) \\right] \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "709d77e53ade9cdb7e160c79b62f567a558ae5c7ccc1590336e5891ff2a1886e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 199, + 94, + 411, + 109 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 111, + 506, + 147 + ], + "lines": [ + { + "bbox": [ + 104, + 109, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 104, + 109, + 133, + 127 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 111, + 156, + 122 + ], + "score": 0.9, + "content": "\\pmb { h } ^ { \\tau - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 109, + 268, + 127 + ], + "score": 1.0, + "content": "is the previous RNN state;", + "type": "text" + }, + { + "bbox": [ + 269, + 113, + 285, + 124 + ], + "score": 0.92, + "content": "Z _ { K } ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 109, + 470, + 127 + ], + "score": 1.0, + "content": "is the set of representation vectors of the top", + "type": "text" + }, + { + "bbox": [ + 470, + 113, + 481, + 122 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 109, + 506, + 127 + ], + "score": 1.0, + "content": "atom", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 208, + 137 + ], + "score": 1.0, + "content": "pairs at the current step;", + "type": "text" + }, + { + "bbox": [ + 209, + 124, + 233, + 136 + ], + "score": 0.91, + "content": "f ^ { \\mathrm { s i g n a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 122, + 318, + 137 + ], + "score": 1.0, + "content": "is a neural network;", + "type": "text" + }, + { + "bbox": [ + 319, + 126, + 326, + 136 + ], + "score": 0.81, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 122, + 506, + 137 + ], + "score": 1.0, + "content": "is a function that maps an unordered set of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 373, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 373, + 147 + ], + "score": 1.0, + "content": "inputs to an output vector. For simplicity, we use a mean function:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 159, + 376, + 195 + ], + "lines": [ + { + "bbox": [ + 234, + 159, + 376, + 195 + ], + "spans": [ + { + "bbox": [ + 234, + 159, + 376, + 195 + ], + "score": 0.95, + "content": "z _ { K } ^ { \\tau - 1 } = g \\left( Z _ { K } ^ { \\tau } \\right) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } W z _ { u _ { k } v _ { k } } ^ { \\tau - 1 }", + "type": "interline_equation", + "image_path": "0e9c7f7c8f57fbffa1573059185dd878f6c44a2b1f423afaa3df4ccddc3b9b61.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 159, + 376, + 177.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 234, + 177.0, + 376, + 195.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 204, + 504, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 219 + ], + "score": 1.0, + "content": "Predicting atom pair At the next sub-step, PN predicts which atom pair changes its bond during", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 215, + 385, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 260, + 229 + ], + "score": 1.0, + "content": "the reaction by sampling from the top-", + "type": "text" + }, + { + "bbox": [ + 260, + 217, + 270, + 226 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 215, + 385, + 229 + ], + "score": 1.0, + "content": "atom pairs with probability:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 242, + 376, + 258 + ], + "lines": [ + { + "bbox": [ + 235, + 242, + 376, + 258 + ], + "spans": [ + { + "bbox": [ + 235, + 242, + 376, + 258 + ], + "score": 0.92, + "content": "p \\left( ( u _ { k } , v _ { k } ) ^ { \\tau } \\right) = \\operatorname { s o f t m a x } _ { K } \\left( s _ { u _ { k } v _ { k } } ^ { \\tau } \\right)", + "type": "interline_equation", + "image_path": "b4fafbedf0de609bab767affb0550c203bd8c905191a98b65789adeeb95db222.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 235, + 242, + 376, + 258 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 260, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 103, + 256, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 103, + 256, + 133, + 278 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 261, + 156, + 273 + ], + "score": 0.9, + "content": "s _ { u _ { k } v _ { k } } ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 256, + 273, + 278 + ], + "score": 1.0, + "content": "is the score of the atom pair", + "type": "text" + }, + { + "bbox": [ + 274, + 260, + 312, + 272 + ], + "score": 0.93, + "content": "( u _ { k } , v _ { k } ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 256, + 505, + 278 + ], + "score": 1.0, + "content": "computed in Eq. 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at the final step", + "type": "text" + }, + { + "bbox": [ + 420, + 143, + 458, + 154 + ], + "score": 0.91, + "content": "\\tau = T _ { \\mathrm { e n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 137, + 509, + 161 + ], + "score": 1.0, + "content": ", the model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 251, + 166 + ], + "score": 1.0, + "content": "receives additional delayed rewards;", + "type": "text" + }, + { + "bbox": [ + 252, + 156, + 259, + 166 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 155, + 363, + 166 + ], + "score": 1.0, + "content": "is the discount factor; and", + "type": "text" + }, + { + "bbox": [ + 364, + 155, + 376, + 167 + ], + "score": 0.89, + "content": "V _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "is the parametric value function.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 165, + 331, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 142, + 178 + ], + "score": 1.0, + "content": "We train", + "type": "text" + }, + { + "bbox": [ + 142, + 166, + 154, + 178 + ], + "score": 0.88, + "content": "V _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 165, + 331, + 178 + ], + "score": 1.0, + "content": "using the following mean square error loss:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 192, + 377, + 227 + ], + "lines": [ + { + "bbox": [ + 234, + 192, + 377, + 227 + ], + "spans": [ + { + "bbox": [ + 234, + 192, + 377, + 227 + ], + "score": 0.94, + "content": "\\mathcal { L } ^ { \\mathrm { v a l u e } } \\ = \\ \\sum _ { \\tau = 0 } ^ { T _ { \\mathrm { e n d } } } \\left\\| V _ { \\phi } \\left( Z _ { K } ^ { \\tau } \\right) - R ^ { \\tau } \\right\\| ^ { 2 }", + "type": "interline_equation", + "image_path": "7388c44aadaef8815777e0517fa2e9879fa86272ec641118d0016718f9e52823.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 192, + 377, + 209.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 234, + 209.5, + 377, + 227.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 236, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 237, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 133, + 240 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 227, + 146, + 237 + ], + "score": 0.87, + "content": "R ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 226, + 226, + 240 + ], + "score": 1.0, + "content": "is the return at step", + "type": "text" + }, + { + "bbox": [ + 227, + 230, + 233, + 237 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 226, + 237, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "Episode termination during training Although the loss defined in Eq. (9) is correct, it is not good", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 260, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 506, + 275 + ], + "score": 1.0, + "content": "to use in practice because: i) If our model selects a wrong sub-action at any sub-step of the step", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 271, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 135, + 287 + ], + "score": 1.0, + "content": "Twrong", + "type": "text" + }, + { + "bbox": [ + 136, + 273, + 197, + 284 + ], + "score": 0.58, + "content": "\\mathit { T _ { \\mathrm { w r o n g } } } < \\mathit { T _ { \\mathrm { e n d } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 271, + 506, + 288 + ], + "score": 1.0, + "content": ", the whole predicted sequence will be incorrect regardless of what will be", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 166, + 297 + ], + "score": 1.0, + "content": "predicted from", + "type": "text" + }, + { + "bbox": [ + 167, + 284, + 208, + 295 + ], + "score": 0.89, + "content": "T _ { \\mathrm { w r o n g } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 283, + 218, + 297 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 219, + 284, + 236, + 294 + ], + "score": 0.9, + "content": "T _ { \\mathrm { e n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 283, + 425, + 297 + ], + "score": 1.0, + "content": ". 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The actor-critic loss in Eq. 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(9) is correct, it is not good", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 260, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 506, + 275 + ], + "score": 1.0, + "content": "to use in practice because: i) If our model selects a wrong sub-action at any sub-step of the step", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 271, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 135, + 287 + ], + "score": 1.0, + "content": "Twrong", + "type": "text" + }, + { + "bbox": [ + 136, + 273, + 197, + 284 + ], + "score": 0.58, + "content": "\\mathit { T _ { \\mathrm { w r o n g } } } < \\mathit { T _ { \\mathrm { e n d } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 271, + 506, + 288 + ], + "score": 1.0, + "content": ", the whole predicted sequence will be incorrect regardless of what will be", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 166, + 297 + ], + "score": 1.0, + "content": "predicted from", + "type": "text" + }, + { + "bbox": [ + 167, + 284, + 208, + 295 + ], + "score": 0.89, + "content": "T _ { \\mathrm { w r o n g } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 283, + 218, + 297 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 219, + 284, + 236, + 294 + ], + "score": 0.9, + "content": "T _ { \\mathrm { e n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 283, + 425, + 297 + ], + "score": 1.0, + "content": ". 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Dataset#reactions#changes#molecules#atoms#bonds
USPTO-15ktrain10,500111112.3112013.641100134.931110134.7
valid1,500111112.3112013.67194134.55199134.2
test3,000111112.3111613.67198134.951102134.7
USPTOtrain409,03511612.2212914.891150139.761165138.6
valid30,00011612.2212514.891150139.671158138.5
test40.00011612.2212214.891150139.871162138.7
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Dataset#reactions#changes#molecules#atoms#bonds
USPTO-15ktrain10,500111112.3112013.641100134.931110134.7
valid1,500111112.3112013.67194134.55199134.2
test3,000111112.3111613.67198134.951102134.7
USPTOtrain409,03511612.2212914.891150139.761165138.6
valid30,00011612.2212514.891150139.671158138.5
test40.00011612.2212214.891150139.871162138.7
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Apart", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 192, + 437, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 437, + 204 + ], + "score": 1.0, + "content": "from “#reactions”, other columns are presented in the format “min | max | mean”.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 257 + ], + "lines": [], + "index": 7, + "bbox_fs": [ + 104, + 222, + 505, + 259 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 271, + 388, + 303 + ], + "lines": [ + { + "bbox": [ + 223, + 271, + 388, + 303 + ], + "spans": [ + { + "bbox": [ + 223, + 271, + 388, + 303 + ], + "score": 0.93, + "content": "\\mathcal { L } ^ { \\mathrm { o v e r } \\mathrm { l e n g t h } } = - \\sum _ { T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } } \\leq \\tau < T _ { \\mathrm { e n d } } } \\log p \\left( \\xi ^ { \\tau } = 0 \\right)", + "type": "interline_equation", + "image_path": "2159a08fa4939f95bf66528b5af1a9f1a1e1e31c7d272bd3e6f5bad26b48707f.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 271, + 388, + 287.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 223, + 287.0, + 388, + 303.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 101, + 306, + 509, + 340 + ], + "spans": [ + { + "bbox": [ + 101, + 306, + 130, + 340 + ], + "score": 1.0, + "content": "wherewhen", + "type": "text" + }, + { + "bbox": [ + 133, + 308, + 150, + 320 + ], + "score": 0.9, + "content": "T _ { \\mathrm { e n d } } ^ { \\mathrm { g t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 306, + 283, + 340 + ], + "score": 1.0, + "content": "end step of the ground-trut. 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Details about these datasets are given in Table 1. The USPTO dataset contains reactant,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "reagent and product molecules represented as SMILES strings. Using RDKit1, we convert the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "SMILES strings into molecule objects and store them as graphs. For each reaction, every atom in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "reactant and reagent molecules is identified with a unique “atom map number”. 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(5). Similar to (Jin et al., 2017), we use Coverage", + "type": "text" + }, + { + "bbox": [ + 465, + 444, + 479, + 454 + ], + "score": 0.51, + "content": "@ k", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "evaluation metric, which is the proportion of reactions that have all groundtruth reaction atom pairs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 270, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 175, + 479 + ], + "score": 1.0, + "content": "appear in the top", + "type": "text" + }, + { + "bbox": [ + 176, + 466, + 182, + 476 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 466, + 270, + 479 + ], + "score": 1.0, + "content": "predicted atom pairs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "We compare our proposed graph neural network (GNN) with Weisfeiler-Lehman Network (WLN)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "(Jin et al., 2017) and Column Network (CLN) (Pham et al., 2017). Since our GNN explicitly uses", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "reagent information to compute the scores of atom pairs, we modify the implementation of WLN and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 515, + 504, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 504, + 528 + ], + "score": 1.0, + "content": "CLN accordingly for fair comparison. From Table 2, we observe that our GNN clearly outperforms", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "WLN and CLN in all cases. We attribute this improvement to the use of a separate node state vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 118, + 549 + ], + "score": 0.87, + "content": "\\boldsymbol { x } _ { i } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 537, + 275, + 551 + ], + "score": 1.0, + "content": "(different from the node feature vector", + "type": "text" + }, + { + "bbox": [ + 275, + 539, + 286, + 549 + ], + "score": 0.83, + "content": "{ \\mathbf { } } v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ") for updating the structural information of a node (see", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "Eq. (21)). The other two models, on the other hand, only use a single vector to store both the node", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "features and structure, hence, some information may be lost. 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Further results are presented in Appendix A.5.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 272, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 274, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 152, + 642 + ], + "score": 1.0, + "content": "3.3 TOP-", + "type": "text" + }, + { + "bbox": [ + 152, + 630, + 162, + 639 + ], + "score": 0.72, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 628, + 274, + 642 + ], + "score": 1.0, + "content": "ATOM PAIR EXTRACTION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 445, + 667 + ], + "score": 1.0, + "content": "The performance of our model depends on the number of selected top atom pairs", + "type": "text" + }, + { + "bbox": [ + 446, + 655, + 456, + 665 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 654, + 505, + 667 + ], + "score": 1.0, + "content": ". 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In addition to the metric Coverage", + "type": "text" + }, + { + "bbox": [ + 490, + 666, + 504, + 676 + ], + "score": 0.77, + "content": "@ k", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 212, + 690 + ], + "score": 1.0, + "content": "in Sec. 3.2, we use Recall", + "type": "text" + }, + { + "bbox": [ + 213, + 677, + 227, + 687 + ], + "score": 0.62, + "content": "@ k", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 676, + 486, + 690 + ], + "score": 1.0, + "content": "which is the proportion of correct atom pairs that appear in top", + "type": "text" + }, + { + "bbox": [ + 487, + 677, + 493, + 687 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 163, + 700 + ], + "score": 1.0, + "content": "find the good", + "type": "text" + }, + { + "bbox": [ + 163, + 688, + 173, + 698 + ], + "score": 0.72, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 688, + 272, + 700 + ], + "score": 1.0, + "content": ". 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ModelUSPTO-15kUSPTO
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WLN* (Jin et al., 2017)81.686.189.189.892.093.3
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CLN (Pham et al., 2017)88.6891.6393.0790.7293.5794.80
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Similar to (Jin et al., 2017), we use Coverage", + "type": "text" + }, + { + "bbox": [ + 465, + 444, + 479, + 454 + ], + "score": 0.51, + "content": "@ k", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "evaluation metric, which is the proportion of reactions that have all groundtruth reaction atom pairs", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 270, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 175, + 479 + ], + "score": 1.0, + "content": "appear in the top", + "type": "text" + }, + { + "bbox": [ + 176, + 466, + 182, + 476 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 466, + 270, + 479 + ], + "score": 1.0, + "content": "predicted atom pairs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 432, + 505, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "We compare our proposed graph neural network (GNN) with Weisfeiler-Lehman Network (WLN)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "(Jin et al., 2017) and Column Network (CLN) (Pham et al., 2017). Since our GNN explicitly uses", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "reagent information to compute the scores of atom pairs, we modify the implementation of WLN and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 515, + 504, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 504, + 528 + ], + "score": 1.0, + "content": "CLN accordingly for fair comparison. From Table 2, we observe that our GNN clearly outperforms", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "WLN and CLN in all cases. We attribute this improvement to the use of a separate node state vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 118, + 549 + ], + "score": 0.87, + "content": "\\boldsymbol { x } _ { i } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 537, + 275, + 551 + ], + "score": 1.0, + "content": "(different from the node feature vector", + "type": "text" + }, + { + "bbox": [ + 275, + 539, + 286, + 549 + ], + "score": 0.83, + "content": "{ \\mathbf { } } v _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ") for updating the structural information of a node (see", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "Eq. (21)). The other two models, on the other hand, only use a single vector to store both the node", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "features and structure, hence, some information may be lost. In addition, using explicit reagent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 407, + 583 + ], + "score": 1.0, + "content": "information boosts the prediction accuracy, which improves the WLN by", + "type": "text" + }, + { + "bbox": [ + 408, + 570, + 431, + 581 + ], + "score": 0.84, + "content": "1 - 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "depending on the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "metrics. The presence of reagent information reduces the number of atom pairs to be searched on and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 592, + 502, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 502, + 605 + ], + "score": 1.0, + "content": "contributes to the likelihood of reaction atom pairs. 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Hence,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 302, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 210, + 734 + ], + "score": 1.0, + "content": "in what follows we select", + "type": "text" + }, + { + "bbox": [ + 210, + 721, + 244, + 731 + ], + "score": 0.91, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 720, + 302, + 734 + ], + "score": 1.0, + "content": "for efficiency.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 654, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 127, + 80, + 484, + 192 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 127, + 80, + 484, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 80, + 484, + 192 + ], + "spans": [ + { + "bbox": [ + 127, + 80, + 484, + 192 + ], + "score": 0.983, + "html": "
ModelUSPTO-15kUSPTO
P@1P@3P@5P@1P@3P@5
WLDN (Jin et al., 2017)76.785.686.879.687.789.2
Seq2Seq (Schwaller et al., 2018)---80.3*86.2*87.5*
GTPN72.311171.26--
GTPN74.5682.6284.2373.2580.5683.53
GTPN474.5683.1984.9773.2584.3185.76
GTPN82.3985.6086.6883.2084.9785.90
GTPN**82.3985.7386.7883.2086.0386.48
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State-of-the-art results from (Jin", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "et al., 2017) are written in italic. Results from (Schwaller et al., 2018) are marked with ? and they are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "score": 1.0, + "content": "computed on a slightly different version of USPTO that contains only single-product reactions. Best", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 230, + 243 + ], + "score": 1.0, + "content": "results are highlighted in bold.", + "type": "text" + }, + { + "bbox": [ + 230, + 230, + 238, + 240 + ], + "score": 0.48, + "content": "\\diamondsuit", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 230, + 367, + 243 + ], + "score": 1.0, + "content": ": With beam search (beam width", + "type": "text" + }, + { + "bbox": [ + 367, + 231, + 387, + 241 + ], + "score": 0.85, + "content": "= 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 230, + 394, + 243 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 394, + 229, + 402, + 240 + ], + "score": 0.35, + "content": "\\lessgtr", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 230, + 506, + 243 + ], + "score": 1.0, + "content": ": Invalid product removal,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 239, + 236, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 236, + 256 + ], + "score": 1.0, + "content": "♣: Duplicated product removal.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 109, + 275, + 276, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 278, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 278, + 288 + ], + "score": 1.0, + "content": "3.4 REACTION PRODUCT PREDICTION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "This experiment validates GTPN on full reaction product prediction against the recent state-of-the-art", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "methods (Jin et al., 2017; Schwaller et al., 2018) using the accuracy metric. The recent method", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "score": 1.0, + "content": "ELECTRO (Bradshaw et al., 2018) is not compatible here because it was only evaluated on a subset", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "of USPTO limited to linear chain topology. Comparison against ELECTRO is reported separately", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "in Appendix A.6. Table 3 shows the prediction results. We produce multiple reaction product", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 344, + 365 + ], + "score": 1.0, + "content": "candidates by using beam search decoding with beam width", + "type": "text" + }, + { + "bbox": [ + 345, + 353, + 378, + 363 + ], + "score": 0.9, + "content": "N = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 353, + 505, + 365 + ], + "score": 1.0, + "content": ". Details about beam search and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 286, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 286, + 376 + ], + "score": 1.0, + "content": "its behaviors are presented in Appendix A.4.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 395, + 393 + ], + "score": 1.0, + "content": "In brief, we compute the normalized-over-length log probabilities of", + "type": "text" + }, + { + "bbox": [ + 396, + 381, + 406, + 390 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "predicted sequences of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 391, + 421, + 405 + ], + "score": 1.0, + "content": "reaction triples and sort these values in descending order to get a rank list of", + "type": "text" + }, + { + "bbox": [ + 422, + 392, + 432, + 402 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "possible reaction", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 400, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 349, + 417 + ], + "score": 1.0, + "content": "outcomes. Given a predicted sequence of reaction triples", + "type": "text" + }, + { + "bbox": [ + 349, + 402, + 394, + 414 + ], + "score": 0.92, + "content": "( \\overline { { u } } , v , b ) ^ { 0 : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 400, + 506, + 417 + ], + "score": 1.0, + "content": ", we can generate reaction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 413, + 375, + 426 + ], + "score": 1.0, + "content": "products from input reactants simply by replacing the old bond of", + "type": "text" + }, + { + "bbox": [ + 375, + 414, + 404, + 425 + ], + "score": 0.92, + "content": "( u , v ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 413, + 425, + 426 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 426, + 414, + 436, + 424 + ], + "score": 0.85, + "content": "b ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 413, + 506, + 426 + ], + "score": 1.0, + "content": ". However, these", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "products are not guaranteed to be valid (e.g., maximum valence constraint violation or aromatic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "molecules cannot be kekulized) so we post-process the outputs by removing all invalid products.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 325, + 459 + ], + "score": 1.0, + "content": "The removal increases the top-1 accuracy by about", + "type": "text" + }, + { + "bbox": [ + 325, + 447, + 340, + 457 + ], + "score": 0.85, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 446, + 361, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 447, + 381, + 457 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 446, + 398, + 459 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 398, + 447, + 449, + 457 + ], + "score": 0.69, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "and USPTO,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "respectively. Due to the permutation invariance of the predicted sequence of reaction triples, some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 482, + 481 + ], + "score": 1.0, + "content": "product candidates are duplicate and will also be removed. This does not lead to any change in", + "type": "text" + }, + { + "bbox": [ + 483, + 469, + 505, + 479 + ], + "score": 0.79, + "content": "P \\ @ { \\cal I }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 405, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 192, + 491 + ], + "score": 1.0, + "content": "but slightly improves", + "type": "text" + }, + { + "bbox": [ + 193, + 479, + 215, + 489 + ], + "score": 0.76, + "content": "P \\ @ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 480, + 232, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 479, + 255, + 489 + ], + "score": 0.72, + "content": "P \\ @ 5", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 480, + 292, + 491 + ], + "score": 1.0, + "content": "by about", + "type": "text" + }, + { + "bbox": [ + 293, + 479, + 323, + 490 + ], + "score": 0.86, + "content": "0 . 5 – 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 480, + 405, + 491 + ], + "score": 1.0, + "content": "on the two datasets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "Overall, GTPN with beam search and post-processing outperforms both WLDN (Jin et al., 2017)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 507, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 507, + 521 + ], + "score": 1.0, + "content": "and Seq2Seq (Schwaller et al., 2018) in the top-1 accuracy. For the top-3 and top-5, our model’s", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 312, + 530 + ], + "score": 1.0, + "content": "performance is comparable to WLDN’s on USPTO-", + "type": "text" + }, + { + "bbox": [ + 313, + 518, + 328, + 528 + ], + "score": 0.33, + "content": "l 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "and is worse than WLDN’s on USPTO. It is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "not surprising since our model is trained to accurately predict the top-1 outcomes instead of ranking", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "the candidates directly like WLDN. It is important to emphasize that we did not tune the model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 425, + 564 + ], + "score": 1.0, + "content": "hyper-parameters when training on USPTO but reused the optimal settings from", + "type": "text" + }, + { + "bbox": [ + 425, + 551, + 474, + 561 + ], + "score": 0.34, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "(which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "is 25 times smaller than USPTO) so the results may not be optimal (see Appendix A.3 for more", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 170, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 170, + 586 + ], + "score": 1.0, + "content": "training detail).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 211, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 213, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 213, + 619 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 632, + 329, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 330, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 330, + 645 + ], + "score": 1.0, + "content": "4.1 LEARNING TO PREDICT CHEMICAL REACTION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "In chemical reaction prediction, machine learning has replaced rule-based methods (Chen & Baldi,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "2009) for better generalizability and scalability. Existing machine learning-based techiques are either", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "template-free (Kayala & Baldi, 2011; Jin et al., 2017; Fooshee et al., 2018) and template-based (Wei", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 700 + ], + "score": 1.0, + "content": "et al., 2016; Segler & Waller, 2017; Coley et al., 2017). Both groups share the same mechanism:", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "running multiple stages with the aid of reaction templates or rules. 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ModelUSPTO-15kUSPTO
P@1P@3P@5P@1P@3P@5
WLDN (Jin et al., 2017)76.785.686.879.687.789.2
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State-of-the-art results from (Jin", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "et al., 2017) are written in italic. Results from (Schwaller et al., 2018) are marked with ? and they are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 231 + ], + "score": 1.0, + "content": "computed on a slightly different version of USPTO that contains only single-product reactions. Best", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 230, + 243 + ], + "score": 1.0, + "content": "results are highlighted in bold.", + "type": "text" + }, + { + "bbox": [ + 230, + 230, + 238, + 240 + ], + "score": 0.48, + "content": "\\diamondsuit", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 230, + 367, + 243 + ], + "score": 1.0, + "content": ": With beam search (beam width", + "type": "text" + }, + { + "bbox": [ + 367, + 231, + 387, + 241 + ], + "score": 0.85, + "content": "= 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 230, + 394, + 243 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 394, + 229, + 402, + 240 + ], + "score": 0.35, + "content": "\\lessgtr", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 230, + 506, + 243 + ], + "score": 1.0, + "content": ": Invalid product removal,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 239, + 236, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 236, + 256 + ], + "score": 1.0, + "content": "♣: Duplicated product removal.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 109, + 275, + 276, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 278, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 278, + 288 + ], + "score": 1.0, + "content": "3.4 REACTION PRODUCT PREDICTION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "This experiment validates GTPN on full reaction product prediction against the recent state-of-the-art", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "methods (Jin et al., 2017; Schwaller et al., 2018) using the accuracy metric. The recent method", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "score": 1.0, + "content": "ELECTRO (Bradshaw et al., 2018) is not compatible here because it was only evaluated on a subset", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "of USPTO limited to linear chain topology. Comparison against ELECTRO is reported separately", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "in Appendix A.6. Table 3 shows the prediction results. We produce multiple reaction product", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 344, + 365 + ], + "score": 1.0, + "content": "candidates by using beam search decoding with beam width", + "type": "text" + }, + { + "bbox": [ + 345, + 353, + 378, + 363 + ], + "score": 0.9, + "content": "N = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 353, + 505, + 365 + ], + "score": 1.0, + "content": ". Details about beam search and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 286, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 286, + 376 + ], + "score": 1.0, + "content": "its behaviors are presented in Appendix A.4.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 297, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 395, + 393 + ], + "score": 1.0, + "content": "In brief, we compute the normalized-over-length log probabilities of", + "type": "text" + }, + { + "bbox": [ + 396, + 381, + 406, + 390 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "predicted sequences of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 391, + 421, + 405 + ], + "score": 1.0, + "content": "reaction triples and sort these values in descending order to get a rank list of", + "type": "text" + }, + { + "bbox": [ + 422, + 392, + 432, + 402 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "possible reaction", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 400, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 349, + 417 + ], + "score": 1.0, + "content": "outcomes. 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However, these", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "products are not guaranteed to be valid (e.g., maximum valence constraint violation or aromatic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "molecules cannot be kekulized) so we post-process the outputs by removing all invalid products.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 325, + 459 + ], + "score": 1.0, + "content": "The removal increases the top-1 accuracy by about", + "type": "text" + }, + { + "bbox": [ + 325, + 447, + 340, + 457 + ], + "score": 0.85, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 446, + 361, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 447, + 381, + 457 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 446, + 398, + 459 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 398, + 447, + 449, + 457 + ], + "score": 0.69, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "and USPTO,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "respectively. Due to the permutation invariance of the predicted sequence of reaction triples, some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 482, + 481 + ], + "score": 1.0, + "content": "product candidates are duplicate and will also be removed. This does not lead to any change in", + "type": "text" + }, + { + "bbox": [ + 483, + 469, + 505, + 479 + ], + "score": 0.79, + "content": "P \\ @ { \\cal I }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 405, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 192, + 491 + ], + "score": 1.0, + "content": "but slightly improves", + "type": "text" + }, + { + "bbox": [ + 193, + 479, + 215, + 489 + ], + "score": 0.76, + "content": "P \\ @ 3", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 480, + 232, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 479, + 255, + 489 + ], + "score": 0.72, + "content": "P \\ @ 5", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 480, + 292, + 491 + ], + "score": 1.0, + "content": "by about", + "type": "text" + }, + { + "bbox": [ + 293, + 479, + 323, + 490 + ], + "score": 0.86, + "content": "0 . 5 – 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 480, + 405, + 491 + ], + "score": 1.0, + "content": "on the two datasets.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 380, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "Overall, GTPN with beam search and post-processing outperforms both WLDN (Jin et al., 2017)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 507, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 507, + 521 + ], + "score": 1.0, + "content": "and Seq2Seq (Schwaller et al., 2018) in the top-1 accuracy. For the top-3 and top-5, our model’s", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 312, + 530 + ], + "score": 1.0, + "content": "performance is comparable to WLDN’s on USPTO-", + "type": "text" + }, + { + "bbox": [ + 313, + 518, + 328, + 528 + ], + "score": 0.33, + "content": "l 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "and is worse than WLDN’s on USPTO. It is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "not surprising since our model is trained to accurately predict the top-1 outcomes instead of ranking", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "the candidates directly like WLDN. It is important to emphasize that we did not tune the model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 425, + 564 + ], + "score": 1.0, + "content": "hyper-parameters when training on USPTO but reused the optimal settings from", + "type": "text" + }, + { + "bbox": [ + 425, + 551, + 474, + 561 + ], + "score": 0.34, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "(which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "is 25 times smaller than USPTO) so the results may not be optimal (see Appendix A.3 for more", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 170, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 170, + 586 + ], + "score": 1.0, + "content": "training detail).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 496, + 507, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 211, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 213, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 213, + 619 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 632, + 329, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 330, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 330, + 645 + ], + "score": 1.0, + "content": "4.1 LEARNING TO PREDICT CHEMICAL REACTION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "In chemical reaction prediction, machine learning has replaced rule-based methods (Chen & Baldi,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "2009) for better generalizability and scalability. Existing machine learning-based techiques are either", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "template-free (Kayala & Baldi, 2011; Jin et al., 2017; Fooshee et al., 2018) and template-based (Wei", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 700 + ], + "score": 1.0, + "content": "et al., 2016; Segler & Waller, 2017; Coley et al., 2017). Both groups share the same mechanism:", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "running multiple stages with the aid of reaction templates or rules. For example, in (Wei et al., 2016)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the authors proposed a two-stage model that first classifies reactions into different types based on the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "neural fingerprint vectors (Duvenaud et al., 2015) of reactant and reagent molecules. Then, it applies", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "pre-designed SMARTS transformation on the reactants with respect to the most suitable predicted", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 295, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 295, + 105 + ], + "score": 1.0, + "content": "reaction type to generate the reaction products.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 655, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "pre-designed SMARTS transformation on the reactants with respect to the most suitable predicted", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 295, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 295, + 105 + ], + "score": 1.0, + "content": "reaction type to generate the reaction products.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "score": 1.0, + "content": "The work of (Jin et al., 2017) treats a reaction as a set of bond changes so in the first step, they", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "predict which atom pairs are likely to be reactive using a variant of graph neural networks called", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Weisfeiler-Lehman Networks (WLNs). In the next step, they do almost the same as (Coley et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "2017) by modifying the bond type between the selected atom pairs (with chemical rules satisfied) to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "create product candidates and rank them (with reactant molecules as addition input) using another", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 393, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 393, + 177 + ], + "score": 1.0, + "content": "kind of WLNs called Weifeiler-Lehman Different Networks (WLDNs).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "To the best of our knowledge, (Jin et al., 2017) is the first work that achieves remarkable results (with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 160, + 205 + ], + "score": 1.0, + "content": "the Precision", + "type": "text" + }, + { + "bbox": [ + 160, + 193, + 174, + 203 + ], + "score": 0.73, + "content": "@ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 192, + 210, + 205 + ], + "score": 1.0, + "content": "is about", + "type": "text" + }, + { + "bbox": [ + 210, + 194, + 237, + 204 + ], + "score": 0.85, + "content": "7 9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 192, + 505, + 205 + ], + "score": 1.0, + "content": ") on the large USPTO dataset containing more than 480 thousands", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 218 + ], + "score": 1.0, + "content": "reactions. Works of (Nam & Kim, 2016) and (Schwaller et al., 2018) avoid multi-stage prediction by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "building a seq2seq model that generates the (canonical) SMILES string of the single product from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "the concatenated SMILES strings of the reactants and reagents in an end-to-end manner. However,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "their methods cannot deal with sets of reactants/reagents/products properly as well as cannot provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 249, + 299, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 299, + 259 + ], + "score": 1.0, + "content": "concrete reaction mechanism for every reaction.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 265, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "The most recent work on this topic is (Bradshaw et al., 2018) which solves the reaction prediction", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "problem by predicting a sequence of bond changes given input reactants and reagents represented", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "as graphs. To handle ordering, they only select reactions with predefined topology. Our method, by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 399, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 399, + 310 + ], + "score": 1.0, + "content": "contrast, is order-free and can be applied to almost any kind of reactions.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 323, + 380, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 382, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 382, + 336 + ], + "score": 1.0, + "content": "4.2 GRAPH NEURAL NETWORKS FOR MODELING MOLECULES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "In recent years, there has been a fast development of graph neural networks (GNNs) for modeling", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 369 + ], + "score": 1.0, + "content": "molecules. These models are proposed to solve different problems in chemistry including toxicity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "prediction (Duvenaud et al., 2015), drug activity classification (Shervashidze et al., 2011; Dai", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "et al., 2016; Pham et al., 2018), protein interface prediction (Fout et al., 2017) and drug generation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 400 + ], + "score": 1.0, + "content": "(Simonovsky & Komodakis, 2018; Jin et al., 2018). Most of them can be regarded as variants of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 403, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 403, + 411 + ], + "score": 1.0, + "content": "message-passing graph neural networks (MPGNNs) (Gilmer et al., 2017).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 423, + 389, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 390, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 390, + 437 + ], + "score": 1.0, + "content": "4.3 REINFORCEMENT LEARNING FOR STRUCTURAL REASONING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "Reinforcement learning (RL) has become a standard approach to many structural reasoning problems2", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "because it allows agents to perform discrete actions. A typical example of using RL for structural", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "reasoning is drug generation (Li et al., 2018; You et al., 2018). Both (Li et al., 2018) and (You", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "et al., 2018) learn the same generative policy whose action set including: i) adding a new atom or", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "a molecular scaffold to the intermediate graph, ii) connecting existing pair of atoms with bonds,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "and iii) terminating generation. However, (You et al., 2018) uses an adversarial loss to enforce", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "global chemical constraints on the generated molecules as a whole instead of using the common", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "reconstruction loss as in (Li et al., 2018). Other examples are path-based relational reasoning in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "knowledge graphs (Das et al., 2018) and learning combinatorial optimization over graphs (Khalil", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 542, + 159, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 159, + 555 + ], + "score": 1.0, + "content": "et al., 2017).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 190, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 192, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 192, + 586 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "We have introduced a novel method named Graph Transformation Policy Network (GTPN) for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "predicting products of a chemical reaction. GTPN uses graph neural networks to represent input", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "reactant and reagent molecules, and uses reinforcement learning to find an optimal sequence of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "bond changes that transforms the reactants into products. 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Experiments on real datasets have demonstrated the competitiveness of our model.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 668, + 504, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "Although the GTPN was proposed to solve the chemical reaction problem, it is indeed generic to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 679, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 692 + ], + "score": 1.0, + "content": "solve the graph transformation problem, which can be useful in reasoning about relations (e.g., see", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "(Zambaldi et al., 2018)) and changes in relation. 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Other examples are path-based relational reasoning in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "knowledge graphs (Das et al., 2018) and learning combinatorial optimization over graphs (Khalil", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 542, + 159, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 159, + 555 + ], + "score": 1.0, + "content": "et al., 2017).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 443, + 506, + 555 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 190, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 192, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 192, + 586 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "We have introduced a novel method named Graph Transformation Policy Network (GTPN) for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "predicting products of a chemical reaction. 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First, to update the graph structure, we simply update the neighbor", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 461, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 131, + 401 + ], + "score": 1.0, + "content": "set of", + "type": "text" + }, + { + "bbox": [ + 131, + 390, + 138, + 398 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 388, + 156, + 401 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 391, + 162, + 398 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 388, + 407, + 401 + ], + "score": 1.0, + "content": "with information from the other atom and the new bond type", + "type": "text" + }, + { + "bbox": [ + 408, + 389, + 414, + 398 + ], + "score": 0.75, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 388, + 461, + 401 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 414, + 393, + 446 + ], + "lines": [ + { + "bbox": [ + 216, + 414, + 393, + 446 + ], + "spans": [ + { + "bbox": [ + 216, + 414, + 393, + 446 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { \\mathcal { N } ^ { \\tau } ( u ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( u ) \\backslash \\left( v , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( v , b ) } \\\\ { \\mathcal { N } ^ { \\tau } ( v ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( v ) \\backslash \\left( u , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( u , b ) } \\end{array}", + "type": "interline_equation", + "image_path": "711fda8a1915bd0c770d210262fb5bf19078bfe031660032107382cf0a3b1810.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 414, + 393, + 430.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 216, + 430.0, + 393, + 446.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 451, + 457 + ], + "score": 1.0, + "content": "Next, to update the node states, our model performs one step of message passing for", + "type": "text" + }, + { + "bbox": [ + 452, + 446, + 458, + 453 + ], + "score": 0.77, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 443, + 477, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 447, + 483, + 453 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 205, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 205, + 467 + ], + "score": 1.0, + "content": "their new neighbor sets:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 479, + 398, + 512 + ], + "lines": [ + { + "bbox": [ + 210, + 479, + 398, + 512 + ], + "spans": [ + { + "bbox": [ + 210, + 479, + 398, + 512 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } { \\pmb { x } _ { u } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { u } ^ { \\tau - 1 } , \\pmb { v } _ { u } , \\mathcal { N } ^ { \\tau } ( u ) \\right) } \\\\ { \\pmb { x } _ { v } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { v } ^ { \\tau - 1 } , \\pmb { v } _ { v } , \\mathcal { N } ^ { \\tau } ( v ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "1037ecb9d4b964ec79c198c33b5dbb951a46c54f5a12e1d58afc79a824035a6f.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 479, + 398, + 495.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 210, + 495.5, + 398, + 512.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 506, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 215, + 522 + ], + "score": 1.0, + "content": "where the MessagePassing", + "type": "text" + }, + { + "bbox": [ + 216, + 510, + 226, + 522 + ], + "score": 0.55, + "content": "( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "function is defined in Eq. (24). For other nodes in the graph to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 233, + 534 + ], + "score": 1.0, + "content": "aware of the new structures of", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 240, + 531 + ], + "score": 0.79, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 520, + 259, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 523, + 266, + 531 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 520, + 506, + 534 + ], + "score": 1.0, + "content": ", we need to perform several message passing steps for all", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 507, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 496, + 545 + ], + "score": 1.0, + "content": "nodes in the graph after Eqs. (28, 29). However, it is very costly to run for every prediction step", + "type": "text" + }, + { + "bbox": [ + 496, + 535, + 503, + 542 + ], + "score": 0.7, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 531, + 507, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Sometimes it is unnecessary since far-away bonds are less likely to be affected by the current bond", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "score": 1.0, + "content": "change (unless the far-way bonds and the new bond are in an aromatic ring). Therefore, in our model,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 564, + 423, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 381, + 577 + ], + "score": 1.0, + "content": "we limit the number of message passing updates for all nodes at step", + "type": "text" + }, + { + "bbox": [ + 382, + 567, + 389, + 575 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 564, + 423, + 577 + ], + "score": 1.0, + "content": "to be 1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 246, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 247, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 247, + 601 + ], + "score": 1.0, + "content": "A.3 MODEL CONFIGURATIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "We optimize our model’s hyper-parameters in two stages: First, we tune the hyper-parameters of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "GNN and the NPPN for the reaction atom pair prediction task. Then, we fix the optimal settings", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "score": 1.0, + "content": "of the first two components and optimize the hyper-parameters of the PN for the reaction product", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 171, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 171, + 654 + ], + "score": 1.0, + "content": "prediction task.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "We provide details about the settings that give good results on the USPTO-15k dataset below. With", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "these settings, we trained another model on the USPTO dataset from scratch. Because training on", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "the large dataset such as the USPTO takes time, we did not tune hyper-parameters on the USPTO,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 694, + 397, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 397, + 705 + ], + "score": 1.0, + "content": "eventhough it is possible to increase model sizes for better performance.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "Unless explicitly stated, all neural networks in our model have 2 layers with the same number of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "score": 1.0, + "content": "hidden units, ReLU activation and residual connections (He et al., 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 376, + 118 + ], + "lines": [ + { + "bbox": [ + 102, + 76, + 329, + 101 + ], + "spans": [ + { + "bbox": [ + 102, + 76, + 133, + 101 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 81, + 154, + 95 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { x } } _ { i } ^ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 76, + 329, + 101 + ], + "score": 1.0, + "content": "is the nonlinear part which is computed as:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 235, + 93, + 375, + 108 + ], + "spans": [ + { + "bbox": [ + 235, + 93, + 375, + 108 + ], + "score": 0.77, + "content": "\\bar { \\pmb { x } } _ { i } ^ { t + 1 } = \\sigma \\left( \\bar { W _ { 1 } } \\left[ \\pmb { x } _ { i } ^ { t } , \\pmb { m } _ { i } ^ { t } , \\pmb { v } _ { i } ^ { t } \\right] + b _ { 1 } \\right)", + "type": "inline_equation", + "image_path": "29f8da63d44c74461484a2fbff27318dbb250ee5f901e01d99162192005dbfee.jpg" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 320, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 123, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 108, + 132, + 115 + ], + "score": 0.78, + "content": "_ \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 106, + 320, + 118 + ], + "score": 1.0, + "content": "is the gate controlling the flow of information:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 102, + 76, + 375, + 118 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 132, + 380, + 147 + ], + "lines": [ + { + "bbox": [ + 230, + 132, + 380, + 147 + ], + "spans": [ + { + "bbox": [ + 230, + 132, + 380, + 147 + ], + "score": 0.91, + "content": "\\pmb { \\alpha } = \\mathrm { s i g m o i d } ( W _ { 2 } \\left[ \\pmb { x } _ { i } ^ { t } , \\pmb { m } _ { i } ^ { t } , \\pmb { v } _ { i } ^ { t } \\right] + b _ { 2 } )", + "type": "interline_equation", + "image_path": "53e933465545dce9b796151fa0986bd5939f3a92542be6e727f34e32b07e067f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 230, + 132, + 380, + 147 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 506, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 441, + 160 + ], + "score": 1.0, + "content": "By combining Eqs. (19,20,22) together, one step of message passing update for node", + "type": "text" + }, + { + "bbox": [ + 441, + 148, + 446, + 157 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 146, + 506, + 160 + ], + "score": 1.0, + "content": "can be written", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 221, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 221, + 171 + ], + "score": 1.0, + "content": "in a generic way as follows:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 146, + 506, + 171 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 183, + 386, + 199 + ], + "lines": [ + { + "bbox": [ + 225, + 183, + 386, + 199 + ], + "spans": [ + { + "bbox": [ + 225, + 183, + 386, + 199 + ], + "score": 0.88, + "content": "\\pmb { x } _ { i } ^ { t + 1 } = \\mathrm { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { i } ^ { t } , \\pmb { v } _ { i } , \\mathcal { N } ( i ) \\right)", + "type": "interline_equation", + "image_path": "b4e04240471ea34e85e0fd90a18a1b3ce676b1d7f72e23c67c9356d0331e60d4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 225, + 183, + 386, + 199 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 210, + 213, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 214, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 214, + 223 + ], + "score": 1.0, + "content": "A.2 UPDATING STATES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 230, + 207, + 242 + ], + "lines": [ + { + "bbox": [ + 107, + 231, + 207, + 243 + ], + "spans": [ + { + "bbox": [ + 107, + 231, + 207, + 243 + ], + "score": 1.0, + "content": "UPDATING RNN STATE", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 107, + 231, + 207, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 504, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "We keep the old representation of the edge that have been modified in the hidden memory of the RNN", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 153, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 153, + 273 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 250, + 505, + 273 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 286, + 355, + 302 + ], + "lines": [ + { + "bbox": [ + 254, + 286, + 355, + 302 + ], + "spans": [ + { + "bbox": [ + 254, + 286, + 355, + 302 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\pmb { h } ^ { \\tau } = \\mathbf { G } \\mathbf { R } \\mathbf { U } \\left( \\pmb { h } ^ { \\tau - 1 } , \\pmb { z } _ { u v } ^ { \\tau } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "e8266b890dd6cec36d096d56b55087b4dc1abae35f9a0f05acfffe6b3708f8dd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 254, + 286, + 355, + 302 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 104, + 300, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 365, + 318 + ], + "score": 1.0, + "content": "where GRU stands for Gated Recurrent Units (Cho et al., 2014);", + "type": "text" + }, + { + "bbox": [ + 366, + 303, + 382, + 315 + ], + "score": 0.9, + "content": "\\boldsymbol { z } _ { u v } ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 300, + 507, + 318 + ], + "score": 1.0, + "content": "is the representation vector of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 161, + 326 + ], + "score": 1.0, + "content": "the atom pair", + "type": "text" + }, + { + "bbox": [ + 161, + 313, + 189, + 325 + ], + "score": 0.93, + "content": "( u , v ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "including its old bond (see Eq. 4). Eq. (25) allows the model to keep track of all", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 458, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 458, + 338 + ], + "score": 1.0, + "content": "the changes happening to the graph so far so it can make more accurate prediction later.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 300, + 507, + 338 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 348, + 319, + 358 + ], + "lines": [ + { + "bbox": [ + 107, + 348, + 319, + 360 + ], + "spans": [ + { + "bbox": [ + 107, + 348, + 319, + 360 + ], + "score": 1.0, + "content": "UPDATING GRAPH STRUCTURE AND NODE STATES", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 240, + 380 + ], + "score": 1.0, + "content": "After predicting a reaction triple", + "type": "text" + }, + { + "bbox": [ + 240, + 366, + 277, + 379 + ], + "score": 0.93, + "content": "( u , v , b ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 366, + 307, + 380 + ], + "score": 1.0, + "content": "at step", + "type": "text" + }, + { + "bbox": [ + 307, + 369, + 314, + 377 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 366, + 505, + 380 + ], + "score": 1.0, + "content": ", we update the graph structure and node states", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "based on the new bond change. First, to update the graph structure, we simply update the neighbor", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 461, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 131, + 401 + ], + "score": 1.0, + "content": "set of", + "type": "text" + }, + { + "bbox": [ + 131, + 390, + 138, + 398 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 388, + 156, + 401 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 391, + 162, + 398 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 388, + 407, + 401 + ], + "score": 1.0, + "content": "with information from the other atom and the new bond type", + "type": "text" + }, + { + "bbox": [ + 408, + 389, + 414, + 398 + ], + "score": 0.75, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 388, + 461, + 401 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 366, + 505, + 401 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 414, + 393, + 446 + ], + "lines": [ + { + "bbox": [ + 216, + 414, + 393, + 446 + ], + "spans": [ + { + "bbox": [ + 216, + 414, + 393, + 446 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { \\mathcal { N } ^ { \\tau } ( u ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( u ) \\backslash \\left( v , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( v , b ) } \\\\ { \\mathcal { N } ^ { \\tau } ( v ) } & { = } & { \\left( \\mathcal { N } ^ { \\tau - 1 } ( v ) \\backslash \\left( u , b ^ { \\mathrm { o l d } } \\right) \\right) \\cup ( u , b ) } \\end{array}", + "type": "interline_equation", + "image_path": "711fda8a1915bd0c770d210262fb5bf19078bfe031660032107382cf0a3b1810.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 414, + 393, + 430.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 216, + 430.0, + 393, + 446.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 451, + 457 + ], + "score": 1.0, + "content": "Next, to update the node states, our model performs one step of message passing for", + "type": "text" + }, + { + "bbox": [ + 452, + 446, + 458, + 453 + ], + "score": 0.77, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 443, + 477, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 447, + 483, + 453 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 205, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 205, + 467 + ], + "score": 1.0, + "content": "their new neighbor sets:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 443, + 505, + 467 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 479, + 398, + 512 + ], + "lines": [ + { + "bbox": [ + 210, + 479, + 398, + 512 + ], + "spans": [ + { + "bbox": [ + 210, + 479, + 398, + 512 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } { \\pmb { x } _ { u } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { u } ^ { \\tau - 1 } , \\pmb { v } _ { u } , \\mathcal { N } ^ { \\tau } ( u ) \\right) } \\\\ { \\pmb { x } _ { v } ^ { \\tau } } & { = } & { \\mathsf { M e s s a g e P a s s i n g } \\left( \\pmb { x } _ { v } ^ { \\tau - 1 } , \\pmb { v } _ { v } , \\mathcal { N } ^ { \\tau } ( v ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "1037ecb9d4b964ec79c198c33b5dbb951a46c54f5a12e1d58afc79a824035a6f.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 210, + 479, + 398, + 495.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 210, + 495.5, + 398, + 512.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 506, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 215, + 522 + ], + "score": 1.0, + "content": "where the MessagePassing", + "type": "text" + }, + { + "bbox": [ + 216, + 510, + 226, + 522 + ], + "score": 0.55, + "content": "( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "function is defined in Eq. (24). For other nodes in the graph to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 233, + 534 + ], + "score": 1.0, + "content": "aware of the new structures of", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 240, + 531 + ], + "score": 0.79, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 520, + 259, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 523, + 266, + 531 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 520, + 506, + 534 + ], + "score": 1.0, + "content": ", we need to perform several message passing steps for all", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 507, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 496, + 545 + ], + "score": 1.0, + "content": "nodes in the graph after Eqs. (28, 29). However, it is very costly to run for every prediction step", + "type": "text" + }, + { + "bbox": [ + 496, + 535, + 503, + 542 + ], + "score": 0.7, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 531, + 507, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "Sometimes it is unnecessary since far-away bonds are less likely to be affected by the current bond", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 566 + ], + "score": 1.0, + "content": "change (unless the far-way bonds and the new bond are in an aromatic ring). Therefore, in our model,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 564, + 423, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 381, + 577 + ], + "score": 1.0, + "content": "we limit the number of message passing updates for all nodes at step", + "type": "text" + }, + { + "bbox": [ + 382, + 567, + 389, + 575 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 564, + 423, + 577 + ], + "score": 1.0, + "content": "to be 1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 510, + 507, + 577 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 246, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 247, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 247, + 601 + ], + "score": 1.0, + "content": "A.3 MODEL CONFIGURATIONS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "We optimize our model’s hyper-parameters in two stages: First, we tune the hyper-parameters of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "GNN and the NPPN for the reaction atom pair prediction task. Then, we fix the optimal settings", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "score": 1.0, + "content": "of the first two components and optimize the hyper-parameters of the PN for the reaction product", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 171, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 171, + 654 + ], + "score": 1.0, + "content": "prediction task.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 609, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "We provide details about the settings that give good results on the USPTO-15k dataset below. With", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "these settings, we trained another model on the USPTO dataset from scratch. Because training on", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "the large dataset such as the USPTO takes time, we did not tune hyper-parameters on the USPTO,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 694, + 397, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 397, + 705 + ], + "score": 1.0, + "content": "eventhough it is possible to increase model sizes for better performance.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 660, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "Unless explicitly stated, all neural networks in our model have 2 layers with the same number of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 399, + 732 + ], + "score": 1.0, + "content": "hidden units, ReLU activation and residual connections (He et al., 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 228, + 80, + 382, + 153 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 228, + 80, + 382, + 153 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 228, + 80, + 382, + 153 + ], + "spans": [ + { + "bbox": [ + 228, + 80, + 382, + 153 + ], + "score": 0.974, + "html": "
Atom attributeData type
Degreenumeric
Explicit valencenumeric
Explicit number of Hsnumeric
Chargenumeric
Part of a ringboolean
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The state vector and the neighbor message vector for an atom both have the size of 99.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 278, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 278, + 267 + ], + "score": 1.0, + "content": "The number of message passing steps is 6.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "score": 1.0, + "content": "Node Pair Prediction Network (NPPN) This component consists of two parts. 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This function is also a neural network with hidden size of 51.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "Policy Network (PN) The recurrent network is a GRU (Cho et al., 2014) with 101 hidden units. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 343, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 165, + 361 + ], + "score": 1.0, + "content": "value function", + "type": "text" + }, + { + "bbox": [ + 166, + 347, + 178, + 359 + ], + "score": 0.88, + "content": "V _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 343, + 420, + 361 + ], + "score": 1.0, + "content": "is a neural network with 99 hidden units. 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(8)) are neural", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 370, + 232, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 232, + 381 + ], + "score": 1.0, + "content": "networks with 81 hidden units.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Training At each step, we set the reward to be 1.0 for correct prediction of signal/atom pair/bond", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "type and -1.0 for incorrect prediction. 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From the rewards and estimated values for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "signal, atom pair and bond type, we define the Advantage Actor Critic loss (A2C) as in Eq. (14). The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 411, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 279, + 461 + ], + "score": 1.0, + "content": "coefficients of components in the final loss", + "type": "text" + }, + { + "bbox": [ + 279, + 448, + 288, + 458 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 448, + 411, + 461 + ], + "score": 1.0, + "content": "are set empirically as follows:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 473, + 452, + 487 + ], + "lines": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "spans": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "score": 0.85, + "content": "\\mathcal { L } = \\mathcal { L } ^ { \\mathrm { A 2 C } } + 0 . 5 \\times \\mathcal { L } ^ { \\mathrm { v a l u e } } + \\mathcal { L } ^ { \\mathrm { a t o m p a i r } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { o v e r l e n g t h } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { i n ~ t o p ~ } K }", + "type": "interline_equation", + "image_path": "d06fb9511efe0c12dce034dae155e1aeb45209904aa758e768521da1a5741b6d.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "We trained our model using Adam (Kingma & Ba, 2015) with the initial learning rate of 0.001", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 143, + 514 + ], + "score": 1.0, + "content": "for both", + "type": "text" + }, + { + "bbox": [ + 143, + 502, + 194, + 512 + ], + "score": 0.58, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "and USPTO. 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The state vector and the neighbor message vector for an atom both have the size of 99.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 278, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 278, + 267 + ], + "score": 1.0, + "content": "The number of message passing steps is 6.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 187, + 507, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "score": 1.0, + "content": "Node Pair Prediction Network (NPPN) This component consists of two parts. 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This function is also a neural network with hidden size of 51.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 103, + 276, + 507, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "Policy Network (PN) The recurrent network is a GRU (Cho et al., 2014) with 101 hidden units. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 343, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 165, + 361 + ], + "score": 1.0, + "content": "value function", + "type": "text" + }, + { + "bbox": [ + 166, + 347, + 178, + 359 + ], + "score": 0.88, + "content": "V _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 343, + 420, + 361 + ], + "score": 1.0, + "content": "is a neural network with 99 hidden units. 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(8)) are neural", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 370, + 232, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 232, + 381 + ], + "score": 1.0, + "content": "networks with 81 hidden units.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 335, + 506, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Training At each step, we set the reward to be 1.0 for correct prediction of signal/atom pair/bond", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "type and -1.0 for incorrect prediction. 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From the rewards and estimated values for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "signal, atom pair and bond type, we define the Advantage Actor Critic loss (A2C) as in Eq. (14). The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 448, + 411, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 279, + 461 + ], + "score": 1.0, + "content": "coefficients of components in the final loss", + "type": "text" + }, + { + "bbox": [ + 279, + 448, + 288, + 458 + ], + "score": 0.79, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 448, + 411, + 461 + ], + "score": 1.0, + "content": "are set empirically as follows:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 393, + 506, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 473, + 452, + 487 + ], + "lines": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "spans": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "score": 0.85, + "content": "\\mathcal { L } = \\mathcal { L } ^ { \\mathrm { A 2 C } } + 0 . 5 \\times \\mathcal { L } ^ { \\mathrm { v a l u e } } + \\mathcal { L } ^ { \\mathrm { a t o m p a i r } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { o v e r l e n g t h } } + 0 . 2 \\times \\mathcal { L } ^ { \\mathrm { i n ~ t o p ~ } K }", + "type": "interline_equation", + "image_path": "d06fb9511efe0c12dce034dae155e1aeb45209904aa758e768521da1a5741b6d.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 158, + 473, + 452, + 487 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "We trained our model using Adam (Kingma & Ba, 2015) with the initial learning rate of 0.001", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 143, + 514 + ], + "score": 1.0, + "content": "for both", + "type": "text" + }, + { + "bbox": [ + 143, + 502, + 194, + 512 + ], + "score": 0.58, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "and USPTO. 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The maximum number of training iterations is", + "type": "text" + }, + { + "bbox": [ + 488, + 534, + 504, + 545 + ], + "score": 0.86, + "content": "1 0 ^ { 6 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 545, + 203, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 203, + 556 + ], + "score": 1.0, + "content": "and the batch size is 20.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 489, + 505, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 273, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 275, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 275, + 582 + ], + "score": 1.0, + "content": "A.4 DECODING WITH BEAM SEARCH", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 463, + 604 + ], + "score": 1.0, + "content": "For decoding, our model generates a sequence of reaction triples (including the stop signal)", + "type": "text" + }, + { + "bbox": [ + 463, + 591, + 505, + 603 + ], + "score": 0.92, + "content": "( \\xi , u , v , b )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 181, + 614 + ], + "score": 1.0, + "content": "by taking the best", + "type": "text" + }, + { + "bbox": [ + 181, + 602, + 204, + 614 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 602, + 223, + 614 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 602, + 229, + 612 + ], + "score": 0.7, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 602, + 400, + 614 + ], + "score": 1.0, + "content": "at every step until it outputs a zero signal", + "type": "text" + }, + { + "bbox": [ + 401, + 602, + 426, + 613 + ], + "score": 0.88, + "content": "\\xi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "). In other words, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 612, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 612, + 201, + 628 + ], + "score": 1.0, + "content": "computes the argmax of", + "type": "text" + }, + { + "bbox": [ + 203, + 613, + 346, + 626 + ], + "score": 0.91, + "content": "p \\left( ( \\xi , u , v , b ) ^ { \\tau } \\mid \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 612, + 398, + 628 + ], + "score": 1.0, + "content": "at every step", + "type": "text" + }, + { + "bbox": [ + 398, + 616, + 405, + 624 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 612, + 506, + 628 + ], + "score": 1.0, + "content": ". However, this algorithm", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "is not robust for the sequence generation task because just a single error at a step may destroy the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 424, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 424, + 650 + ], + "score": 1.0, + "content": "entire sequence. 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Algorithm 1 describes beam search in detail. Some notable technicalities are:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 131, + 345, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 132, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 132, + 344, + 256, + 358 + ], + "score": 1.0, + "content": "• We only do beam search for", + "type": "text" + }, + { + "bbox": [ + 257, + 346, + 280, + 357 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 344, + 298, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 346, + 304, + 356 + ], + "score": 0.77, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 344, + 459, + 358 + ], + "score": 1.0, + "content": "if the prediction is ongoing, i.e., when", + "type": "text" + }, + { + "bbox": [ + 459, + 346, + 488, + 357 + ], + "score": 0.91, + "content": "\\xi ^ { \\tau } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 344, + 505, + 358 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 140, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 140, + 356, + 319, + 369 + ], + "score": 1.0, + "content": "keep track of this, we use a boolean vector", + "type": "text" + }, + { + "bbox": [ + 320, + 357, + 329, + 367 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 356, + 369, + 369 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 369, + 357, + 379, + 367 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 356, + 402, + 369 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 402, + 357, + 415, + 367 + ], + "score": 0.89, + "content": "C ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "is initialized to be all", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 367, + 164, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 164, + 380 + ], + "score": 1.0, + "content": "true.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 132, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 132, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "• To avoid beam search favoring short sequences, we normalize the log probability scores", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 392, + 397, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 397, + 404 + ], + "score": 1.0, + "content": "over sequence lengths. 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It suggests that the proper use of reagent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "information will lead to better prediction. In our model, before computing the scores for all atom", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "pairs, we append to the representation vector of every atom a binary scalar indicating whether this", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 358, + 588 + ], + "score": 1.0, + "content": "atom comes from a reagent molecule or not. 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Thus, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 234, + 252, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 252, + 245 + ], + "score": 1.0, + "content": "decompose the first term as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 221, + 506, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 262, + 476, + 312 + ], + "lines": [ + { + "bbox": [ + 132, + 262, + 476, + 312 + ], + "spans": [ + { + "bbox": [ + 132, + 262, + 476, + 312 + ], + "score": 0.92, + "content": "\\begin{array} { l l l } { \\log p \\left( ( \\xi , u , v , b ) ^ { \\tau } | \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) } & { = } & { \\log p \\left( \\xi ^ { \\tau } | \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) + \\mathstrut } \\\\ & & { \\log p \\left( ( u , v ) ^ { \\tau } | \\xi ^ { \\tau } , \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) + \\mathstrut } \\\\ & & { \\log \\left( b ^ { \\tau } | ( \\xi , u , v ) ^ { \\tau } , \\mathcal { G } , ( \\xi , u , v , b ) ^ { 0 : \\tau - 1 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "3102af7b53b8fc38e42d9ff2d6978fe328c3fc7aa6f92308b75b606cab663d15.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 132, + 262, + 476, + 278.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 278.6666666666667, + 476, + 295.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 132, + 295.33333333333337, + 476, + 312.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 504, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 137, + 327 + ], + "score": 1.0, + "content": "At step", + "type": "text" + }, + { + "bbox": [ + 137, + 316, + 144, + 323 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 311, + 282, + 327 + ], + "score": 1.0, + "content": ", we do beam search for the signal", + "type": "text" + }, + { + "bbox": [ + 282, + 314, + 293, + 325 + ], + "score": 0.88, + "content": "\\xi ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 311, + 372, + 327 + ], + "score": 1.0, + "content": ", then the atom pair", + "type": "text" + }, + { + "bbox": [ + 373, + 313, + 401, + 325 + ], + "score": 0.92, + "content": "( u , v ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 311, + 506, + 327 + ], + "score": 1.0, + "content": "and finally the bond type", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 323, + 432, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 117, + 334 + ], + "score": 0.84, + "content": "b ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 323, + 432, + 336 + ], + "score": 1.0, + "content": ". Algorithm 1 describes beam search in detail. Some notable technicalities are:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 311, + 506, + 336 + ] + }, + { + "type": "list", + "bbox": [ + 131, + 345, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 132, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 132, + 344, + 256, + 358 + ], + "score": 1.0, + "content": "• We only do beam search for", + "type": "text" + }, + { + "bbox": [ + 257, + 346, + 280, + 357 + ], + "score": 0.92, + "content": "( u , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 344, + 298, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 346, + 304, + 356 + ], + "score": 0.77, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 344, + 459, + 358 + ], + "score": 1.0, + "content": "if the prediction is ongoing, i.e., when", + "type": "text" + }, + { + "bbox": [ + 459, + 346, + 488, + 357 + ], + "score": 0.91, + "content": "\\xi ^ { \\tau } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 344, + 505, + 358 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 140, + 356, + 319, + 369 + ], + "score": 1.0, + "content": "keep track of this, we use a boolean vector", + "type": "text" + }, + { + "bbox": [ + 320, + 357, + 329, + 367 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 356, + 369, + 369 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 369, + 357, + 379, + 367 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 356, + 402, + 369 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 402, + 357, + 415, + 367 + ], + "score": 0.89, + "content": "C ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "is initialized to be all", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 367, + 164, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 164, + 380 + ], + "score": 1.0, + "content": "true.", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 132, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "• To avoid beam search favoring short sequences, we normalize the log probability scores", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 392, + 397, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 397, + 404 + ], + "score": 1.0, + "content": "over sequence lengths. This is shown in lines 10, 17, 32 and 47", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + } + ], + "index": 14, + "bbox_fs": [ + 132, + 344, + 505, + 404 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 421, + 210, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 211, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 211, + 433 + ], + "score": 1.0, + "content": "BEAM WIDTH ANALYSIS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "list", + "bbox": [ + 107, + 441, + 506, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 507, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 507, + 454 + ], + "score": 1.0, + "content": "Table 5 reports how beam width affects the decoding performance on the USPTO-15k dataset.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "Surprisingly, the top-1 accuracy in case of beam width3 of 1 is higher than the those when beam", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "widths range from 2 to 15. It means that large beam width is not always good in our situation.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 474, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 496, + 486 + ], + "score": 1.0, + "content": "However, at beam width of 20, our beam search achieves the best results for different values of", + "type": "text" + }, + { + "bbox": [ + 496, + 474, + 503, + 484 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 474, + 507, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 484, + 356, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 356, + 498 + ], + "score": 1.0, + "content": "Thus, we set the beam width to 20 in subsequent experiments.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 20, + "bbox_fs": [ + 105, + 440, + 507, + 498 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 511, + 324, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 324, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 324, + 524 + ], + "score": 1.0, + "content": "A.5 USING REAGENT INFORMATION EXPLICITLY", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "As can be seen from Table 6, reagent molecules account for about a half of the input molecules on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 157, + 556 + ], + "score": 1.0, + "content": "average and", + "type": "text" + }, + { + "bbox": [ + 157, + 543, + 191, + 554 + ], + "score": 0.87, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "of all reactions containing reagents. It suggests that the proper use of reagent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "information will lead to better prediction. In our model, before computing the scores for all atom", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "pairs, we append to the representation vector of every atom a binary scalar indicating whether this", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 358, + 588 + ], + "score": 1.0, + "content": "atom comes from a reagent molecule or not. Then, at the top-", + "type": "text" + }, + { + "bbox": [ + 358, + 577, + 369, + 586 + ], + "score": 0.75, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "atom pair selection step, we also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "exclude all atom pairs that have either atoms belong to a reagent molecule. The improvement in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 598, + 408, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 281, + 610 + ], + "score": 1.0, + "content": "prediction accuracy on the validation set of", + "type": "text" + }, + { + "bbox": [ + 281, + 598, + 331, + 609 + ], + "score": 0.51, + "content": "U S P T O – I 5 k", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 598, + 408, + 610 + ], + "score": 1.0, + "content": "is shown in Fig. 4.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 533, + 506, + 610 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 624, + 267, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 268, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 268, + 637 + ], + "score": 1.0, + "content": "A.6 COMPARISON WITH ELECTRO", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "In method Both GTPN and ELECTRO (Bradshaw et al., 2018) are able to explain the mechanism", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "behind a reaction. ELECTRO regards a reaction as an ordered sequence that alternates between", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "removing and adding a single bond. Our model, on the other hand, assumes no specific order of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "transformations as well as the amount of valences that a bond can change. Thus, our model is more", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 689, + 384, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 384, + 702 + ], + "score": 1.0, + "content": "generic than ELECTRO and can cover a much larger set of reactions.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 646, + 505, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 45, + 507, + 723 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 105, + 45, + 507, + 723 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 45, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 45, + 507, + 723 + ], + "score": 0.739, + "html": "
Algorithm 1 Reaction triple prediction using beam search.
Input: A multi-graph G consisting of reactant and reagent molecules, number of bond types E, max prediction steps T,beam width N
1: P0=[(-1,-1,-1,-1),] DThe best N subsequences of(ε,u,v,b)
2: s0 =[0,..] DThe length-normalized log joint probabilities of the best N subsequences 3: Co=[True,.] DThe continuation indicator of the best N subsequences
4: Perform L steps of message passing for all nodes using Eq. (1)
5:x=xii∈V DThe initial states of all nodes before decoding 6: N(i)=N(i) ∀i ∈V DThe initial neighbor set of all nodes before decoding
7: h° is loaded from the saved model DThe initial RNN hidden state before decoding
8: for T from 1 to T do
9: Find the top K atom pairs {(uk, Uk)T |k = 1,K} using Eqs. (4,5)
10: ST-1:0 = ST-1× T-1 Superscript O denotes the sub-step 0
PT-1:0 = PT-1;CT-1;0 =CT-1 11:
12: Beam search for continuation signals
13:
14: Rsignal =@ DStores the log joint probabilities for N × 2 possible signals
15: for n from 1 to N do 16: Computep (|Pn-1:0）usingEq. (6)
17: AddCT-i0× 1 logp(εT =δ| Pπ-10)+ S-1:0 to Rsignal forδ∈ {True,False}
18: end for
19: Sort Rsignal in descending order
20:
21:
22:
Pr-1;1 =extract (-, 23: CT-1;1 = extract (CT-1;0,[-1;1)
24: 25:
26:
27: Beam search for atom pairs 28:
29: Ratom pair = @ DStores the log joint probabilities for N × K possible atom pairs
30: for n from 1 to N do 31: Compute p (u,u)|Sn,P-1;1) using Eq. (7)
Add CT-i1 ×1logp((uU)sn,P-1;1)+S-1:1 to Ratompairk ∈1,K 32:
33: end for
Sort Ratom pair in descending order 34:
35:
36:
37:
Pt-1;2 = extract (P-1;1,[-1;2) 38:
39: CT-1;2 = extract (C-1;1,;2)
40: = extract (,-1;2)
41:
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Algorithm 1 Reaction triple prediction using beam search.
Input: A multi-graph G consisting of reactant and reagent molecules, number of bond types E, max prediction steps T,beam width N
1: P0=[(-1,-1,-1,-1),] DThe best N subsequences of(ε,u,v,b)
2: s0 =[0,..] DThe length-normalized log joint probabilities of the best N subsequences 3: Co=[True,.] DThe continuation indicator of the best N subsequences
4: Perform L steps of message passing for all nodes using Eq. (1)
5:x=xii∈V DThe initial states of all nodes before decoding 6: N(i)=N(i) ∀i ∈V DThe initial neighbor set of all nodes before decoding
7: h° is loaded from the saved model DThe initial RNN hidden state before decoding
8: for T from 1 to T do
9: Find the top K atom pairs {(uk, Uk)T |k = 1,K} using Eqs. (4,5)
10: ST-1:0 = ST-1× T-1 Superscript O denotes the sub-step 0
PT-1:0 = PT-1;CT-1;0 =CT-1 11:
12: Beam search for continuation signals
13:
14: Rsignal =@ DStores the log joint probabilities for N × 2 possible signals
15: for n from 1 to N do 16: Computep (|Pn-1:0）usingEq. (6)
17: AddCT-i0× 1 logp(εT =δ| Pπ-10)+ S-1:0 to Rsignal forδ∈ {True,False}
18: end for
19: Sort Rsignal in descending order
20:
21:
22:
Pr-1;1 =extract (-, 23: CT-1;1 = extract (CT-1;0,[-1;1)
24: 25:
26:
27: Beam search for atom pairs 28:
29: Ratom pair = @ DStores the log joint probabilities for N × K possible atom pairs
30: for n from 1 to N do 31: Compute p (u,u)|Sn,P-1;1) using Eq. (7)
Add CT-i1 ×1logp((uU)sn,P-1;1)+S-1:1 to Ratompairk ∈1,K 32:
33: end for
Sort Ratom pair in descending order 34:
35:
36:
37:
Pt-1;2 = extract (P-1;1,[-1;2) 38:
39: CT-1;2 = extract (C-1;1,;2)
40: = extract (,-1;2)
41:
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45:for n from1 to N do
46:Compute p (bT 1(E,u,o)n,P-1) using Eq. (8)
47:AddCT-i;2× ↓logp (bT =β|(ξ,u,u)，Pπ-1) + S[-1 to Rbond ∀β ∈1,B
48:end for
49: Sort Rbond in descending order
50:ST-1;3 = RO:N Rbond
51:
52:
53:PT-1;3 = extract (Pr-1;2, [T-1;3)
54:CT-1;3 = extract (CT-1;2, [T-1;3)
55:T = extract (T,IT-1;3)
56:(u,u)T = extract (u,u)T,IT-1;3)
57:
58:ST = ST-1;3;CT = CT-1;3
59:P =append(P-1:3，(,u,U,b)n)
60:for n from1 to N do
61:Update the NT(un) and NT(Un) for all n = 1,N using Eqs. (26,27)
62:Update xun and xn using Eq. (1)
63:Perform m steps of message passing for all nodes in the graph
64:Update husing Eq. (25)
65:end for
66: end for Output: PT,ST
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Dataset%reactionscontaining reagents%reagents over input molecules
USPTO-15ktrain63.1%41.3%
valid65.3%42.3%
test63.6%40.9%
USPTOtrain79.7%54.0%
valid80.0%54.4%
test79.9%54.2%
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45:for n from1 to N do
46:Compute p (bT 1(E,u,o)n,P-1) using Eq. (8)
47:AddCT-i;2× ↓logp (bT =β|(ξ,u,u)，Pπ-1) + S[-1 to Rbond ∀β ∈1,B
48:end for
49: Sort Rbond in descending order
50:ST-1;3 = RO:N Rbond
51:
52:
53:PT-1;3 = extract (Pr-1;2, [T-1;3)
54:CT-1;3 = extract (CT-1;2, [T-1;3)
55:T = extract (T,IT-1;3)
56:(u,u)T = extract (u,u)T,IT-1;3)
57:
58:ST = ST-1;3;CT = CT-1;3
59:P =append(P-1:3，(,u,U,b)n)
60:for n from1 to N do
61:Update the NT(un) and NT(Un) for all n = 1,N using Eqs. (26,27)
62:Update xun and xn using Eq. (1)
63:Perform m steps of message passing for all nodes in the graph
64:Update husing Eq. (25)
65:end for
66: end for Output: PT,ST
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Dataset%reactionscontaining reagents%reagents over input molecules
USPTO-15ktrain63.1%41.3%
valid65.3%42.3%
test63.6%40.9%
USPTOtrain79.7%54.0%
valid80.0%54.4%
test79.9%54.2%
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ModelProcessed USPTO
P@1P@3P@5
WLDN (Jin et al., 2017)84.091.192.3
ELECTRO (Bradshaw et al., 2018)GTPN**87.094.595.9
87.3590.2290.68
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Meanings of markers in our model:", + "type": "text" + }, + { + "bbox": [ + 290, + 160, + 298, + 169 + ], + "score": 0.56, + "content": "\\diamondsuit", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 160, + 432, + 172 + ], + "score": 1.0, + "content": ": With beam search (beam width", + "type": "text" + }, + { + "bbox": [ + 433, + 161, + 454, + 171 + ], + "score": 0.84, + "content": "= 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 160, + 461, + 172 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 461, + 160, + 469, + 170 + ], + "score": 0.31, + "content": "\\spadesuit", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 160, + 504, + 172 + ], + "score": 1.0, + "content": ": Invalid", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 170, + 302, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 176, + 184 + ], + "score": 1.0, + "content": "product removal,", + "type": "text" + }, + { + "bbox": [ + 176, + 171, + 185, + 182 + ], + "score": 0.28, + "content": "\\mathbf { \\hat { \\psi } } _ { \\mathbf { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 170, + 302, + 184 + ], + "score": 1.0, + "content": ": Duplicate product removal.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 115, + 208, + 495, + 337 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 208, + 495, + 337 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 208, + 495, + 337 + ], + "spans": [ + { + "bbox": [ + 115, + 208, + 495, + 337 + ], + "score": 0.97, + "type": "image", + "image_path": "aebc50a959e34bea76abae1e557fd21e1b1f51ece918963e01707e0931304628.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 115, + 208, + 495, + 251.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 115, + 251.0, + 495, + 294.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 294.0, + 495, + 337.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 344, + 506, + 399 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "Figure 5: Performance with respect to different numbers of bond changes. (a) Top-1 accuracy. (b)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 367 + ], + "score": 1.0, + "content": "Errors grouped by length. In (a), blue: all reactions having that sequence length; orange: correct", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "predicted reactions. In (b), red: the predicted sequence is shorter (than the groundtruth sequence);", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "green: the predicted and the groundtruth have the same length; blue: the predicted sequence is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 282, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 282, + 402 + ], + "score": 1.0, + "content": "longer; number indidate the average length.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "In performance To do a fair comparison with ELECTRO (Bradshaw et al., 2018), we follow their", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "procedure described in the paper to prepare a new test set that contains only reactions with linear", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "chain topology and single-valence bond changes. 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All the results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "below are computed on the USPTO-15k dataset by using beam search decoding with the beam width", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 601, + 238, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 602, + 140, + 612 + ], + "score": 0.89, + "content": "N = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 601, + 238, + 615 + ], + "score": 1.0, + "content": "and no post-processing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "Errors grouped by number of bond changes Fig. 5 shows the top-1 accuracies for reactions with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "different number of bond changes. 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We conjecture that this problem mainly comes from the fact that GTPN was not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 531, + 335, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 335, + 543 + ], + "score": 1.0, + "content": "optimized on the compatible training and validation sets.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 421, + 506, + 543 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 558, + 210, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 211, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 211, + 570 + ], + "score": 1.0, + "content": "A.7 ERROR ANALYSIS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "In this section, we analyze several error types that our model makes during prediction. All the results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "below are computed on the USPTO-15k dataset by using beam search decoding with the beam width", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 601, + 238, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 602, + 140, + 612 + ], + "score": 0.89, + "content": "N = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 601, + 238, + 615 + ], + "score": 1.0, + "content": "and no post-processing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 580, + 505, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 683 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "Errors grouped by number of bond changes Fig. 5 shows the top-1 accuracies for reactions with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "different number of bond changes. Our model performs poorly on reactions with many bond changes.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "However, those kinds of reactions only accounts for a small proportion in the dataset. From Fig. 5b,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "we see that the lengths of the error sequences tends to be shorter than the lengths of the groundtruth", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 673, + 152, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 152, + 685 + ], + "score": 1.0, + "content": "sequences.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 628, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "Errors caused by signal/atom pair/bond type We define a sub-action causing error as the first", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "sub-action that our model makes a wrong decision. In Fig. 6a, we plot the the proportion of errors", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "with respect to the three kinds of sub-actions. 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Therefore, more effort should be put on improving the prediction of atom pairs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 331, + 505, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "Errors caused by symmetry There exists cases in which different sequences of bond changes can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "result in the same products due to symmetric graph structures. Errors caused by symmetry account", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 385, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 120, + 401 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 388, + 142, + 398 + ], + "score": 0.87, + "content": "5 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 385, + 506, + 401 + ], + "score": 1.0, + "content": "of the top-1 errors on the USPTO-15k dataset as shown in Fig. 6b. For better understanding,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "we provide a short list of wrong reaction triple predictions caused by symmetry in Fig. 7. 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Dataset#reactions#changes#molecules#atoms#bonds
USPTO-15ktrain10,500111112.3112013.641100134.931110134.7
valid1,500111112.3112013.67194134.55199134.2
test3,000111112.3111613.67198134.951102134.7
USPTOtrain409,03511612.2212914.891150139.761165138.6
valid30,00011612.2212514.891150139.671158138.5
test40.00011612.2212214.891150139.871162138.7
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ModelUSPTO-15kUSPTO
C@6C@8C@10C@6C@8C@10
WLN* (Jin et al., 2017)81.686.189.189.892.093.3
WLN (Jin et al., 2017)88.4591.6593.3490.9793.9895.26
CLN (Pham et al., 2017)88.6891.6393.0790.7293.5794.80
Our GNN88.9292.0093.5791.2494.1795.33
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P@1P@3P@5P@1P@3P@5
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Algorithm 1 Reaction triple prediction using beam search.
Input: A multi-graph G consisting of reactant and reagent molecules, number of bond types E, max prediction steps T,beam width N
1: P0=[(-1,-1,-1,-1),] DThe best N subsequences of(ε,u,v,b)
2: s0 =[0,..] DThe length-normalized log joint probabilities of the best N subsequences 3: Co=[True,.] DThe continuation indicator of the best N subsequences
4: Perform L steps of message passing for all nodes using Eq. (1)
5:x=xii∈V DThe initial states of all nodes before decoding 6: N(i)=N(i) ∀i ∈V DThe initial neighbor set of all nodes before decoding
7: h° is loaded from the saved model DThe initial RNN hidden state before decoding
8: for T from 1 to T do
9: Find the top K atom pairs {(uk, Uk)T |k = 1,K} using Eqs. (4,5)
10: ST-1:0 = ST-1× T-1 Superscript O denotes the sub-step 0
PT-1:0 = PT-1;CT-1;0 =CT-1 11:
12: Beam search for continuation signals
13:
14: Rsignal =@ DStores the log joint probabilities for N × 2 possible signals
15: for n from 1 to N do 16: Computep (|Pn-1:0）usingEq. (6)
17: AddCT-i0× 1 logp(εT =δ| Pπ-10)+ S-1:0 to Rsignal forδ∈ {True,False}
18: end for
19: Sort Rsignal in descending order
20:
21:
22:
Pr-1;1 =extract (-, 23: CT-1;1 = extract (CT-1;0,[-1;1)
24: 25:
26:
27: Beam search for atom pairs 28:
29: Ratom pair = @ DStores the log joint probabilities for N × K possible atom pairs
30: for n from 1 to N do 31: Compute p (u,u)|Sn,P-1;1) using Eq. (7)
Add CT-i1 ×1logp((uU)sn,P-1;1)+S-1:1 to Ratompairk ∈1,K 32:
33: end for
Sort Ratom pair in descending order 34:
35:
36:
37:
Pt-1;2 = extract (P-1;1,[-1;2) 38:
39: CT-1;2 = extract (C-1;1,;2)
40: = extract (,-1;2)
41:
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Dataset%reactionscontaining reagents%reagents over input molecules
USPTO-15ktrain63.1%41.3%
valid65.3%42.3%
test63.6%40.9%
USPTOtrain79.7%54.0%
valid80.0%54.4%
test79.9%54.2%
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45:for n from1 to N do
46:Compute p (bT 1(E,u,o)n,P-1) using Eq. (8)
47:AddCT-i;2× ↓logp (bT =β|(ξ,u,u)，Pπ-1) + S[-1 to Rbond ∀β ∈1,B
48:end for
49: Sort Rbond in descending order
50:ST-1;3 = RO:N Rbond
51:
52:
53:PT-1;3 = extract (Pr-1;2, [T-1;3)
54:CT-1;3 = extract (CT-1;2, [T-1;3)
55:T = extract (T,IT-1;3)
56:(u,u)T = extract (u,u)T,IT-1;3)
57:
58:ST = ST-1;3;CT = CT-1;3
59:P =append(P-1:3，(,u,U,b)n)
60:for n from1 to N do
61:Update the NT(un) and NT(Un) for all n = 1,N using Eqs. (26,27)
62:Update xun and xn using Eq. (1)
63:Perform m steps of message passing for all nodes in the graph
64:Update husing Eq. (25)
65:end for
66: end for Output: PT,ST
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HO:4NH:1[(1,6),(5,6)] [(1,10),(10,11)]HCI:5OH:4prob: 0.852 prob: 0.852
[(1,6),(5,6)]HCI:5prob: 0.852[(1,10),(10,11)]OH:4prob: 0.852
[(3,4), (3,18)]HCI:4prob: 0.917[(3,4), (3,15)]prob: 0.917
Mg:10:10[(10,11), (6,7), (6,11)]HBr:Mg:1NH:22prob: 0.901[(10,11), (2,3), (3,11)]HBr:2Mg:1NH4:22prob:0.901
H,O:17C:14C:12[(10,11), (13,14), (10,13)]H,O:17C:19c:CH,:14prob: 0.917[(10, 13), (11,12), (10, 11)]HO:17C:19C:18CH,:12prob: 0.917
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